Shintani Zeta Functions (London Mathematical Society Lecture Note Series)

LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor J.W.S. Cassels, Department of Pure Mathematic...

Author: Akihiko Yukie

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London Mathematical Society Lecture Note Series. 183

Shintani Zeta Functions

Akihiko Yukie Oklahoma State University

CAMBRIDGE UNIVERSITY PRESS

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo

Cambridge University Press The Edinburgh Building, Cambridge C132 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521448048

© Cambridge University Press 1993

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1993

A catalogue record for this publication is available from the British Library ISBN 978-0-521-44804-8 paperback Transferred to digital printing 2008

To my parents Kenzo and Fumiko Yukie

Table of contents Preface

Notation Introduction §0.1 What is a prehomogeneous vector space? §0.2 The classification §0.3 The global zeta function §0.4 The orbit space Gk \ Vks §0.5 The filtering process and the local theory: a note by D. Wright §0.6 The outline of the general procedure Part I The general theory Chapter 1 Preliminaries §1.1 An invariant measure on GL(n) §1.2 Some adelic analysis Chapter 2 Eisenstein series on GL(n) §2.1 The Fourier expansion of automorphic forms on GL(n) §2.2 The constant terms of Eisenstein series on GL(n) §2.3 The Whittaker functions §2.4 The Fourier expansion of Eisenstein series on GL(n) Chapter 3 The general program §3.1 The zeta function §3.2 The Morse stratification §3.3 The paths §3.4 Shintani's lemma for GL(n) §3.5 The general process §3.6 The passing principle §3.7 Wright's principle §3.8 Examples

Part II The Siegel-Shintani case Chapter 4 The zeta function for the space of quadratic forms §4.1 The space of quadratic forms §4.2 The case n = 2 §4.3 /3-sequences

§4.4 An inductive formulation §4.5 Paths in 31 §4.6 Paths in `3, 3X34

§4.7 The cancellations §4.8 The work of Siegel and Shintani Part III Preliminaries for the quartic case Chapter 5 The case G = GL(2) x GL(2), V = Sym2k2 ® k2 §5.1 The space Sym2k2 0 k2 §5.2 The adjusting term §5.3 Contributions from a1, 03 §5.4 Contributions from 02,04 §5.5 The contribution from Vstk §5.6 The principal part formula

Table of Contents

viii

Chapter 6 The case G = GL(2) x GL(1)2, V = Sym2k2 ® k §6.1 Reducible prehomogeneous vector spaces with two irreducible factors §6.2 The spaces Sym2k2 ® k, Sym2k2 ®k2

§6.3 The principal part formula Chapter 7 The case G = GL(2) x GL(1)2, V = Sym2k2 ®k2 §7.1 Unstable distributions §7.2 Contributions from unstable strata §7.3 The principal part formula Part IV The quartic case Chapter 8 Invariant theory of pairs of ternary quadratic forms §8.1 The space of pairs of ternary quadratic forms §8.2 The Morse stratification §8.3 /3-sequences of lengths > 2 Chapter 9 Preliminary estimates §9.1 Distributions associated with paths §9.2 The smoothed Eisenstein series Chapter 10 The non-constant terms associated with unstable strata §10.1 The case Z = (/34) §10.2 The cases a = ()35), ()3io,,3io,i) §10.3 The cases 0 _ (,136), (a8, /38,1)

§10.4 The case d=(/37) §10.5 The case d=(/38) §10.6 The cases 0 = (/38i /38,2), (/39)

Chapter 11 Unstable distributions §11.1 Unstable distributions §11.2 Technical lemmas

Chapter 12 Contributions from unstable strata §12.1 The case Z) = (/31)

§12.2 The case a=()32) §12.3 The case 0=(/33) §12.4 The case b = (/34)

§12.5 The case d=(/35) §12.6 The case a = (/36) §12.7 The case a = (/37) §12.8 The case d=(/38) §12.9 The case 0 = (/39) §12.10 The case l = (/31o)

Chapter 13 The main theorem §13.1 The cancellations of distributions §13.2 The principal part formula §13.3 Concluding remarks Bibliography List of symbols Index

Preface The content of this book is taken from my manuscripts `On the global theory of Shintani zeta functions I-V' which were originally intended for publication in ordinary journals. However, because of its length and the lack of a book on prehomogeneous vector spaces, it has been suggested to publish them together in book form. It has been more than 25 years since the theory of prehomogeneous vector spaces

began. Much work has been done on both the global theory and the local theory of zeta functions. However, we concentrate on the global theory in this book. I feel that another book should be written on the local theory of zeta functions in the future. The purpose of this book is to introduce an approach based on geometric invariant theory to the global theory of zeta functions for prehomogeneous vector spaces. This book consists of four parts. In Part I, we introduce a general formulation based on geometric invariant theory to the global theory of zeta functions for prehomogeneous vector spaces. In Part II, we apply the methods in Part I and determine the principal part of the zeta function for Siegel's case, i.e. the space of quadratic forms. In Part III, we handle relatively easy cases which are required to handle the case in Part IV. In Part IV, we use the results in Parts I-III to determine the principal part of the zeta function for the space of pairs of ternary quadratic forms. We expanded the introduction of the original manuscripts to help non-experts to have a general idea of the subject. What we try to discuss in the introduction is the history of the subject, and what is required to prove the existence of densities of arithmetic objects we are looking for. Even though the theory of prehomogeneous vector spaces involves many topics, we concentrate on two aspects of the theory, i.e. the global theory and the local theory, in the introduction. Parts I-III of this book correspond to Parts I-III of the above manuscripts, and Part IV of this book corresponds to Parts IV and V of the above manuscripts. Since the manuscripts were originally intended for publication in ordinary journals, certain changes were made to make this book more comprehensible and self-contained. However, it is impossible to make this book completely self-contained, and we have to require a reasonable background in adelic language, basic group theory, and geometric invariant theory. For this, we assume that the reader is familiar with the following four books and two papers [1] A. Borel, Some finiteness properties of adele groups over number fields, [2] A. Borel, Linear algebraic groups, [28] G. Kempf, Instability in invariant theory, [35] F. Kirwan, Cohomology of quotients in symplectic and algebraic geometry, [46] D. Mumford and J. Fogarty, Geometric invariant theory, [79] A. Weil, Basic number theory. Weil's book [79] is a standard place to learn basic materials on adelic language.

Since we do not depend on class field theory, it is enough for the reader to be familiar with the first several chapters of Weil's book. Borel's paper [1] is a place to learn properties of Siegel domains. We need two facts in geometric invariant theory. One is the Hilbert-Mumford criterion of stability, and the other is the rationality of the equivariant Morse stratification. Mumford-Fogarty [46] and Kirwan [35] are the

x

Preface

standard books to learn geometric invariant theory and equivariant Morse theory. The rationality of the equivariant Morse stratification was proved by G. Kempf in his paper [28]. However, even though the proofs of the above two facts are technically involved, the statements of these facts are fairly comprehensible and do not require a special background to understand. Therefore, if the reader is unfamiliar with these subjects, I recommend the reader not to worry about the proofs of the statements in this book which we quote from geometric invariant theory and look at the above documents later if necessary. We have three original results in this book. One is a generalization of `Shintani's lemma' to GL(n) concerning estimates of the smoothed Eisenstein series. Shintani proved this lemma for GL(2) in [64]. The statement of the result is Theorem (3.4.31). The second result is the determination of the principal part of the zeta function for the space of quadratic forms. The statement of the result is Theorem (4.0.1). Shintani himself studied this case and determined the poles of the associated Dirichlet series for quadratic forms which are positive definite in [65]. The last and the main result of this book is the determination of the principal part of the zeta function for the space of pairs of ternary quadratic forms. The statement of the result is Theorem (13.2.2). We discuss the relevance of these results in the introduction. D. Wright contributed to this book in many places. He suggested the use of 'Wright's principle' in §3.7 after he read the first manuscript of my paper [86]. Also §0.5 is largely from his note. He also found the reference concerning Omar Khayyam when we wrote our paper [84], and helped me to find some references in this book. I would like to give a hearty thanks to him. As I mentioned above, this book is based on geometric invariant theory. For this, I owe a great deal to D. Mumford for teaching me geometric invariant theory and equivariant Morse theory. I was staying at Institute for Advanced Study during the academic year 1989-1990, and at Sonderforschungsbereich 170 Gottingen during the academic year 1990-1991 while I was writing the manuscript of this book. I would like to thank them for their support of this project. This work was partially supported by NSF Grants DMS-8803085, DMS-9101091.

Akihiko Yukie February 1992, Stillwater, Oklahoma, USA

Notation For a finite set A, the cardinality of A is denoted by #A. If f, g are functions on a set X and If (x) I < Cg(x) for some constant C independent of x E X, we denote f (x) << g(x). If x, y E R, we also use the classical notation x << y if y is a much larger number than x. Since we use this classical notation only for numbers, and not for functions, we hope the meaning of this notation will be clear from the context. Suppose that G is a locally compact group and IF is a discrete subgroup of G contained in the maximal unimodular subgroup of G. For any left invariant measure dg on G, we choose a left invariant measure dg (we use the same notation, but the meaning will be clear from the context) on X = G/I' so that IG f(9)d9 =

f

f(9Y)d9

X ryEr

We denote the fields of rational, real, and complex numbers by Q, R, C respectively. We denote the ring of rational integers by Z. The set of positive real numbers

is denoted by ]R+. For any ring R, Rx is the set of invertible elements of R. Let k be a number field, and ok its integer ring. Let 931, 9R,, 9J1k, 9J1c, 93f be the set of all the places, all the infinite, real, imaginary, finite places of k respectively. Let Af (resp. A f) be the restricted product of the kv's (resp. k,X 's) over v E Of . Let k,, (resp. be the product of the kv's (resp. k,X 's) over v E 9R.. Then A = k,,. x Af, Ax = kI x A! . If x E A or Ax , we denote the finite (resp. infinite) part of x by x f (resp. x.). If V is a vector space over k, we define VA, Vim, Vf similarly. Let .5"(V, ), '(Vo,,), 9'(V1) be the spaces of Schwartz-Bruhat functions. For any place v, kv is the completion at v. If v E 9J1f, o, C k is, by definition, the integer ring of kv. Let I I be the adelic absolute value. The absolute value of kv is denoted by 11,,. For x E Ax, we denote the product of the I xI v's over all v E 9X f

(resp. v E 931.) by Ixi f (resp. Ixloo). For v E Of, let irv be the prime element, Note that if v is imaginary and Ixl is the usual absolute value, and Iirvly = Ixly = Ixl2 Let r1, r2 be the numbers of real and imaginary places respectively. Let h, R, and e be the class number, regulator, and the number of roots of unity of k respectively. Let Ak be the discriminant of k. Let (Ek = 2T1(21r)r2hRe-1. We choose a Haar measure dx on A so that fAlk dx = 1. For any finite place v, we choose a Haar measure dxv on kv so that fov dxv = 1. We use the ordinary Lebesgue measure dxv for v real, and dxv A dxv for v imaginary. Then dx = IokI-2 11v dx, (see [79, p. q,-,1.

91]).

For A E R+, let .\ be the idele whose component at v is b°Q if v E 9)1,, and 1 if v E 9J1f. Clearly, JAI = A. We identify II8+ with a subgroup of Ax by the map A -> A. Let A' = {x E Ax IxI = 1}. Then Ax = Al/kx x ][B..F, and Al/kx is compact. We choose a Haar measure dxtl on Al so that fAl/kx dxtl = 1. Using this measure, we choose a Haar measure dxt on Ax so that I

fAX f(t)dxt o= f f f(tl)dx.dxtl, 1

Notation

xii

where d',\ = )c 1dA. For any finite place v, we choose a Haar measure dxty on kv so that f0 dxty = 1. Let dxtv(x) = IxI;1dx if v is real, and dxtv(x) = IxI- ldxnd:Tc if v is imaginary. Then dxt = (tk 1 !lv dxty (see [79, p. 95]). Let <>= fiv <>v be a character of A/k. Let v be a finite place. Suppose that <>v is trivial on 7rv'°o,, and non-trivial on 7rv'°-lov. Then we define a,, = irv°. Let If v is a real place, there exists av E kv such that < x >v= e(avx), e(x) = and if v is an imaginary place, there exists a,, E kv such that < x >,= e(avx+a,,x). For almost all v, cv = 0. Let a = (av),, E A'. Then dal = jAkI-1 (see [79, p. 113]). The idele a is called the difference idele of k. Let (k(s) be the Dedekind zeta function. As in [79], we define Zk(s) = jAkl2 (7r

2r(2))"

((2-r)-e T(s))TZ SIC ($).

We define tk = Res,=1 Zk(s).

For a character w of Ax /kx, we define 6(w) = 1 if w is trivial, and 6(w) = 0 otherwise.

Introduction §0.1 What is a prehomogeneous vector space? One contribution of Gauss to number theory in the early nineteenth century was the discovery of the correspondence between equivalence classes of integral binary quadratic forms and ideal classes of quadratic fields. This correspondence can be described as follows. Let f (v) = f (vl, v2) = xovi +x1v1v2 +x2v2 be a binary quadratic form such that x0i x1, x2 are rational integers. We define an action of the group {±1} x GL(2, Z) on the set of integral binary quadratic forms so that if g = (t, gl) where t = ±1, 91 E GL(2, Z), g f (v) = t f (vgl). We consider equivalence classes of integral binary qua-

dratic forms with respect to this action. It is easy to see that the discriminant x1- 4x0x2 is invariant under such an action. On the other hand, let m be a square free integer, and consider a non-zero ideal a of the ring of algebraic integers in the field k = Q(\/m). The discriminant Ak of k is m if m =_ 1 mod 4 and 4m if m - 2 or 3 mod 4. As a module over 7G, a is generated by two elements, say a, , 3, because a is a torsion free rank two module over Z. Consider the binary quadratic form fa(v) = N(a)-1N(avl + 13V2), where N(a), N(avl +,(3V2) are the norms. It is easy to see that fa depends only on the ideal class of a. Moreover, it turns out that ideal classes of k correspond bijectively to equivalence classes of primitive integral binary quadratic forms with discriminant Ak by the map a -+ fa. Gauss established this correspondence in [16], and the reader can see a modern proof in Theorem 4 [3, p. 142]. Here, we consider a natural question: why do we consider such a correspondence? One conceptual reason is that it gives us a parametrization of ideal classes of quadratic fields in terms of a group action on

a vector space. We can use this parametrization to actually compute the class numbers of quadratic fields. But what we are interested in in this book is a more analytic question. In order to illustrate our purpose, let us describe the conjecture of Gauss. Let hd be the number of SL(2, 7G)-equivalence classes of primitive integral binary quadratic forms which are either positive definite or indefinite. Then Gauss conjectured the asymptotic property of the average of hd. However, an integral form in the sense of Gauss is a form xovi + 2x1v1v2 + x2v2 such that x0, x1, x2 are integers. Here we consider xovi + x1v1v2 + x2v2 such that x0, X1, x2 are integers. With this understanding, we have the following asymptotic formula

0<-d<x

hd

18(3)x2' 2

0
hdlogEd - 18(3)x2'

where Ed = (t + uTd) and (t, u) is the smallest positive integral solution of the equation t2 -2due = 1. This conjecture was first proved by Lipschitz [42] for d < 0, and by Siegel [69] for d > 0, and much work has been done on the error term estimate also (see [65, pp. 44,45] for example). However, we are allowing all integers d here, and if d = m2d' and d' is a square free integer, hd, hd, are related by a simple relation. Therefore,

Introduction

2

we are counting essentially the same object infinitely many times in (0.1.1). If k is a quadratic field over Q, let hk, Rk be the class number and the regulator respectively. If d is a square free integer, hd is the number of ideal classes with respect to multiplication by elements with positive norms. Therefore, this hd is slightly different from hk of a number field of discriminant d even though they are closely related. The problem of counting hkRk of quadratic fields k was first settled by GoldfeldHoffstein [17] and was slightly generalized by Datskovsky [9] from our viewpoint recently. Here, we quote Datskovsky's statement for the simplest case.

hk-

(0.1.2)

0<-ok<= [k:Q1=2

hkRk ti

H(1-p-2-p _

2 37r

p

-4)2

3+p 4)x2'

3) rT(1 _ p_2 _ p_3 + p-4)x2

O
p

[k:Q)=2

where k runs through quadratic fields and Ak is the discriminant. Note that s3 = 18. Therefore, (0.1.1) and (0.1.2) are very similar except for the difference between p-3 + p-4). S() and fp(1 - p 2 Statements like (0.1.2) are the kind of density theorems we are looking for. We discuss the difference between (0.1.1) and (0.1.2) later in the introduction, and we go back to the space of binary quadratic forms again. The main ingredients of the above correspondence were the group G = GL(2) acting on the vector space V of binary quadratic forms, and the polynomial 0(x) = x2 - 4xox2 (x = (xo, xl, x2)) which satisfies the property A(gx) = det gA(x) for g E GL(2), x E V. Moreover, if we consider this vector space over an algebraically closed field, the generic point is a single G-orbit. The fundamental reason why one can prove results like (0.1.1), (0.1.2) is that we can use the Fourier analysis on the vector space V. Also when we consider the averages as (0.1.1) or (0.1.2), we can use the value of A(x) to average over. The fact that the generic point is a single orbit assures us that there is essentially one choice of such a polynomial. Sato and Shintani introduced the notion of prehomogeneous vector spaces in [60] and generalized the situation as the above example. We now state the definition of prehomogeneous vector spaces from our viewpoint.

Let k be an arbitrary field. Let G be a connected reductive group, V a representation of G, and XV an indivisible non-trivial rational character of G, all defined over k.

Definition (0.1.3) The triple (G, V, Xv) is called a prehomogeneous vector space if the following two conditions are satisfied. (1) There exists a Zariski open orbit. (2) There exists a polynomial A E k[V] such that A(gx) = X(g)A(x) where X' is a rational character and X' = Xv for some positive integer a.

Note that if AliA2 are two polynomials as in the above definition, there exist positive integers a, b such that o is a G-invariant rational function. Since there 2

exists an open orbit, this implies that °¢ o2 is a constant function. Therefore, for any

Introduction

3

k-algebra R, the set VRS = {x E VR I A(x) 0} does not depend on the choice of A, and we call it the set of semi-stable points. A polynomial A which satisfies the property (2) is called a relative invariant polynomial. If V is an irreducible representation, the center of the image G -> GL(V) has split rank one by Schur's lemma. Therefore the choice of Xv is unique. So we call (G, V) a prehomogeneous vector space also. For the space of binary quadratic forms, let G = GL(1) x GL(2), and Xv(t, g) _ t det g for g = (t, g). Then (G, V, Xv) is a prehomogeneous vector space in the above sense.

Before we start the discussion on prehomogeneous vector spaces, let us make one more historical remark. There is no doubt that Gauss was the first mathematician who recognized the group theoretic approach to number theory. But one particular prehomogeneous vector space had appeared already in the eleventh century. The solution of cubic and quartic equations by radicals has been known for a long time. But before the solution was found, there was a poet-mathematician Omar Khayyam in medieval Persia who worked on this problem. He did not think it was possible to solve cubic equations by radicals, and instead he tried to express the solutions to cubic equations geometrically. For example, the solution of the equation x3 = N can be realized as the intersection of two parabolas y = x2, y2 = Nx. After the solution by radicals was found, his work has long been forgotten. However, it is surprisingly related to the theory of prehomogeneous vector spaces. What makes the space of binary quadratic forms so interesting is that we can associate a quadratic field to a generic point of the vector space. More precisely, if G is GL(1) x GL(2), V is the space of binary quadratic forms and Xv(t, g) = t det g, there is a map from GQ \ VAS to the set of isomorphism classes of fields of degree less than or equal to 2 over Q. This map is clearly surjective, and this surjectivity is the reason why we count the class number of all the quadratic fields in (0.1.2).

Now, let us consider the group G = GL(3) x GL(2), and the vector space V of pairs of ternary quadratic forms. If we define Xv(91,92) = (detgl)4(detg2)3, the triple (G, V, Xv) is a prehomogeneous vector space (see [59] or Chapter 8). If x = (Q1, Q2) E Vk and Q1, Q2 are ternary quadratic forms, we can consider the set Zero(x)={v E P2 Q1(v) = Q2(v) = 0}. We call Zero(x) the zero set of x. We will show in Chapter 8 that (G, V, Xv) is a prehomogeneous vector space and VSS consists of points whose zero sets are four distinct points in p2. It is easy to see that the field generated by the residue fields of points in Zero(x) is a splitting field of a quartic equation. Now the question is if we can get all such fields from pairs of ternary quadratic forms. This is easy because any quartic equation x4 + a1x3 + a2x2 + a3x + a4 = 0 can be written as an intersection of two conics as I

follows. V = x2,y2 + alxy + a2x2 + a3x + a4 = 0.

But this is what Omar Khayyam did about 900 years ago, and he essentially proved the surjectivity of the map from GQ \ VAS to the isomorphism classes of splitting fields of quartic equations. For the works of Omar Khayyam, the reader should see [74]. The analytic theory of this prehomogeneous vector space is the main topic of this book, and we handle the global zeta function for this case in Part IV.

Introduction

4

§0.2 The classification In this section, we discuss the classification of irreducible prehomogeneous vector spaces over an algebraically closed field. Throughout this section, k is an algebraically closed field of characteristic zero. First, we show that from a given prehomogeneous vector space, we can make infinitely many prehomogeneous vector spaces which are essentially the same as the original prehomogeneous vector space.

Let G be a reductive group, and V a representation of G. Suppose that the dimension of V is n. For an integer 0 < m < n, we consider two representations (G x

GL(m),VOkt), (GxGL(n-m),VOkn--). Then genericGL(m)k-orbits of VkOktm correspond bijectively with generic GL(n - m)k-orbits of Vk 0 kn because the Grassmann variety of m planes in V can be identified with the Grassmann variety of n-m planes in V. Therefore, (G x GL(m), V ok') is a prehomogeneous vector space if and only if (G x GL(n - m), V 0 kn-n,.) is a prehomogeneous vector space, and the sets of generic orbits coincide. If two prehomogeneous vector spaces are related in this way, we identify two such representations and consider the equivalence relation determined by this identification. If the dimension of (G, V) is the smallest among prehomogeneous vector spaces which are equivalent to (G, V), we say that (G, V) is reduced. Also we identify two prehomogeneous vector spaces (G, V), (G', V) if the images of G, G' in GL(V) are the same. Sato and Kimura proved in [59) that the following is the list of all the irreducible reduced prehomogeneous vector spaces. (1) G = GL(n) x H, V = M(n, n)k where H C GL(n) is any reductive subgroup such that the kn is an irreducible representation of H.

(2) G = GL(1) x GL(n), V = Sym2kn. (3) G = GL(1) x GL(2n), V = (4) G = GL(1) x GL(2), V = Sym3k2. (5), (6), (7) G = GL(1) x GL(n), V = A3kn where n = 6,7,8. (8) G = GL(3) x GL(2), V = Sym2k3 ® V. (9) G = GL(6) x GL(2), V = n2k6 0 (10), (11) G = GL(n) x GL(5), V = kn 0 A2k5 where n = 3, 4. (12) G = GL(3) x GL(3) x GL(2), V = k3 ®k3 0 k2 (13) G = GSp(2n) x GL(2m), V = ken 0 k2m where n > 2m > 2. (14) G = GL(1) x GSp(6), V is a 14 dimensional representation of G. (15) G = GO(n) x GL(m), V = kn 0 k'' where n > 3, 2 > m > 1. (16), (17), (18) G = GSpin(7) x GL(n), V = spin? 0 kn where n = 1, 2,3 and spin7 is the spin representation. (19), (22) G = GSpin(n), V = spine where n = 9, 11. (20), (21) G = GSpin(10) x GL(n), V = halfspinlo 0 kn where halfspinlo is the halfspin representation and n = 2, 3. (23), (24) G = GL(1) x GSpin(n), V = halfspinn where n = 12,14. (25), (26) G = G2 x GL(n), V = k7 0 kn where k7 is a representation of G2 and n = 1,2. (27), (28) G = E6 x GL(n), V = k27 ® kn where k27 is a representation of E6 and n = 1, 2. (29) G = GL(1) x E7, V is a 56 dimensional representation of E7 (30) G = GSp(2n) x GO(3), V = ken 0 k3 n2k2n.

V.

Introduction

5

The cases (1)-(29) are what we call regular prehomogeneous vector spaces. For the definition of the regularity, the reader should see [59]. Even though it does not make any difference over an algebraically closed field, we have included the GL(1) factor in (2)-(5) etc. and used groups like GSp(2n), GO(n) instead of Sp(2n), O(n) etc., because it is more natural number theoretically. Most of these representations are what we call prehomogeneous vector spaces of parabolic type classified by Rubenthaler in his thesis [52]. This is the kind of prehomogeneous vector spaces which one can construct from parabolic subgroups of reductive groups as follows. Let G be a reductive group, and P = MU a standard parabolic subgroup where M is the Levi component and U is the unipotent radical. The reductive part M acts on U by conjugation, and therefore on V = U/[U, U] also. Since V can be considered as a vector space, we have a representation of a reductive group M. Vinberg [75] proved that there is a Zariski open orbit. Therefore, if there exists a relatively invariant polynomial, (M, V) is a prehomogeneous vector space by choosing a relative invariant polynomial and is called a prehomogeneous vector space of parabolic type. For example, if we consider the Siegel parabolic subgroup P of GSp(2n), M = GL(1) x GL(n) and V is the space of quadratic forms in n variables. If G is a type Cn group etc., we say that (M, V) is of type C,,, etc. Then (2) is Cn type, (3) is Den type, (4) is G2 type, (5), (6), (7) are of E6, E7, E8 types, (8) is F4 type, (9) is E7 type, (10), (11) are E7, E8 types, (12) is E6 type, (13) is Cn+.,,,, type, (14) is F4 type, (15) is B, D type, (16) is F4 type, (20), (23), (25), (27) are E7 type, (21), (24), (26), (28) are E8 type (29) is E8 type. (1) is not always of parabolic type. (17), (18) (19), (22), (25), (26) are not in Table 1 [52, pp. 35-38]. For the details on prehomogeneous vector spaces of parabolic type, the reader should see [52].

§0.3 The global zeta function In this section, we discuss the meromorphic continuation and the functional equation of the zeta function, restricting ourselves to irreducible prehomogeneous vector spaces (G, V, Xv) for simplicity. The reader should see §3.1 for the general definition of the zeta function. For the rest of this section, k is a number field. For simplicity, we assume that there exists a one dimensional split torus To GL(1) in the center of G acting on V by the ordinal multiplication by tep E GL(1) and Xv(t) = to for t E To where eo, e > 0 are positive integers. Let A be a relative d invariant polynomial, and d the degree of A. Then I A(gx) _ Xv(g)I IA(x)I. Let N be the dimension of V. We assume that the representation G -+ GL(V) is faithful. Therefore, in terms of the list in §0.2, we are considering (GIT,V) where T is the kernel of the homomorphism G -> GL(V). We fix a Haar measure dg on GA. Moreover, we assume that dg is of the form dg = fv dg,, where dg, is a Haar measure on Gky for v E 912. Let L C VV' be a Gk-invariant subset. For 4 E .'(VA) and a complex variable s, we define (0.3.1)

ZL(`1', s) = f A /Gk

IXv(g)I E 4'(gx)dg, xEL

Introduction

6

-

ZL+(4),s) = f GA/Gk IXv(9)II E-P(9x)d9, Ixv(9)1>_1

xEL

if these integrals are well defined. We say that (G, V, xv) is of complete type if Zvka (iP, s) converges absolutely for Re(s) >> 0 and Zv, +(4b, s) is an entire function for all 4 E .Y(VA). It is very likely that the first condition implies the second condition. We say that (G, V, xv) is of

incomplete type if it is not of complete type. If the stabilizer G., contains a split torus in its center for some x E Vks, (G, V, xv) is of incomplete type. This applies to the cases (2) n = 2, (12), (15) m = 1, 2, (17) in §2. If Gx does not contain a split torus in its center for any x E Vks, it is very likely that it is of complete type even though this has yet to be proved. Some examples are known. Siegel showed that the case (2) in §0.2 for n > 3 is of complete type. In general, if dim G = dim V, it is of complete type, as we show in this book. This applies to the cases (4), (8), (11) in §0.2 (the case (4) is due to Davenport and Shintani). If (G, V, xv) is an irreducible prehomogeneous vector space of complete type, we choose Vks as L in the definition of the zeta function and use the notation Z(4), s), Z+(4), s). We fix a measure dx = Fv dxv on VA. Then Io(9x)I = Ixv(9)I1! IA(x)l.

d(9x) =

First we show that ZL(4), s) decomposes into a summation over rational orbits. E Y(VA). Let x E Vks. Let Gx be the stabilizer of x, and Suppose 4) = ®t, G° its connected component of 1. Let o(x) = [Gxk : G°k]. Then by an obvious consideration, ZL(4), s) XEGk\L

o(x) fGA/Gzk I Xv(9)I

,P(9x)d9

Let x E L, and v E fit. We choose a left Gkv-invariant measure dg' ,v on Gk /G°ku

and a Haar measure dg',,, on G°kv so that dg,, = dg's vdg'",v for all v and dg' _ rjv dg'x,v, dgx' = ft, dg'' ,v are well defined. Then there exists a constant bx,,, > 0 such that q, (9x,vx)dgxv = bx,v Gkv /Gzk

JGkx 1'(yv)ID(yv)Iv d dyv

for any measurable function tW on Gk x. We discuss the choice of the measures dg.' v, dg' ,v for some cases in §0.5.

Let µ(x) be the volume of G°A/G°k with respect to the measure dg'. We define N

Xx,v(-P,s) _

Gk,x

Zx,v(4D, s) = f

v(yv)ID(yv)IV-- dyv,

k/Gzk

I Xv(gxvI v(9xvx)d9xv

Introduction

7

Note that these distributions depend only on the orbit Gk x and D,,. These distributions are related as follows.

f(x)Ilxv(x)dg

Zx,v(4,S) = I (x) I-i ID(x)I

df

ID(g',vx)I d'v(gX,vx)dgx,v

Gk,. /Gxk

= bx,vIA(x)IVo1Xv(.D, S)

We choose an infinite place vo. Then the above decomposition becomes

(0.3.2) ZL(-', s) _

o(x) 11 Zx,v(4), s) xEGk\L

(

"Em

u(x) Let V1,

, U be

11

bx,voXX,v,(4), s)

xEGk\L o(x)IO(x)I o

Zx,v(,D, S)

vcm\{vo}

-orbits of Vkvo . Note that even though the group G is assumed

to be connected as an algebraic group, Gk .0 may not be connected in general, for , U, example GL(1). The distribution Xx,,,o(4D, s) only depends on the sets V1i and w e denote it by Xl((D, s), , X, (4), s). We define (0.3.3)

CCX((D, s) _

(X1(`),s),... ,X1(4),s)),

S) _ E xEGk\V,nL

i(x)bx vo

f

,Zix,v(,D, S),

o(x)I(x)I0 vE97t\{{vo}

SL, 1(', S)

L(D,S) = L,(P, S) Then ZL(D, S) = X ('P, S)eL(4), S)

-

If L = VkB, we drop L and use the notation 6(4), s) etc. Let V* be the dual space of V and (x, y) the natural pairing of x E V, y E V*.

For g E G, let g* E GL(V*) be the element such that (gx,y) = (x,g*y) for all x E V, y E V*. Then g . (g*)-1 is a representation of G on V* and is called the contragredient representation of (G, V). If (G, V) is a prehomogeneous vector space, (G, V*) is also a prehomogeneous vector space, and there exists a relative Also invariant polynomial A* of the same degree d. Moreover Xv(g) V*o has the same number of Gkvo-orbits Vi , , V*. If L* C Vk is a Gk-invariant subset, we define functions Xi (-D*, s), 6L. i(V, s) for i = 1, , l and X * (4D*, s), eL. (4)*, s) similarly as in (0.3.3) for V*. Let A* (a), A(a) be the differential operators with constant coefficients on V, V* which correspond to A*, A. The local theory at infinite places is based on the following fact.

Lemma (0.3.4) There exists a polynomial b(s) (resp. b*(s)) such that A*(a)A(x)3 = b(s)A(x)s-1 (resp. A(a)A*(x)* = b*(s)A*(x)s-1)

Introduction

8

for any natural number s. This lemma follows from the fact that A(8), A*(8) are invariant differential operators. These polynomials b(s), b*(s) are called the b-functions for A, A* respectively. For these facts, the reader should see [60], [64]. Once we know (0.3.4), the meromorphic continuation of Xi(4>, s) follows by the relations Xi(O*(8)4), s) = +b(3dN)Xi(4), s - d) X (O*(8)0*(8)4i, s) = fb(dN)b(dN )X (4), s - d) For 41 E So(VA), we define its Fourier transform 4i

v° E erg, v° E 931c.

E 9(V;) by

4)(y) = fV' 4i (x) < (x, y) > dx.

For V E 9'(V;), we define its Fourier transform 4)* E .9(VV) similarly. If -D has a product form 4) = ®v4i,,, 1i has the product form where 4iv is the Fourier transform of 4i with respect to < (x) y) >,,. The following theorem was proved by Sato and Shintani in [60].

Theorem (0.3.5) (Sato-Shintani) The distributions Xi((D, s), Xt (4i*, s) can be continued meromorphically to the entire s E C for all i, and satisfy a functional equation X* (4), N - s) = X(41, s)C(s),

where C(s) = (cij(s))1
dXi((D s), Xi ($1, N - s) = IXv(g)I eX, (4i, N - s)

for all i.

Choose xi E V for i = 1, ,1. Then Vi = Gkvo /G,;;,k,, . So if 4i E is compactly supported, (0.3.5) follows from the uniqueness of left Gkvo-invariant the reader , V1. For the proof of (0.3.5) for general measures on orbits V1i should see [60]. A similar statement to the above theorem for reducible prehomogeneous vector spaces was proved by F. Sato. As we will see below, the functional equation of the global zeta function follows from (0.3.5) for complete types. Therefore, one can deduce a similar global functional equation under the assumption that the zeta function is defined for V. For this, the reader should see [56]-[58]. For finite places, a similar statement to the above theorem was proved by Igusa in [22]. Suppose that (G, V), (G, V*) are of complete type. We denote the zeta function for V* by Z*(4D*, s), Z+(4i*, s). Let G°I = {g E GA I IXv(g)I = 1}. For 4; E 9'(VA) and A E R}, we define 4a E 9(VA) by the formula 4ia(x) = 4i(Ax). We choose a Haar measure dg° on G°, so that

Z(4i, s) = JO f

GA/Gk

c(g°x)d" Adg°.

As

xEVk"

Introduction

9

For any (D E 9'(V,), we define

°

J(D,g) = (lx(g)I

D(gx) xEVk \Vk

,

xEVk \Vk'

J(,9°)d9°,

I0((.b) = f

A/Gk 1

I(,D,s) = fSIO()dXAThen by the Poisson summation formula, (0.3.6)

Z(4, s) = Z+(ID, s) +

N - s) + I(1, s).

Sato and Shintani proved the following theorem under a strong assumption in [60]. Shintani later used a slightly different argument and proved that the meromorphic continuation and the functional equation of the global zeta function follow from Theorem (0.3.5). The reader can see his argument in [65] where he only considers the space of quadratic forms, but his argument works for the general case as follows.

Theorem (0.3.7) Suppose that (G, V), (G,V*) are of complete type. Then the zeta functions Z(4), s), Z*(4)*, s) can be continued meromorphically everywhere for all (D E 9'(VA), V E 9(1) and satisfy a functional equation Z(4', s) = Z*($, N - s). Proof. Consider vo E 9BBOO in (0.3.2). Suppose v° E 931k. We fix 1 < i < l and prove that Si(4), s) can be continued meromorphically everywhere. We choose 'D = so that (Dv0 (x) E Co (V ). Let 4)1 = ®v(D1v E .O(VA) be the function such that and (D1v = lDv for all v E 931 \ {vo}. Then 1vo(x) = so (D1vo(x) = 0, $lv0(y) = 0 for x E Vk,o \ Vky y E Vk o \ s)+Z+($1, N-s). Similarly, Z*($1, s) _ Vk* . Then by (0.3.6), Z(4D1, s) = o Z+(4)1i s) + Z+(4)1, N - s). Therefore, Z(-D1, s), Z*((D1, s) are entire functions and Z(-I)1i S) = Z*($1, N - s). Since Xi(D1, s) = ±b(S aN)Xi(,D, s - d), we can choose ,D ,a so that Xi(451i s) 0. By the choice of Dvo, Xj((D1, s) = 0 for j # i. Therefore, Xi(,D 1, s)t i(,D, s) is an entire function. This proves the meromorphic continuation of Si(4), s). The meromorphic continuation of 1;'*((D*, s) is similar. By (0.3.5) and what we have just shown, X * (4)1, N -

N - s) = X(4)1, s)(4), s) = X (,D1,

Since Xj(,D 1, s) = 0 for j

i, l

(0.3.8)

i(4), s) _

Cij j=1

N - s).

N - s).

Introduction

10

Therefore, C(
Since C(4), s) and e*( , N - s) do not depend on the vo part, the above equation is true for any 4) E So(VA).

Now we consider an arbitrary 4). The meromorphic continuation of Z((D, s), Z*(
follows from that of X(
Z(
N=Z*($,N-s).

s)

Thus the functional equation for general
If vo E 911, the proof is similar, except that we consider either <1vo (x) _ Then

or

(Dlvo(x) = (21r Yi)dA*(x)
It is easy to see that d

d ND(yvo)

..

X(iD1 s) = J

dyvo or

d-N

fb(dN)fVva
(Yvo)Tdyvo.

Therefore, X(
z(P, s) =

Z(
-

Z(
d)

and Z(
Introduction

11

case of pairs of ternary quadratic forms. Therefore, the theory of b-functions does not predict the correct pole structures of zeta functions in general. Let f (s) be a meromorphic function of s. Suppose a

f (s)

s

so)n

+ ... +

a-1

s-so + g(s),

where g(s) is holomorphic around so. We call a_n

(s - so)n

+ ... +

a_1

s - so

the principal part of f around so. If (G, V), (G, V*) are of complete type, the set of poles of Z(4), s) is contained in the set of roots of the b-function. Therefore, Z(4D, s) has a finite number of finite order poles. An easy consideration as in the following proposition shows that we can expect a principal part formula if (G, V), (G, V*) are of complete type.

Proposition (0.3.9) Suppose that (G, V), (G, V*) are of complete type. Let s = s1i , sn be the poles of Z(,D, s), and fi((D, s) the principal part of s) around si. Then n

fi(4, s).

Z(41, s) = Z+(`1, s) + Z+($, N - s) + i=1

Proof. We define

n

I (D, S) = I (D, s) - > fi(41, s). i=1

We also define I*(V, s) etc. for V* similarly. Then it is easy to see that I(1, s) _ IJ(iD,g)I is an integrable function if I*(D,N - s). By assumption, IXv(g)I Re(s) >> 0. Moreover if IXv(g)I < 1, this function is decreasing with respect to Re(s) and R.LU

Re(im

IXv(g)I

0

if IXv(g)I < 1. Therefore, by the bounded convergence theorem, I(4D, s) is bounded

ifRe(s)>>0and lim

Re(s)-goo

I(q), s) = 0.

The function a 1 fi(4), s) clearly satisfies similar properties. Therefore, I(4D, s) is bounded if Re(s) >> 0 and lim

Re(s)-.oo

I(4b,s)=0.

The same is true for I*(4)*, s) also for any V E 9(V;). Since

I(4, s) = I*( , N - s), I(1D, s) is bounded if Re(s) << 0 and lim

Re(s)T-.

I(4D, s) = 0

Introduction

12

also.

Therefore, I(4), s) is an entire function, and is bounded if Re(s) >> 0 or Re(s) <<

So, by Lindelof's theorem, I(4), s) is bounded everywhere. This implies that I(4), s) is a constant function by Liouville's theorem. Since 0.

lim I(4i, s) = 0, Re(s)-oo

I(4), s) = 0. Q.E.D.

We call a formula of the form (0.3.9) a principal part formula. The statement of the above proposition is very likely to be false for prehomogeneous vector spaces of incomplete type. For example, in the case of the space of binary quadratic forms, the zeta function is defined for the set of forms without rational factors and it has infinitely many poles from the zeros of the Riemann zeta function. We will discuss this case in §4.2. Suppose that there is an involution g --4g' of G and a bilinear form [ , ] defined over k such that [gx, g`y] = [x, y]. We use this bilinear form to identify V with V*. Then t composed with the homomorphism G --+ GL(V*) gives us the contragredient representation of V. Therefore, we can consider 4? E 9'(VA) under this condition. So if (G, V) is of complete type, (G, V*) is of complete type also, and we can consider Z' ($, s) = Z($, s). For groups over R, the existence of such an involution is known (see [64] for example). If G is a product of GL(n)'s like G = GL(ni) x x GL(nf), there is a canonical

involution g = (gl,

.

, gf)

t9f 1). Then the weights of the

.+ g" = (tgi

representation (g, x) -* g`x and the weights of V* are the same. Therefore, these representations are equivalent. This implies the existence of a bilinear form as above. For the rest of this section, we assume that G is a split group over Q for simplicity. Suppose 4) = ®4>, and 4 v is the characteristic function of 1/ for all v E 9Jt f. Then

4? is clearly invariant under the action of Kf =

11vEJRf

Gov. Since Kf \ GAIGQ

GR/GZ,

IXV()l(goox)dg..

ZL(P, s) = f

xEL

R/Gz

But since goo E GR, 4 (gonx) = 0 unless x E Vx. Therefore,

ZL('D, s) = f

Doo(goox)dgoo.

IXV(9oo)I 00 xEVznL

G R/Gz

This is the classical definition of the zeta function as was considered by Shintani in [64], [65]. We can apply a similar argument and this time, we can write Z(4i, s) in terms of Dirichlet series. Let X1(41, s), , X1(41, s) be as before. Let µ00(x) be and oon(x) = [Gz; GO]. the volume of G° R/G° z with respect to the measure Then (0.3.10)

ZL('P, S) _

Xi (4, s) =1

poo(x)bx,ao xEGz\V,'nVznL Ooo(x)IA(x)Ioo

Introduction

13

This implies that

µ.(x)bx,°°

a(p, s) =

xEGz\V,nVznL ooo(x)IO(x)I

for all i. Many statements in analytic number theory are based on the following theorem.

Theorem (0.3.11) (Tauberian Theorem) Let f (s) = E,a n be a Dirichlet series such that an > 0 for all n. Suppose that f (s) has an analytic continuation to Re(s) > a and is holomorphic everywhere except for a pole at s = 1. Moreover, we assume that f (s) = g(s)(s - 1) a for some positive integer a where g(s) is a function which is holomorphic around s = 1. Then

n=1

X(logX)a-1. an - P(l) a

This is the plain Tauberian theorem. For the proof of this theorem, the reader should see Theorem I in [46, p. 464]. The version of Sato and Shintani in [60] was specially designed for prehomogeneous vector spaces and is much stronger. One can use Sato-Shintani's version to actually give some error term estimates in some cases. For this, the reader should see [65]. Let us first consider the associated Dirichlet series, and discuss what we can deduce for Gz-equivalence classes of generic integral orbits. If (G, V), (G, V-) are of complete type, the poles of 1j2(4),s)'s are contained in the set of poles of Z(1), s). Therefore if Z(4i, s) satisfies the condition of (0.3.11), we obtain the existence of constants C, a, b such that

E

a EGZ\v;nvznL Io(=)I0o<x

l_too(x)bx,oo o 00 (x)

CXa(logX)b.

§0.4 The orbit space Gk \ Vks In the last section, we saw two decompositions of the zeta function into summations over rational and integral orbits. The main purpose of the analytic theory of prehomogeneous vector spaces is to use these decompositions and the Tauberian theorem and study the distribution of arithmetic objects represented by these orbit spaces. Therefore, it is natural to compare the difference between these two decompositions, and consider the meanings of these orbit spaces. Let k be any field of characteristic zero. Let (G, V, x) be a prehomogeneous vector space over k. We choose a point w E Vks. This is possible because k is an

infinite field and VSS C V is a Zariski open set. Let x E V. Then x is in the G-orbit of w over the algebraic closure k. We choose gx E Gk such that gw = x. For o, E Gal(k/k), we define Tx,Q = g; 1g°. Then cx = {-r,,,,} defines an element of H1(k/k, G,,,) (the first Galois cohomology set). Let p : H'(k/k, Gx) -+ H'(k/k, G) be the natural map. The following proposition was proved by Igusa in [24].

Proposition (0.4.1) (Igusa) By the map x - cx, Gk \ Vks corresponds bijectively with p 1 (1).

14

Introduction

Proof. Let x, y E Vks. Suppose ex = cy. Then there exists an element h E G,,,k such that g. -19a = h-1gy lgy h'. This implies that the element h' = ghg;-1 is invariant under the action of the Galois group. Therefore, h E Gk and hx = y. Following the argument backward, the converse is true also. This shows that the map Gk \ VkS -> p-'(1) is well defined and is injective. Let E p -1(l). This means that there exists g E GE such that To = g-1g° for all v E Gal(k/k). Then (gw)° = g°w = grow = gw for all Therefore gw E Vk. This proves the surjectivity. Q.E.D.

We consider some examples. If G is a product of GL(n)'s, H1(k/k, G) = 0 (see [38, p.16] for example). Therefore, if w E Vks, the quotient space Gk \ Vks corresponds bijectively with H1(k/k,G,,,). We consider the cases (2) n = 2, (4), (8), (9), (11), (12), (15) n = 4, 6, m = 2 of the classification. For these cases, D. Wright and the author proved in [84] that there exists a point w E VkS such that the connected component G° of 1 is a product of GL(n)'s and the quotient G,,,/G°, is isomorphic to 67i for some i where 67i is the permutation group with the trivial Galois group action. The number i is 2 for the cases (2) n = 2, (15) n = 4, 6, 3 for the cases (4), (9), (12), 4 for the case (8), and 5 for the case (11). Since the Galois group action on 67i is trivial, H1 (k/k,67i) is the set of conjugacy classes of homomorphisms from Gal(k/k) to 63i. If x E Vks, let p,,, be the homomorphism which corresponds to x. The kernel of px is a closed subgroup of finite index, and therefore determines a Galois extension of k, which we denote by k(x). We also proved in [84] that k(x)'s are splitting fields of degree i equations without multiple roots and all such fields arise in this way. We already discussed the construction of k(x) for the case in §0.1. As is illustrated by this example, the construction of k(x) for other cases was geometric also. This consideration shows the following theorem.

Theorem (0.4.2) Consider the cases (2) n = 2, (4), (8), (9), (11), (12), (15) n = 4, 6, m = 2 of the classification. Let x, y E VkS. Then GkX = GkY if and only if there exists a permutation r such that px = rpyr-1.

For the details, the reader should see our paper [84]. Next, we consider the case (15) in general (i.e. not necessarily split). We fix a

quadratic form Q on W = k"` and consider the group GO(Q) as GO(n) in (15). with We identify V with the space of linear forms in m variables y = (yl, , , w,,,. E W, coefficients in the space W. To x = ylwl + + y,,,,w,,,, where wl, we associate a quadratic form in m variables Q(ylwl + + ymwm). This means that we are considering quadratic forms which can be expressed in terms of a fixed quadratic form Q. So this is the situation of the Siegel-Weil formula. However, the zeta function theory may give us a different statement from the Siegel-Weil formula. In [24], Igusa considered some exceptional cases and proved that points in Gk\Vks correspond with division algebras. Igusa also proved in [24] that the orbit space Gk \ VkS consists of one point for the cases (1), (3), (13), (15) n even and m = 1, (16), (19), (20), (27). If k is a number field, (6) falls into this category also. We call these cases single orbit cases. As far as rational orbits are considered, these are the cases where the counting problem is trivial. The reader can probably see from these examples that the space Gk\Vks parametrizes interesting arithmetic objects. But what are we counting? In the case of

Introduction

15

Gz-orbits, we were counting the quantity o- (X) If we make an analogy when k is a number field, we may be counting the quantity In the case G = GL(1) x GL(2), V = Sym2k2, µ(x) gives us the class number times the regulator by the natural choice of the measures dg.' dg'X as was proved by Datskovsky in [9]. If we consider the group G = GL(1) x SL(2) and the space of binary quadratic forms, we obtain the statement (0.1.1) by the consideration of Gz-orbits. Therefore, the difference between rational orbits and integral orbits is roughly speaking the difference between (0.1.1) and (0.1.2). So by considering GZ \ V/9, we may be counting essentially the same object infinitely many times. For the above reason, we are more interested in rational orbits. However, one big problem arises now. In the decomposition (0.3.2), the coefficient of is not a constant, and the quantity to average over is not so clear. Therefore, we cannot directly apply the Tauberian theorem. If the rightmost pole is a simple pole, Datskovsky and Wright [11] and Datskovsky [9] formulated a process called the `filtering process' to deduce density theorems for Gk-orbits. We discuss this process in the next section.

§0.5 The filtering process and the local theory: a note by D. Wright The content of this section is largely from D. Wright's note, with a slight modification according to Datskovsky's formulation in [9]. In this section and §0.6, we assume that k is a number field. We have a function Z(4), s) called the zeta function. It has some group theoretic properties. However, the filtering process depends only on the formal properties of Z(4D, s). So we formulate the filtering process in an abstract way. We consider an arbitrary tempered distribution Z(4), s), i.e. a continuous linear functional on .'(VA). We have to make certain assumptions. The first assumption is the following.

Assumption (0.5.1) (1) The function Z(4), s) has an analytic continuation to Re(s) > rc > 0 and is holomorphic everywhere except for a simple pole at s = K with the residue R((P). (2) The function Z(4), s) has the following decomposition C(x)

Z(,D, s) _ E

xEI IOxI

8

Lx (4), s),

where I is some index set, c(x) > 0, Ox E Z are constants depending on x. This Ox should not be confused with 0(x). (3) If 4) = ®v4Dv E .S°(VA), the function Lx(4D, s) has a product form Lx (4),

fl Lx, (4),, 8). V

(4) If v E 93 f and iDv is the characteristic function (which we denote by s) has a form

V0,,,

Lx,v(IDv) s) = 1 + E an,v9'v "s m> 1

of

Introduction

16

where am,,, > 0 for all m. We define L.,,,,(s) = LX,,,(,Do,,, s). For any finite set 9JtO C S C 9R, we define

4)S = II opv, vES

Lx,s(4', s) = rl

s),

vos

s) = ri Lx,v(4v, s), vES

L.,s(s) = fj Lx,v(s). vos

By the assumption (0.5.1) (4), we may write ax's(m).

Lx,s(4), s) = 1 + E

r, s

m>2

Obviously, ax,s(m) > 0 for all x, S, m. Let Av be the index set of all the possible types of Euler factors Lx,v(oDv, s), I« indexed by Av such that if x, y E av, i.e. I has a decomposition I = Lx,v('v, s) = Ly,v(4)v, s). For av E Av, let Lay s) be the corresponding Euler

factor. Let La (s)

L1,ES Av.

Let As =

For a = (a,,) E As, we

define

(bs, s) =

HL.,

(cv, s).

vES

For x E I, we say that x E a E As if xeIav for allvES. Suppose that 4) = ®v,Dv and (Dv = %, for v S. Then

s) E p-XISLx5(s). xEa

S)

aEAs

We define

ea,s(s) = E

(xLx,s(s)

sEa IA- IS

Assumption (0.5.2) (1) For any S and a E As, 4's may be chosen so that (a(4)s, s) # 0 while (p(4)s, s) _ 0 for all /3 0 a.

(2) There is a constant ra > 0 for each a such that

R(4) = E ra(«('s,'i) aEAs

These two assumptions imply that Res,=,. ,s(s) = ra.

Introduction

17

Let S C T C 9A be another finite set. For

=

E AT, we define

Is =

E As. For a E As, we define

IA7Lx,T(S)

a,T(S) _ xEa

x

c(x) LX(s) I: I: IOls x

pEAT,1Is=a xE,Q

E

ea,T(S)

IEAT,/I S=a

We need another condition of uniformity.

Assumption (0.5.3) There is a Dirichlet series 1 + 2m>2 a;,,;" which is holo morphic in Re(s) > n such that (ax,T(m)I < aT(m) for all x E I. Also, for any N > 0, we may choose T sufficiently large so that aT(m) = 0 for 1 < m < N. Finally, aT(m) < as(m) if S C T. Under these assumptions, the following proposition was proved in [9], [11].

Proposition (0.5.4) 1"

lim

X-00 X

E c(x) = 1k lim T-M IOxI<_X

rp

I

OEAT,PIs=a

if the right hand side converges.

Proof. By the Tauberian theorem, 1

(0.5.5)

E

c(x)ax,T(m) - 1

X"

xEa

rp.

PEAT,/I s=a

Iox Im<x

An easy consideration shows that (0.5.6)

c(x) + E c(x)ax,T(m).

E c(x)ax,T(m) _

xEa

xEa

E-

Azj-<X

1<-
IoxI<x

By the assumption (0.5.3),

1<"

t,

c(x)aT(m)

c(x)ax T(m)

xE

-E- X

1< n
E aT(m) >xE-c(x).

1<m<X

1ox1< x

Since ax,T(m) > 0, by (0.5.5), (0.5.6), there exists a constant C > 0 such that

c(x) < CX" IoxI<x

Introduction

18

for all X > 0. So applying this to Xm, a.T(m)

E aT(m) E c(x) < CX" 1<m<X

1<m<X

IA.1:5m

n

< CX"(LT(rl) - 1), where LT(s) = 1 + Em°°-2 amm'

.

Considering the limit T -* 9)1, we get the statement of the proposition. Q.E.D.

From this consideration, it is clear that in order to apply this process, we have to know the residue R(F) because otherwise we have no control of the value r0. This means that without knowing R(f), we do not even know the existence of a statement of the form (0.1.2). Therefore, the computation of the principal part of the global zeta function is not just for determining the constant in the asymptotic formula for Gz-orbits for which we basically know the existence. As far as the local theory is concerned, we have to answer the following questions. (1) Estimate s) on the domain Is I Re(s) > v} for some o < k and study its q-expansion. (2) Find the value b,,,, for sufficiently many x, v. (3) Find the value ra for sufficiently many a. Therefore, we do not necessarily have to know the poles of Z,,,,, (4P, s), because it is very likely that Z,,,, (4P, s) is holomorphic at the rightmost pole of the global

zeta function. However, we may have to explicitly compute it in order to deduce the assumptions (0.5.1), (0.5.3). Except for the single orbit cases, this process has been carried out for only two cases. One is the space of binary cubic forms where the statement for Gk-orbits is a slight generalization of the theorem of Davenport and Heilbronn: X

lokl<'

((3) '

[k:Q]=3

where k runs through all the cubic fields and Ok is the discriminant. The other case is the space of binary quadratic forms where the statement for Gk-orbit is the theorem of Goldfeld-Hoffstein-Datskovsky (0.1.2). Except for these two cases and the single orbit cases, we do not even know if an asymptotic formula like (0.1.2) exists or not. These cases are good examples to illustrate the filtering process, so we review these two cases. The representations we consider are (GL(1) x GL(2)/T, V) where T is the kernel of the homomorphism GL(1) x GL(2) -> GL(V) and V is the space of binary cubic or quadratic forms. We consider L = Vk 5 in the case of binary cubic forms, and the set of forms without rational factors in the case of binary quadratic forms. Let v E 9)1. We

choose the index set A as the set of

in V. For each a E A,,, we

choose a `good' representative in Vkv and use the same notation a,,. This element a corresponds to a field k of degree < 3 or < 2 over k and has the property that is the local discriminant of In the case of binary cubic forms, G° is trivial. In the case of binary quadratic forms, we choose a measure dg' on G°v so that the volume of G°°0 0 is 1 if v E 9J . 01

Introduction

19

If v E Vc., we choose the `canonical measure' on G°v, which is isomorphic to either ", a circle, or C'. This formulation is due to Datskovsky. If x E L is in the orbit of a,,, we choose an element h,, E Gkv so that x = hxa,,.

Clearly, G°k = hzG°vk.h-1. We define dg',,, to be the measure induced from G°yk.. Then br,,, = b«,,,,,. (We defined b.,,, for x E L, but we can extend the definition to elements of Vk".) Therefore,

Z.,,(-,D, ',

s) =

bx,"

.

lo(x)Iv

X"(-P,X'v , X.,"(4), s) = Io(x)Iv

IV

Z«o,v(4, s)

IA(x)Iv

in the case of binary quadratic forms, and

s) = I°av I e I0(x)I"

S)

in the case of binary cubic forms by a similar consideration. Let k(x) be the field generated by a root of the form, and Ax the discriminant of the field k(x). Since IA(x)l = 1, ZX,"(-I>, s) V

IAXI

4

R,

Z,

(4',s), or

1A.I 9 11 Za,,.v(b, s),

for all v. We where k(x) is the field which corresponds to x and Gk.,av = choose Z«,,,v(4P, s) as Lx,v(oDvI s), and o(') as c(x). Then all the assumptions are satisfied.

The computation of r0 reduces to the computation of (1 - qt-, ')(1 - gv2)IA«vlv in the case of binary cubic forms and vol(Kvav) in the case of binary quadratic forms. In [11], Datskovsky and Wright proved that if v E 9J f,

1-qv3, where o(av) is the order of the stabilizer in Gkv in the case of binary cubic forms, and in [9], Datskovsky proved that if v E 9Jl f,

vol(Kav) = 1 -qv2 -qv3+qv4 in the case of binary quadratic forms. This explains the Euler factor in (0.1.2) or in the theorem of Davenport and Heilbronn. In both cases, we have Ck(2) from the residue of the zeta function. In the case of binary cubic forms, it cancels out with (1 - q,-') in the above consideration. For the details, the reader should see [9]-[11]. As far as the principal part of the zeta function is concerned, Shintani handled two cases: (a) the space of binary cubic forms; (b) the space of binary quadratic forms and the space of positive definite quadratic forms in n > 3 variables. In this book, we prove principal part formulas for two more cases: (c) the space of quadratic

forms in n > 3 variables (not necessarily positive definite); (d) the space of pairs

Introduction

20

of ternary quadratic forms. Igusa studied the Euler factors of the single orbit cases and the zeta function is basically a product of Dedekind zeta functions. In case (c), the rightmost pole of the zeta function is simple. So if one can work out the local theory, it is possible to deduce a density theorem for Gk-orbits. However, this may follow easily from Siegel's Mass formula without using zeta function theory. In case (d), the rightmost pole is of order 2, so the filtering process has to be improved in the future. Besides the single orbit cases, the above cases are all the split irreducible reduced prehomogeneous vector spaces for which the principal part of the global zeta function is known. Since the cases (12), (15) m = 1, 2, (17) are prehomogeneous vector spaces of incomplete type, the meromorphic continuation of the zeta function (or the definition of the zeta function) is unknown and it does not follow from (0.3.7). The case (2) n = 2 is of incomplete type, but was handled by Shintani in [65]. In many cases, integrals of the form

J

IO(yv)Jvdyv ov

were computed. For this, the reader should see a survey by Igusa [26] or KimuraSato-Zhu [34]. However, in order to apply the filtering process, one has to consider integrals of the form

JGkx where x E Vky and 'Iv is a Schwartz-Bruhat function. This kind of integrals are not known for many cases. Also we have yet to answer to questions (2), (3).

§0.6 The outline of the general procedure Before we start the discussion on global zeta functions, it may help the reader to give an outline of the general procedure in this book. Let (G, V, Xv) be an irreducible prehomogeneous vector space of complete type. Roughly speaking, we will try to write the principal part of Z(4), s) in terms of special values of similar integrals for different prehomogeneous vector spaces. One can immediately notice that special values of zeta functions are homogeneous in the sense that Z((DA, s) = )-8Z(4), s). Therefore, let us consider a slightly more abstract but simpler situation. , 4 such Let fi ((D, s) be a meromorphic function on the entire plane for i = 1, that fi((D,\, s) = )ai8+6ifi((D, s) where ai, bi E C. Moreover, we assume the existence of an entire function gi(4), s) such that (0.6.1)

fi((D, s) = gi(4P, s) + f Asfi+i(-D a, ci)CdxA 1 0

for i = 1, 2, 3 where ci E C. Then by the homogeneity of fi(0, s), we get the principal part formula (0.6.2)

fi+1(D,ci)

fi(4)>s)=gi((D's)+ s + ai+1ci + bi+1

Introduction

21

provided that ci is not a pole of fi+i (4), s). If these formulas are true, by substituting successively, we get (0.6.3) f2(4), Cl) = 92(4), Cl) +

=92(4,,cl)+

f3(4), C2) cl + a3c2 + b3

93(b,C2) Cl + a3c2 + b3

+ (C1 + a3c2 + b3)(C2 + a4C3 + b4)

Life would be easier if we could prove (0.6.1) for i = 1 directly without using other cases. But unfortunately, what we can prove is a formula like (0.6.1) except f2(4, cl) is replaced by the right hand side of (0.6.3). Then only after proving that the right hand side of (0.6.3) is f2(4), cl) can we determine the principal part of f1(,P, s). For the sake of the filtering process, we only need the rightmost pole. However, this argument shows that we pretty much have to determine all the poles simultaneously in order to determine one of the poles. Let us be slightly more explicit. Our approach is based on the use of the Poisson summation formula as is often the case in number theory. Consider (0.3.6). Here, the issue is the term I°(4)). Now we face the following two questions. (1) Can we find a nice stratification of the singular set Vk \ (Vks U {0})? (2) If the answer to the question (1) is yes, can we separate the integral I°(0) according to this stratification? Kimura and Ozeki determined the orbit decompositions of prehomogeneous vector spaces over an arbitrary algebraically closed field of characteristic zero by representation theoretic methods (see [31]-[33] for example). However, they did not consider the rationality question or the inductive structure of orbits. Question (1) can now be answered by a branch of geometric invariant theory called equivariant Morse theory. In the early 1980s, Kempf [28], Kirwan [35], and Ness [48] established the notion of equivariant Morse stratification for algebraic situations. It was mainly intended for non-prehomogeneous representations, because their interest was moduli. However, if we restrict ourselves to prehomogeneous vector spaces, it gives us a systematic way of handling orbit decompositions of prehomogeneous vector spaces. For example, it is possible to write a computer program to find all the necessary data. Moreover, the rationality of the inductive structure of rational orbits is guaranteed by Kempf's theorem [28]. This is one of the reasons why we chose the formulation of geometric invariant theory in this book. We discuss invariant theoretic background in §3.2.

Consider question (2). Suppose that we have a `nice' stratification

Vk \ {0} = Vks JJ lSik i=1

The main analytic difficulty is that the theta series EXES:k 4(gx) is not integrable on G' = {g E GA I IXv(9)I = 1} in general. For GL(n) let GL(n)) = {g E GL(n)A I

Idetgl=1}. Let G = GL(n). In order to handle this difficulty for the space of binary cubic forms, Shintani [64] constructed a function '(g, w) on G°A x C with the following properties for the case n = 2.

Introduction

22

(1) If f (g°) is an integrable function on G°°/Gk, fGA/Gk f (g°)'(g°, w)dg° becomes A

a meromorphic function of w and the residue at a certain pole gives the original integral fGA/Gk f(go)dgo

(2) If f (g°) is a slowly increasing function, fGA/Gk f (g°)e(g°, w)dg° converges absolutely for Re(w) sufficiently large.

By the property (1), by considering this kind of integrals, we can recover the original integral. By the property (2), if we have a finite number of slowly increasing functions f, (g), , f, (g) on GA/Gk such that the sum f(g) = fi(g) + + fi(g) is integrable, t

fG/Gk f(g°)e(g°, w)dg° _

JG/Gk f(g°),9(g

w)dg°.

Using this formula, we can associate a certain distribution to each stratum Si of Vk \ (V2S U {0}).

We prove in Part I of this book that the smoothed version of Eisenstein series satisfies the above properties when the group is a product of GL(n)'s. For this reason, we have to restrict ourselves to such groups. However, there are many interesting cases where the group is a product of GL(n)'s, and we hope this is not such a heavy restriction. In a way, we are going to establish cancellations of divergent integrals indirectly in terms of the smoothed Eisenstein series. We will see this feature in Chapters 4, 10, 12, 13.

Now we start our discussion on global zeta functions, and we concentrate on the global theory for the rest of this book.

Part I The general theory Chapter 1 Preliminaries §1.1 An invariant measure on GL(n) In this section, we choose an invariant measure on G = GL(n). For the rest of this book, we assume that k is a number field unless otherwise stated. For any group G over k, let X*(G),X*(G) be the groups of rational characters and of one parameter subgroups (which we abbreviate to '1PS' from now on) respectively. For any split torus T = GL(1)h, T+ is the subset of TA which corresponds to R+ by the above identification.

Let G = GL(n) for the rest of this section. Let T C G be the set of diagonal matrices, and N C G the set of lower triangular matrices whose diagonal entries are

1. Let N° be the set of upper triangular matrices with diagonal entries 1. Then B = TN is a Borel subgroup of G. We use the notation (tl (1.1.1)

an(t1,...,tn)

t =

t1i...,tnEAx

= tn

nn(u) =

U = (ui.i)i>.i E A

2

for elements in TA, NA respectively. We also use the notation a(tl, , tn), n(u) when there is no confusion. Clearly, T+ = It = an(t1, , tn) I t1i , to > 0} for the above T.

Let K =

r1,,:cm

K,,, where K = O(n) if v E 931R, K = U(n) if v E 'JRc,

and Kv = GL(n, o,,) if v E Of. The group GA has the Iwasawa decomposition GA = KTANA. So any element g E GA can be written as g = k(g)t(g)n(u(g)), where k(g) E K, t(g) = an(tl(g), ... , tn(g)) E (Ax )n, and u(g) E A 1. Let t = X * (T) 0 R, t" = X- (T) ® R, and t = t* ® C etc. For s E *, we define is in the usual manner. We identify tt with C' so that tZ = It1Iz1 . . . Itnlzn for z = (z1, zn) E Cn, t = an(t1i ,tn) ETA. Also let tv = Itllvl Itnlvn for a place v E 931. The Weyl group of G is isomorphic to the group of permutations of n numbers {1, , n}. For two permutations Tl,T2 of n numbers {1, , n}, we define the product Trr2 by T1T2(i) = T2(TI(i)). For a permutation T, we consider a permutation matrix whose (i, T(i))-entry is 1 for all i, and denote it by T also. For a permutation r and z as above, we define Tz E Cn by rzi = z,-(j) . Let p be half the sum of the weights of N with respect to conjugations by elements

of T. Let du = lli>j duij. This is an invariant measure on NA. We choose an invariant measure dk on K so that fK dk = 1. Let dxt = dxtl . . . dxtn (t = , tn)). Let db = t-2Pdxtdu (t-2p = Hi
Part I The general theory

24

measurable function f (g) on GA,

f f (g)dg = GA

nfI 0

Adg°,

GA GO

where dx A = A-ldX. We define invariant measures on Gkv, K,,, Bk,,, Nk,,, Tk similarly, and denote them by dg,,, dk,,, db,,, due, dx t respectively. Then du = I Ok I -1' " n du,, and d't = (Ek n fl, dx t,,. Let (1.1.2)

0(s) =

Zk(s) , 0.(s) = 110(s + i - 1) for n > 2. Zk(s + 1) i=1

-

We define g = Res,-, 0(s). Let

01 --

Zk(2)... Zk(n)

cnk _1

The constant 3n is the volume of G°A/Gk with respect to the measure dg°. We define 931 = 1 for convenience.

Let Sl C BA be a compact set. For a constant g > 0, we define T., = fan (t1, ... tn) ETA I I titi+lI

'q},

and T,7+ = T+ n T,7. Let TO, = G°A n T,, and T°+ = T+ n TO. Then for suitable Q, ,q, E = KT,7+SZ surjects to GAIGk. Let 6° = 6 n G. Then 670 also surjects to GAIGk. We call E3 (resp. E,°) a Siegel domain for GAIGk (resp. G°A/Gk). There also exists a compact set Si C GA such that Ey C S2T,7+,13° C S2T°+. Note that since 12 is a compact set, Iti(g)ti+i(g)-11 is bounded below for all i for g E E5.

§1.2 Some adelic analysis For later purposes, we prove some elementary estimates of various theta series. Let m be a positive integer. For x = (xi, , x,n) E (koo)"", we define

xi[k: +

IIXII- = i

(EMR

k:Q

Ixil 2 VE9XC

The following lemma follows from the integral test.

Lemma (1.2.1) Let m be a positive integer, N > in, and v E I[8+. Let L C k. be a lattice. Then there exists a constant C(N, m, L) such that (1)

E (1 + VIIXIIm)-N < C(N, m, L) sup(1, v-''), x E L'"

(2)

: (1 + UIIXIIm.)-" < C(N, m, L) inf(v-"', v-N). xEL"\{0}

1 Preliminaries

25

If m < N' < N, then inf(v-m, v-N) < v-N' . Therefore, if a function f (v) is bounded by a constant multiple of functions of the form (1) for all N > m, then f (V) « v-N for all N > m. When we consider estimates on Siegel domains, a typical situation is that we have an action of elements which are products of diagonal elements and elements from a fixed compact set. Weil proved many useful statements in [77] to prove certain estimates in this kind of situation. The following lemma is Lemma 4 in [77, p. 193].

Lemma (1.2.2) For any sequence (at)tEN of real numbers at > 0, there exists a Schwartz-Bruhat function 4k E 9(R) such that nff(at(1 + IxD)-t) < 4)(x) IEN

for all x E R.

Proof. Let f (x) = infzEN(at(1 + Ixl)-t). We choose a function 0 < g E C°°(R) so that it is supported on [-1, 1] and fR g(x)dx = 1. Let h be the convolution h = f *g. Then f(x - 1) > h(x) > f(x + 1) for x > 1,

f(x-1) 0, so Ix'DPh(x) I is bounded for

all n, p > 0. Therefore, h c 9(R). Choose ho E 9(R) so that ho(x) > 0 for all x and ho(x) > ao on [-2,2]. Then f (x) < h(x - 1) + h(x + 1) + ho(x). Q.E.D.

The following lemma is a special case of Lemma 5 in [77, p. 194].

Proposition (1.2.3) Let C C GL(VA) be a compact set, and (D E Y(VA). Then there exists T > 0 such that 14) (gx) I < IF (x) for all g E C,x E VA. Let 4bf = ®VEm f 4v. Proof. We can assume that 4) has a product form D = Since C is a compact set, there exists a compact open subgroup L C Vf such that 4Pf (gx) = 0 for g E C, x f E V1 \ L. Therefore, there exists T f E 9(Vf) such that

I4'f(gx)I «f(x) for g E C, x f E Vf \ L. We consider infinite places. We identify V., = Rn for some n, and write xoo = x?. We choose at in (1.2.2) as follows. (xl, , xn). Let r(x,,) at =

sup

(1 + r(x.))tI4>°o(g°oxoo)I

XE VA,gEC

Since C is compact, {g-1 I g E C} is also a compact set. Therefore, at < oo for all 1. Then by (1.2.2), there exists a Schwartz-Bruhat function P. E 9(R) such that inf at(1 + r(x,,,))-t < l IEN

It is easy to see that I4oo(gooxoo)I < oo(r(xoo))

Part I The general theory

26

for x E VA,g E C. Let %P,,. (x) = lY,,, (r (x.)). Then T = JAW f satisfies the condition of our lemma. Q.E.D.

In particular, for any (D E .9(VA), there exists IF > 0 such that I(D(x)I < qf(x) for all xE VA. Now we can easily prove the following two lemmas using (1.2.2), (1.2.3).

Lemma (1.2.4) For any P E 9(R) and a positive integer n, there exists 0 < T E 9(R) such that'P(-x) = T (x), I-D(x)l n < WY(x), and W is decreasing on [0, 00).

Proof. We can assume that 4)(-x) = -D(x). In (1.2.2), we choose a, = supa, (1 + IxI)`I,D(x)I*. Then there exists W1 E 9(R) such that inf((1 + IxI)-` sup(1 + IyD)`I4>(y)I n) <- W1(x)

IEN

YER

for all x. But (D(x)I n < inf((1 + IxI)-` supp(1 + IyI)`ID(y)I ° EN

x

for all x E R. Since 'Y1 E 9(R),

`I`1(x) = - f

We choose2 E 9(

f-hI1i(y)dy.

so that I d i (x) I < W2 (x) for all x. We define

q'3(x) = f

yd W2(y)dy

for x > 1. Then W3(x) > W1(x) for x > 1 and T3 is decreasing on [1,00). Now we can extend `W3 to an even Schwartz-Bruhat function on R decreasing on [0, oo).

Then'Y3(x) > I4(x)In for x > 1 and x < -1. Since Ib(x)I* is bounded on [-1, 1], we can choose 'W3 so that 'Y3(x) _> I4>(x)In on [-1,1] also (for example, add a a-,2 constant multiple of the function ). Q.E.D.

Lemma (1.2.5) Suppose 4) E 9(A'). Then there exist Schwartz-Bruhat functions

'n > 0 such that I((xi,... xn)I <

4)1(x1)...-D,n(xn)-

It is easy to see that Proof. We can assume that 4 has a product form 4P = the finite part 4)f of 4) is bounded by a product of Schwartz-Bruhat functions on Af. So we consider infinite places. Since the argument is similar, we only consider real places. Since O(n, R) is a compact group, there exists 'P E 9(R) such that IFi > 0 and I'b(x1,...

forallx=(xi,,xn).

xn)I < T 1(

X12+

"' +X2

1 Preliminaries

27

We choose 'P2 E 9(R) so that T2 > 0, T2 is decreasing on [0, oo), and IF, (r) <

'I2(r)" for all r E R. Then

x1+...+xn)GW2( x1+...+xn)' G'F2(x1).W2(x,i). Q.E.D.

In the following three lemmas, we consider the following situation. Let Viji be a vector space of dimension mij, > 0 for i = 1, 2, ji = 1, , ki. Let V = ®72-1 V2jy. We write an element of V in the form (y, z), where y = (yl, , ykl) E Uiki) for etc. Let vil, , Viki E R+ for i = 1, 2. Let vi = (vi1, We define v2z i = 1,2. We define vly = (v_11y1,... vlkiyki) for y E similarly. Let v = (vl, v2). We define v(y, z) = (vly, v2z). We identify Viji = k'iii I and write yl = (y11, . . , ylmii) etc. Lemma (1.2.6) Let -P be a Schwartz-Bruhat function on VA. Then for any Nij > mij, <

(VI Y,

1<ji
1<j1
Sllp(1' V2j2 2j2 ).

11

V2z)I

1<j2
Biz E V2iz k

Proof. There exist a lattice L C k., and 0 < 4 E 9(V,,.) such that

E

Y', 1<j2
E

I'D (vly,v2z)I << E

1<jl
-D.(Vly, V24

1<jl
zj2EV2j2k

zj2 EV2j2L

where V1j1L,V2j2L are the subsets of points whose coordinates belong to L. Let M be a sufficiently large number. Since 4>. is rapidly decreasing,

E

'Doo(Vly, V2z)

1<ji
«E

L

(1 + Vii, Ilyjl IIm1j1 )

-M

1<ji
sj2 EV2i2 L

(1 + V2j2IIzj2IIm2;2) M .

X

1<j2
Then (1.2.6) is clear from (1.2.1) and the remark after that. Q.E.D.

Lemma (1.2.7) Let 4D E Y(VA). Let pj2 = p72 (v, y, z) E A be a function which does , k2. Then there exists a Schwartz-Bruhat not depend on z72, . . , zj2 for j2 = 1,

function 4)l > 0 such that

1:

1:

1<j2
s32 E V2j2k

1<ji
«

(D(Vly,t 21(zl

E 1<j1
(D1(Vly, v24

+pl),...

,

V2k2(zk2 +Pk2))I

Part I The general theory

28

Proof. We may assume that 4) > 0. We fix a non-degenerate bilinear form [ V2j2 defined over k for j2 = 1, , k2. By the Poisson summation formula,

E 'D(v1y,v21(zl

,

]j2 on

+pl),... ,v2k2(zk2 +pk2))

Zk2 EV2k2k

is equal to v2k2 2k2 times the following sum W(v1y,v21(z1 +p1), ... v2k2-l(Zk2-1 +pk2-1), U2k2zk2) < -zkzpk2 >, Zk2 E V2k2 k

where W(y, z) is the Fourier transform of -D with respect to zk2 and [ , ]k2. We can find a Schwartz-Bruhat function 'Y1 > 0 so that W1 > DTI. Then the above sum is bounded by a constant multiple of v2k22k2

l(vly,!21(zl+pl),...

v2k2-1(zkz-1+pkz-1)>v-1Zk2)

Zk2 EV2k2k

By applying the Poisson summation formula successively, there exists a Schwartz-

Bruhat function '2 > 0 such that

1:

1:

'P(v1y,

21(21 +

pl),...

v2k2(Zk2 + Pk2))I

1<jl
sj2 EV2j2k

<< v21 21 ... v2kz

E

2k2

'F2(vly,

v21z).

1<j1
Let 'Y3 be the Fourier transform oft with respect to

]k2. Then if

'1 >- 031, 4b1 satisfies the statement of (1.2.7). Q.E.D.

The following lemma follows from (1.2.6) and (1.2.7).

Lemma (1.2.8) Let 4P, y, z, v = (vl, v2), pj2, j2 = 1, for any Nlj, ? m1j. , j1 = 1, ... kl,

E

I"D(v1y,V21(z1+pl),...

1<j1
sj2 E V292 k

<< f v-N1j1 lil 1<jl
fi

1<j2
sup(1, v2jZ 2'2 ).

, k2 be as in (1.2.4). Then

,1!2k2(zk2 +pk2))I

Chapter 2 Eisenstein series on GL(n) This chapter and the next chapter are the technical heart of this book. In this chapter, we estimate the Eisenstein series on GL(n) for the Borel subgroup B. This

reduces to the estimate of Whittaker functions on GL(n). The estimate of the Whittaker functions consists of two problems. One is at finite places, and the other is at infinite places. We prove the explicit formula for the p-adic Whittaker functions in §2.3 following Shintani [66]. Even though some estimate of Whittaker functions are known, the uniformity of the estimate is a big issue in this book, because we intend to consider contour integrals. Therefore, it is necessary to prove an estimate which is of polynomial growth with respect to the parameter of the Lie algebra. For this purpose, we generalize Shintani's approach in [64], which is the naive use of integration by parts. As far as GL(3) is concerned, Bump's estimate [5] is the best possible estimate. Our estimate, even though it is enough for our purposes for the moment, may not be the optimum estimate, and we may have to prove a better estimate in the future to handle more prehomogeneous vector spaces.

§2.1 The Fourier expansion of automorphic forms on GL(n) In this section, we review the notion of the Fourier expansion of automorphic functions on GL(n) following Piatecki-Shapiro in [51]. We assume that G = GL(n) in this section. Let K C GA be the maximal compact subgroup which we defined in §1.1. We use the notation a E (A" )n-1, n(u) E NA, W«(n(u)) =< alU21 > ... < < an-1unn-1 >v for a E (k,, )n-1, n(u) E Nk,,. (n(u)) =< a1u21 >v

i,b«,v

If a E kn-1, « is a character of NA/Nk. Let f be an automorphic form on GA/Gk, and a = (al,

, an-1) E

kn-1

Definition (2.1.1) .f«(g) = .f«1i...

n.-. (9)

=

JNA/Nk

f (gn( u))(n(u))du.

Definition (2.1.2) For 1 < i < n - 1, we define

ri=S( A I02 )AEGL(i)k}l`\ I'°°= where In

A0 c

d

0 0

1(0

0

In-i

AEGL(i-1)k, cEki-1, dEk"

is the unit matrix of dimension n - i.

One difficulty of harmonic analysis on GL(n) is that the unipotent radical N of the Borel subgroup is not abelian. So we cannot directly use the the Fourier expansion of functions on affine spaces over A. However, in the case of GL(n), it

Part I The general theory

30

turns out that one can reduce the consideration to integrations with respect to a character of NA/Nk as the following proposition shows.

Proposition (2.1.3) Let f (g) be a continuous function on GA/Gk. Then

E

f(9)=

fa(9ryn-1"'1'1)

Eri/r°O if i#0

aEkn-1

7i=1 if i=O

if the right hand side converges absolutely and locally uniformly.

Proof. For un = (unl,

,

unn_1) E An-1, we define

1

n(1)(un) = unn-1

Let N1 C N be the subgroup which consists of elements of the form n(1)(un). We define a measure dun on N1A in the usual manner. For a(1) = (ant) ann_1) E '1-1

we define

faa) (9) =

f

f (9n(l)(un)) < anlunl + ... + ann-lunn-1 > dun.

NIA/N1k

Since N1 C N is an abelian subgroup, f has the following Fourier expansion with respect to N1:

f(9) _ E f«(1)(9) «(1)Ekn-1

If we just assume the continuity of f, this equality is true for almost all g, and the convergence may not be uniform. However, each Fourier coefficient ff(1) (g) is a continuous function again.

Let fl ann_1(g) = f... ,o,a n_,)(9) Consider the matrix ry =

Ol)

l

where

AE GL(n - 1)k. Then fl,a,,,,-1(9'Y) = f

f (91'n(1)(un)) C ann-lunn-1 > dun

NIA/N1k

= N1A/Nlk

(91'n(1)(un)1'

f

1)

C ann-lunn-1 > dun,

because f is a function on GA/Gk. An easy computation shows that n(1) (un) where u;, = unA-1. Therefore, unn-1 = aln-lunl + + an-ln-lurtn_1

yn(1)(un)ry-1 =

where t(ain-l, ''' ,an-in_1) is the (n - 1)-st column of A. This implies that f1,ann_1(97) =

f(aIn-1ann-1,"',an_In-1ann-1)(9) Hence,

f (9) _

fl,ann-1

'Yn-1 Ern-1/r°° 1 nn-1Ekxn

(91'n-1) + fl,o(9)

2 Eisenstein series on GL(n)

For un_1 = (un_11i

31

, un_ln_2) E An-2, we define

/

1

n(2)(un-1) =

1

un-ln-2

1

J

Let N2 C N be the subgroup which consists of elements of the form n(2)(un_1). We define a measure dun-1 on N2A in the usual manner. Let f2 -1401 -11,...,an-in-2)(9) =JN2A1N2k .fl,onn-i(9n(2)(un-1))

X < an-llun-11 +

' + an-ln-2un-1n-2 > dun-1

Since N2k C I'n' 1, fl,ann_1 (gun-1) = f1,ann_1 (g) for u._1 E kn-2. So the Fourier expansion of fl,ann_1(9) with respect to N2 is as follows.

E

fl,ann-1(9) _

fan.-1,(an-11,...,an-1n-2)(g)'

an-11, ,an-1n_2Ek

As in the previous step, we have f1,ann-1(9) =

E

f2,(an-1n-2,ann-1)(g Yn-2) + f2,(O,ann-1)(9),

'Yn-2Ern-2/r te 2

n-1n-2Ekx

where

The equality (2.1.3) for almost all g follows by continuing this process. Since we are assuming that the right hand side of (2.1.3) converges absolutely and locally uniformly, both sides of (2.1.3) are continuous functions. Therefore, the equality (2.1.3) is true for all g. Q.E.D.

We are going to apply this proposition to Eisenstein series on GL(n) in §§2.32.4. It turns out that the assumption in (2.1.3) is satisfied for Eisenstein series on GL(n). Therefore, we do not have a problem with the convergence of (2.1.3) for our purposes. Next, we consider the representative yn_1 ^yl more explicitly.

Definition (2.1.4) For a permutation T, we define

NT = N n 7N7--1, N, = N n TN-7--1.

Note that NT n N, = {1}. Definition (2.1.5) We call 1(r) = dimN/Nr the length of T.

Part I The general theory

32

For a permutation T, let

IT = {(i, j) 11 < j < i < n, T(i) < T(j)}. For a pair of numbers (i, j), we define T(i, j) = (T(i),T(j)). An element n(u) E N belongs to NT (resp. N,) if uij = 0 for (i, j) E IT (resp. (i, j) IT). Therefore, 1(T) = #IT. We use induction with respect to l(T) very often in Chapters 2 and 3. For that purpose, we have to know how l(T) behaves when T is a product of two elements.

Lemma (2.1.6) Suppose T = TlT2. Then (1) 1(T) < 1(Tl) + 1(T2) if T = 7172,

(2) 1(T) = 1(Tl) + 1(T2) if and only if IT = IT, Ij T1 1,2.

Proof. Suppose T = rl-2. Then 1(7-)< dimN/N n T1NT1 1 n TNT-1

= 1(Tl)+dimNT,/NT1

nrNT2T-

There is an imbedding of NT1/NT1 n rNT2T-1 into T1NTj 1/T1NT2T1 1. This proves that l (T) < l (T1) + 1(72) in general, and 1(r) = 1(Ti) + 1(7-2) if and only if NT C NT1 and NT1 /NT is isomorphic to 7-1NT1 1IT,NT2Tj 1. The first condition is equivalent

to the condition IT, C IT. If this is true, the second condition is equivalent to the condition IT = IT1 UTl 1IT2. But since l(r) < l(Tl)+l(T2), if this happens, the union has to be a disjoint union. Q.E.D.

Lemma (2.1.7) (1) For all T,N=N,NT. (2) Suppose that r = Ti-r2 and l(T) = I(Ti) + l(r2). Then N; = N,f, (TiNT2T1 1)

Proof. If 1(T) = 1, (1) is clear. Suppose that T = T1-r2, rl,T2 # 1, and l(T) _ l(Ti) + l(T2). Let n E N. By induction on l(r), we can assume that n = n1n2,

n2 = T1n3T1 1, and n3 = n4n5 where n1 E N,f,, n2 E NT1, n4 E NT2, and n5 E NT2.

Since IT = IT, l ri 1IT2, N,,,-r N,2Ti 1 C N,-. Therefore, nl(Tln4T1 1) E N,-. This implies that T1n5T1 1 E N. Since n5 E NT2 C T2NT2 1, T1n5T1 1 E 7172Nr 1T1 1 = TNT-1. So T1n57-1 1 E NT. If n E N,-, TlnsTj 1 E NT n NT = 1. Therefore, N,- = N,- (TiN,r2Ti 1). This proves the lemma. Q.E.D.

Remark (2.1.8) If r = T1T2T3 and l(T) = 1(Ti) + l(T2) + l(r3), then 1(7172) = 1(71) + l(7-2), 1(7-27-3) = l(T2) + 1(7-3),

using (1) of the above lemma. In (2.1.9), (2.1.10), we consider a subset I = {il,

,

ih} C {1,

, n -1},

where

Proposition (2.1.9) Let f (g) be as in (2.1.3). Then f (g) has the following expansion

f (g) =

1 T,

Ekn-1 «i#0 iE1

-0 i¢I

T ryEN.,k

mg-YT),

2 Eisenstein series on GL(n)

33

where r runs through all the permutations satisfying r-1(i) < T-1(j) if i < j, j

I,

provided the right hand side converges absolutely and locally uniformly.

Lemma (2.1.10) (1) Suppose that T,-.l fixes ij + 1,

, n, and does not change the order of ml < m2

for m2 # i.j. Then r = Th - Tl satisfies the condition in (2.1.9), and l(r) _ l(Th) + ... + l(Tl). (2) Conversely, any T satisfying the condition in (2.1.9) can be written uniquely in the above form.

Proof. We use induction on #I. Let I' = {il, ,ih_l}, and r' = Th_1

T1. Suppose that (i, j) E ITh. Then j = Th 1(ih). SO T(i,i) = T'(i - 1,ih). Since i - 1 < 2h, T'(i - 1) < T'(ih) = ih. Therefore, ITh CIT. Suppose that (i, j) E Th 1IT,. Then (Th (i), Th (j)) E IT,. So T'(Th(i)) < T'(Th(j))

< ih_1. Therefore, ITh n Th1IT' = 0, and r(i) < r(j). If i < j, since Th(i) > rh(j), (j,i) E ITh. This implies that i = Th1(ih). Therefore, (ih,Th(j)) E I,,. This is a contradiction. Hence, Th'IT, C IT, and 1(7-) = l(Th) + l(T').

Suppose that i < j,7-1(i) > T-1(j). Then (r-1(i),T-1(j)) E IT. This means that if (T-1(i),7--1(j)) E ITh, T-1(j) = Th 1(T'-1(j)) = Th1(ih). This implies that j = ih. If (7-1(i),T-1(j)) E Th 1IT', (rt-1(i),T'-1(j)) E I,,. Therefore, by induction, j E P. This proves (1). Let Th be a permutation such that (i) Th 1(i) = i for i > ih, (ii) if i < j < ih, Th 1(2) < Th 1(2), and (iii) Th l(ih) = T-1(ih). Let T' = Th 1T. Then T'(ih) = ih.

Suppose that ih > i > ih_1 and i > j. Then T-1(i) > --1(j). If Th(T-1(i)) < Th(T-1(j)), Ti-1(j) = Th(T-1(j)) = ih, so j = ih. This is a contradiction. Therefore, T'-1(i) > i. This implies that r'(i) = i for i > ih_1. By induction, this proves (2). Q.E.D.

Proof of (2.1.9). Take T = Th is a subgroup of I ". Consider the natural map

Tl as in (1) of (2.1.11). Let N,,,, = Nk n r,.

NijT,Nij /Nij -> Nij rI' /I'°°. We show that this map is bijective. Suppose that n(u) E Ni,, and n(u)Tjr _ TjI'°°. Then n(u) E Ni, n r I''°r 1. This means that uTi 1(ij) = 0 for i < ij. If i < i' < ij, TI-1(i) < T,-1(i') because of the condition on T,j. So u Therefore, Nii n TAI '°T 1 This means that n(u) E Ni, n

0.

= Ni, n

T,jxij TT-

The set Nii/Nij n rjNijr3 corresponds bijectively with the set Nf,k. Since Ni, \ rid /I'°° is represented by the collection of such Ti's we get (2.1.9) by (2.1.8), (2.1.10). Q.E.D.

Part I The general theory

34

§2.2 The constant terms of Eisenstein series on GL(n) In order to make this book accessible to non-experts, we include a discussion on the constant terms of Eisenstein series on G = GL(n) in this section. We start with the definition. We fix a place v E 931. For u E k,,, we consider the Iwasawa decomposition of the element 'n2(U)-

Definition (2.2.1) For u E k, we define

ey(a) _

(1 + lulu) 2, sv(u) = ucv(u) cv(u) _ (1 + IuIy) 2, sv(u) =

v E 9AR, V E 9RC,

cv(u)

v E Tt f.

t(u1 uEov}sv(u)={? u ov J 1

Let hv(u) be the matrix

\

Cs,(u)

(

Cs'()

U)

Cv(u)

-so(u)

ev(u)

ev(u)

for v E 93tR, TIIC, 91f respectively. Then it is easy to verify the following equality 1

CC,0(U)

tn2(u) = hv(u)

c,(O)-1) tc (u)sv(u)

0) 1

Definition (2.2.2) We define (1) av(uv) = Iev--(uv)Iv for uv E ky, (2) aoo(uoo) = HVE'-moo av(uv) for uoo = (uv) E koo,

(3) af(uf) = HVEn 1 a,, (uv) for of = (uv) E Af, (4) a(u) = fly av(uv) for u = (uv)0 E A. If u = (uv)v E A, we use the notation a,),, (u), af(u) for a. (u.), a f(u f) also. The function av(uv) has the following explicit description

(1+luyly) av(uv) _

2 luvIV)-1

(1 + sup(l, luvly)-1

VG9AR,

v E Mc, v E 1931f.

We now define the Eisenstein series for the Borel subgroup of G. For g E GA, let g = k(g)t(g)n(u(g)) be the Iwasawa decomposition as before. Consider the space + zn = 0}. We can consider p (half the sum of the , z,,,) E Cn I z1 + {z = (z1, positive weights) as an element of this space. Definition (2.2.3) For g° E G°A and z as above, we define

EB(9°, z) _

t(9°'Y)z+v 7EGk/Bk

Our first task is to prove the convergence of EB(g°, z). We do it by reducing the problem to the case n = 2. First, observe that Nk \ Gk/Bk is represented by Weyl

2 Eisenstein series on GL(n)

35

group elements. If T is a permutation, an element n E Nk fixes the coset TBk if and only if n E NTk. Therefore, the Eisenstein series has the following decomposition (2.2.4)

E

EB(g°, Z) =

t(g°'YT)Z+n

T -YENk/Nrk

We are going to estimate each sum in (2.2.4). For this purpose, we need a lemma.

Lemma (2.2.5) Let e > 0 be a constant. Then there exists a continuous function c(a) of o, E R such that for any t E A', u E A, E a(t(u + -y))a+1 < C(a) sup(1, Itl-1) -YEk

foro > 1+E. Proof. We choose compact subsets U, C A', U2 C A so that U1, U2 surjects to A'/k', A/k respectively. Since we are trying to prove an estimate which is uniform

with respect to u, we can assume that t = At' and t' E U1, u E U2. We write y = ab-1 where a E ok, b E ok \ {0}. The inequality a f(At'(u + y)) < 1 is always satisfied. Therefore, a(J t'(u + -y))a+1 < a. (.At'(u + 7))°+1

and this is bounded by a constant multiple of

Ib'f+1 fJ (IbIv+lat'(u+a)Iv)

2

1I

vE flc

vE9)IR

2

(k(a + 1) 1I (IbIv + IAt'(u + a)I v)

1I (IbIv + LAt'(u + vEmc

vE9AR

Since U, is compact, there exist constants 0 < C, < C2 such that C1 < It'Iv < C2 for t' E U1. Also It'Iv = 1 for almost all v and for all t' E U1. Therefore, the above function is bounded by a constant multiple of

2 11 (IbIv + IA(u +

(k (o, + 1) II (IbIv + I A(u + a)I v)

V EM.

a)Iv)-(a+1).

v E 9Rc

Now (2.2.5) follows by the integral test. Q.E.D.

Let tETA,yEN,rk,and n(u)ENA. Lemma (2.2.6) There exists a constant M > 0 such that the sum t(tn(u)yt-1T)z -YENrk

converges absolutely and locally uniformly with respect to t, z, uniformly with respect to u, if for any pair (i, j) E IT, Re(zT(ti) - zr(j)) > M. Moreover this sum is bounded by a finite sum of functions of the form c(z)

JJ (i,A)EI,-

It.t71I`0,

Part I The general theory

36

where cij is a constant independent of u, t, z, and c(z) is a function of polynomial growth. (So n(u) can depend on t.)

Proof. We can assume that z is real. We use induction on l(r). Let r = Ti-r2, 1(T) = 1(n1) + 1(T2), and l(Tl) = 1. We can write 'y = 'ylTl'72T1 1, where ryl E NTik,'y2 E NTk. Let tn(u)'yltl 1Ti = kanl, where

k E K, a = a(t, u,'yl) ETA, nl = nl(t, u,'yl) E NA. For t E TA, let tT = 7--1tr. Then tn(u)'Y171'Y2T1 It-1T = tn(u)'ylt-1Tltn1'y2T1 it-1T

=

kanltT1'Y2T2(t-1)T = katTln2'Y2T2(t-1)T

for some n2 = n2(t,u,7i) E NA. This is equal to katnln2'y2a-1(t-1)T1T2aT2(tT1)T2(t-1)T = katTln2'y2a-1(t-1)T1T2aT2. So t(atT1n2'y2a-1(t-')TI

t(tn(u)'yt-1T)Z =

72)ZaT2Z.

Suppose that for any (i,j) E IT, ZT(i) - ZT(j) >> 0. Since I T = IT, l ni 1IT2, if (i, j) E IT2, then 7-1 1(i, j) E IT. Since T(Tl 1(i)) = T2(i), the condition on z is satisfied for r2. So by induction,

1:

t(atTln2'y2a-1(t-1)T17*2)zaT2Z

72ENT2k

is bounded by a finite sum of functions of the form cl(z)ar2=

II

laiaj 1t,1 1(i) t-11 T1

(7)

ICij

(i,j) E 42

where cl(z) is a function of polynomial growth and c2j is a constant for all (i, j) E 1,2- If (i, j) E IT2, (Tl 1(i),Ti 1(j)) E IT. So we can ignore I tT1 '1-' (i)

Ti

U)

-2(i)<-2(j)

By definition, (72z)T(i) = Zr1(i). So the condition on T2z is satisfied for Tl. Note that cij does not depend on z. So by induction, laZjaj -il.j

E a72Z 71ENT1k

i>j

T2(i)<,-2(j)

is bounded by a finite sum of functions of the form c2(z)

I i>j

*1(i) <*1(j )

Itit,1l 'j,

2 Eisenstein series on GL(n)

37

where C2(z) is a function of polynomial growth. Since 1, C IT, we have reduced the problem to the case 1(T) = 1. If 1(T) = 1, NT is a normal subgroup of N. So n(u)y = n(ul)-yn(u2(Y)), where n(ul) E N,-A, n(u2(y)) E N,-A. Therefore, t(tn(u)yt-1T) = t(tn(ul)yt-1T),

and we have reduced the problem to the case n = 2. But the case n = 2 is (2.2.5). Q.E.D.

Now the convergence of the Eisenstein series is an easy consequence of (2.2.6).

Lemma (2.2.7) The sum EB(g°, z) converges absolutely and locally uniformly if Re(zi - zi+1) >> 0 for all i. Proof. We modify each term of (2.2.4) as follows.

t(goyr)z+P =

t(t(g°)n(u(g°))yr)z+P

^yE Nk /Nrk

-rE Nk I Nr k

E t(t(g°)n(u(g°))'Yt(g°)-lt(g°)T)z+P -rENk/Nrk

E

t(go)Tz+TPt(t(g°)n(u(g°))yt(g°)-l,r)z+P

yE Nk /N, k

Then the convergence of this sum follows from (2.2.6). Q.E.D.

It turns out that EB(g°, z) is holomorphic if Re(zi - zi+1) > 1 for all i. But since we do not need such a convergence, we do not try to prove the convergence

EB(g, z) on the above domain. The Fourier expansion of the Eisenstein series EB(9°, z) and subsequent estimates are the issue of this chapter. For the rest of this section, we consider the constant terms of EB(g, z).

Definition (2.2.8) For z E C, we define MM(z) = f I cv(uv)I v+lduv. v

It is easy to verify that this integral converges absolutely for Re(z) > 0. We evaluate this integral explicitly here.

Proposition (2.2.9) (1) If v E 9fR, Mv(z) =

7r

r(z) 2

(2) If v E 9Jtc, Mv(z) =

(27r)zr(z) (27 )z+lr(z + 1) .

1 - q, (z+l)

(3) If v E 9Jtf, Mv(z) =

1-qvz

Part I The general theory

38

Proof. Let v E fits. Then

r() M(z) = f

J e-tt 2 (1+u)-Pd"tdu

f

=

+ oz

e-t(1+-2)t-2 dxtdu =

JR+

e-tt2d"t

(2) This proves (1).

The case v E Rc is similar to the real case and is left to the reader. Again, notice that Ixly is the square of the usual absolute value.

Let v E Of. Then

M(z) = f du, + i=1

ov

f

o

qv i(z+1)duv

00

=1+

- Qv

i=1 1 - q,,, (z+1) 1

(1-Qvl)Qvz -z 1

-z - Qv

Q.E.D.

We define M(z) = f a(u)z+ldu.

(2.2.10)

W

Then by the choice of the measure on A, M(z) _ jAkI 2 Hv Mv(z) where Ak is the discriminant of k. Therefore, M(z) = Zk(z+1) _ q5(z) (see (1.1.2)). Since we know that the functions 7 2r 7r_

2

(2

)

(21r)zr(z)

r(1)'2 (27r)z+lr(z + 1)'

1

Ck(z),

(k(z + 1)

are at most of polynomial growth on any vertical strip contained in {z I Re(z) > 0} when Im(z) >> 0, the function M(z) is at most of polynomial growth on such a vertical strip when lm(z) >> 0. We fix a Haar measure du on N,fA so that the volume of NrAINTk is 1. For a permutation T, we define (2.2.11)

MT(z) = fN,A

Lemma (2.2.12) If t E TA° and r is a Weyl group element, f t(tn(u)T)z+Pdu = tTZ+PMT(z). NrA

2 Eisenstein series on GL(n) Proof.

Let t = an(tl,

39

, tn). Then

t(tn(u)T)z+Pdu = JNr_n l t(tn(u)t-1TT-1tT)z+Pdu l 1 l l / NrA

= tTz+Tp

t(tn(u)t 1T)z+Pdu N,A

(tit, l1-1

tTz+Tp

/

t(n(u)T)z+Pdu.

JN;A l

t>ii

*(o<,-(i)

We simplify the coefficient of the integral as follows. 2

Ittt7ll-1

tTP i>j

rIItT 1(i)tr11(i)I

ItZti

1I-1

i>i

i

°z

H

1I

Itit,1l-1

I tit? r(i)
1

2

i<j

I titA

126

H>i

I

I tit .7 l

By exchanging i, j in the second factor, this is equal to V. This proves the lemma. Q.E.D.

Proposition (2.2.13)

H

MT(z) =

M(zT(i) - Z,

(j)).

+>i

Proof. By induction on the length l(T). We have already considered the case l(T) =

1. Suppose that r = T1T2, T1, T2 # 1, and l(T) = l(n) + l(T2). This implies that NTA = N A(T1NT1Anj 1). Therefore, M,(Z) _ J

t(n(ul)(Tln(u2)Tl 1)T)z+Pduldu2

NT1AXNrlA

_ f

N

=

f

t(n(ul)Tln(u2)T2)z+Pduldu2

t(t(n(ul)T1)n(u2)T2)z+Pduldu2

t(n(ui)Tl)T2z+Pdu1 J

Nrl A

= MTl (72z)MT2 (z),

t(n(u2)T2)z+Pdu2

NT2 A

Q.E.D.

Part I The general theory

40

By (2.2.4) and (2.2.13), JEB(g°n(u) z)du =

r

jN®INrk

t(g°n(u)r):+Pdu

= E J Nw/NrAt(t(g°)n(u)r)z+Pdu r

Ef

NfA t(t(g°)n(u)r)Z+Pdu _ _ EMr(z)t(g°)"+P

r Therefore, we have the following proposition.

Proposition (2.2.14)

f

NA/Nk

EB(g°n(u), z)du = , M7(z)t(g°)TZ+P r

§2.3 The Whittaker functions We estimate the Whittaker functions on G = GL(n) in this section. They appear as the Fourier coefficients of Eisenstein series, which we will consider in §2.4.

Part of our computation is similar to Stade [71]. However, our computation is based on a naive use of integration by parts. Let rG be the longest element among . Let K = fv K the permutations of {1, , n}, i.e. rG = (1, n)(2, n - 1) be the maximal compact subgroup of GA which we defined in §1.1. Consider z = (zl, , zn) E Cn such that zl + + zn = 0.

Definition (2.3.1) Let v be a place of k. We define for a E (kv )n-1,

(1)

Wn,v(av,z) _

(2)

Wn,oo(a., z) _ JJ Wn,v(av, z) for aoo = (av)vEfito, E (A' )n-1,

N,G k, vE9Ji,

(3)

Wn,f(af, z) = rl Wn,v(av, z) for a f = (av)vEot1 E (Af )n-1, VE.mj

(4)

11 Wn,v(av, z) for a = (av)v E

Wn(a, z)

(A>)1.

v EO1t

If a = (av)v E A", we also use the notation Wn,o,,(a,z) etc. We fix v E'93t. For the rest of this section, we drop the index v from cv(u), sv(u) etc. if there is no confusion. 1,

We first prove an inductive formula for Wn,v(a, z). Let zi = zi + n"l for i = Clearly, zi - zj = zi - zj for all i, j < n, and , n and z = (zl,

z1 +...+2n-1=0.

2 Eisenstein series on GL(n)

41

Proposition (2.3.2) Let v E One. Then Wn,v(a, Z)

IC(u21)Ivn...

kv -1

x Wn-l,v(a2C(u21)C(u31)-1, ... ,

an-lC(un-ll)C(unl)-1,

x < a1u21 + a2s(u21)u3l + ... + an-ls(un-ll)unl >v du21 ... dun1, Proof. Let Tl = (1, 2), T2 = (2, 3), ... , Fn-1 = (n - 1, n), vn_l = TGL(n_l).

Then

TG = Fl ...Tn-lvn-1,

l(TG) = l(Fl) + ... + l(Tn_i) + 1(vn_1). Let Ail(u), Ai2(u), Ai3(u), Ai4(u) be the following matrices C(U)

1

u

c(u)

S(U)

1

In-i Ii_2 c(u)

1

c(u) s(u)

c(u)-1

1

In-i respectively. Let pi = Ti

Tn_lvn_l. Also let

1

U32

ui-12 Bi(u) =

' ' '

ui2

... ...

un2

...

0

ui-li-2 1 ... 0 ... nii-1

...

uni-1

1

nil

un1

uni

...

unn-1

1

...

unn-1

1

Then 1

u32

ui-12 Ail(uil)

... ui- li-2

0 ui2

...

...

1

0

1

uii-1

0

1

ui+li k un2

...

...

uni-1

uni

uni

= Ail(uil)Ti-Bi+l(u)/.ti = hAi2(uii)Bi+i(u)/zi = hAi3(uil)Ai4(uil)Bi+i(u)pi = hAi3(ui1)Bi+1(u)/zin(u ),

Part I The general theory

42

where h E K,,, n(u') E NA, and n(u) is obtained by replacing uji by

uji - c(uil)S(uil)ujl for j = i + 1,

, n.

Let

Oi(a u) =< a2C(u21)C(u31)-1u32 + ... + ai-2C(ui-21)C(ui-11)-lui-li-2 + ai-1C(ui-11)uii-1 + a1u21 + a2s(u21u31) + .'. + ai-ls(ui-11)uil >v

Then by the above relation, IC(u21)

Ivn-i -z,.+l ... IC(ui-11) IZUn-i+2-Z,.+lt(B.

x

Jk-I

(u)µi-1)Z+P

Wi(a,u)du21...dun1

)IZry-i-Zn+1

IC(u21

.. C( uil I

X 11i+1(a, u)du21

)Iz"-i+l-Zn+1 t Bi+1 (

(u

. dun1

for all i. Now (2.3.2) follows by induction. Q.E.D.

We first consider finite places. The explicit formula for p-adic Whittaker functions was first proved for GL(n) by Shintani in [66], and was generalized to arbitrary quasi-split groups by Casselman-Shalika in [71 . We only consider GL(n), for which we use an explicit computation to compute the p-adic Whittaker functions.

Let v E 9Jtf. Let a =

be the difference idele. Then the character x a character of index zero on k,,, so we assume that < >, is of index zero to start with.

Lemma (2.3.3) (1) u > du = 0 if i < 0.

if i<-1.

(2)fr;1ov

Proof. Since the character < > is non-trivial on the compact group 7rvo, , (1) follows. Since 7r,' ,ox, = 7r,,o \ 7rv+lo,,, (2) follows from (1). Q.E.D.

We first consider the case n = 2.

Proposition (2.3.4) Let v E Wl f, and a E V. Then (1) if 01 o,,,W2,,,(a,z)=0,

-(z+1) (2) if a E 1r;,o;, for i _> 0, W2, (a , z) = 1 - qv z (1 _ qv (i+1)z) 1-qv_

Proof. Since lc(u)I depends only on IuI,,, we assume that a = 7rv for i E Z. Suppose

i < 0. Then r

(2.3.5)

W,,,, (a, z) =

J

< 7rvu > du +

00

Ic(u)Iv+l < lrvu > du.

2 Eisenstein series on GL(n)

43

Both terms are zero by (2.3.3). So (1) follows.

Suppose i > 0. The equation (2.3.5) is still valid, and the first term is 1 this time. If 1 > i + 2, Ic(u)Iv+1 < 7r'vu >v du = 0

for a similar reason as above. Therefore, q,-')qv-'z + qv (Z+1)(=+1) k-(i+0,,X < 7r;,u > v du.

Wn,,,(a, z) = 1 + By (2.3.3),

k-(Z+1)'v

vdu =0.

This implies that the last term is equal to -qv (i+1)(z+1)vol(irv iov) = -qv (i+1)z-1 So i

Wn,v(a, z)

= 1 + E(1 - qv 1)qv tz -

qv(i+1)z-1

1=1

Now it is easy to verify that this is the right hand side of (2). Q.E.D.

We continue to assume that < >v is of index zero. We choose t = an(t1i ... , t,) E

Tk so that a1 = t1t2 1

(2.3.6)

an-1 = t i_1tn 1

Let Cn(t, z) be the n x n matrix whose (i, j)-entry is Iirn-jtjlvn-i+1 i.e. l7rv 1t11,

I7r v 2t2lvn

...

ItnlVz

Cn(t, z) _ i7rv

t11v1

l7rv-2t21v'

Itnlvl

We define (zs-z,,+1)

(2.3.7)

An,v(t, z) = t-rczqv

det Cn(t, z)7fi1 - qv -(zi-zj ) i<j 1 - qv

Note that this function depends only on a. Also .n,v(t, z) = 0 if ai E irv 1o,' for some i. The following theorem is due to Shintani [66], and Casselman-Shalika [7].

Theorem (2.3.8) (1) If ai ov for some i, Wn,v(a, z) = 0. (2) If ai E ov for all i, Wn,v(a, z) _An,v(a, Z). Proof. We use induction on n. We prove (1) first.

Part I The general theory

44

Lemma (2.3.9) Wn,v(a, z) = 0 if ai 0 ov for some i. Proof. By induction on n. Suppose ai 0 o,,. If uil E ov, aic(ui1)c(ui+ii)-1 0 ov. Therefore, by induction, (2.3.2) is equal to the integral over the set {(u21,

, unl) E

ov}. This implies that s(uit) = 1. By definition,

kv-1 I uii

I

Ic(ui+ii)Iv"-i-z

+1 < aiui+l + ai+1s(ui+11)ui+21 >

X Wn-1,v(a2C(u21)e(u31)-1, ... an-1c(un-11)C(un1)-1, z)dui+11

o

< aiu >v du x W.-,,,(.. , aic(uil), ai+1c(ui+11)-1, ...

00

+ T -i(=n-i-zn+1) Qv Jf < az7rv u >v du < ax+lux+21 >v °

i=1

X Wn_1 v(... c c(21i1)7rv 3, 7rvai+1C(ui+21)-1, ...

Z),

Since ai V ov, all these terms are zero by (2.3.3). Q.E.D.

Lemma (2.3.10) Let ai E ov for all i. Then in (2.3.2),we can replace s(uji) by 1 for all i without changing the value of the integral. Proof. By induction on i. Suppose that we can replace s(uit) by 1 for all j < i without changing the value of (2.3.2). Then W., (a, Z)

IC(u21)Ivn-i-z

Ik_1

+1

... IC(unl)Ivl-z

+1

X Wn-1,v(a2C(u21)C(u31)-1,...,an-1C(un-11)C(un1)-1, Z)

X < a1u21 + ... + ai-1ui1 + ais(uil)ui+11 + ... >v du21 ... dung. ov, s(uit) = 1. So we compare the integral over the set {(u21,

,

unl) I

uil E ov}. If aiui+11 E ov, < aiui+ll >v=< ais(uii)ui+ii >v= 1. If aiui+11

ox,,

If ui1

ov, because Ic(uii)Iv = 1. Therefore,

aic(uil)c(ui+11)-1

Wn-l,v(a2c(u21)C(u31)-1,... ,an-iC(un-11)C(unl)-1,N/ = 0. Thus, we can replace s(uit) by 1. Q.E.D.

We continue the proof of (2.3.8). Let (tl(u),...,tn-1(u))

t(u) =

=

(t2c(u21),...,tnc(un1))

Then tl(u)t2(u)-1 = a2C(u21)C(u31)-1, ...,tn-2(u)tn-l(u)-1 = an-1C(un-11)C(u'n1)-1

Lemma (2.3.11) Let ai E o, for all i. Then Wn,v(a, Z) = j

IC(u21)Ivry-i-Z,.+1 ...

An-1,v(t(u), Z)

v-1

X < a1u21 + x22131 + ... + an-lung >v du21 ... du,nl.

2 Eisenstein series on GL(n)

45

Proof. We can assume that al = irv1, , an_1 = 7rva-1. Let D71, x 7rvn 1 ov . We compare integrals over the set Dj..... .jn-1 If aic(ui1)c(ui+11)-1 E ov for i = 2, , n - 1,

,7n-1 = 7ri1o v v x

Wn_1,v(a2C(u21)c(u31)-1,... ,an-1C(un-11)C(un1)-1,Z) _.n-l,v(t(u), ) on D1,... j _1. Suppose that aic(ui1)c(ui+11)-1 E 7r;, where 1 < 0 for some i. Then Wn-1,v(a2C(u21)C(u31)-1, ... , an-1C(un-11)C(un1)-1, ) = 0.

Since c(ull) is always integral, hl + ji+i < 1. If l < -1, Ic(u21)Ivn-l-zn+1

...

IC(unl)Iv1-zn+1'n_1,v(t(u), Z)

X < a1u21 + a2u31 + ... + an-lunl >v du21 ... dunl = 0

by applying (2.3.3) to ui+11. If 1 = -1, and ui1 ov, hl + ji+l < -1, so again, the above integral is zero. If 1 = -1, and uil E ov, h1+ji+1 = -1, and )n_1,v(t(u), z) _ 0 by the remark after (2.3.7). Q.E.D.

Now we can complete the proof of (2.3.8). By definition, (Ixv-7-1tj+lc(uj+11)Ivn

Cn-1(t(u),z) =

2t2c(u21)Ivn

17r

...

i)i,j,

ItnC(unl)Ivn

Cn_1(t(u), z) _ Irv-2t2C(2121)Ivl

...

ItnC(unl)I v'

Let vn_1 = TGL(n_1). An easy computation shows that IC(u21)Ivn-zn+1

n-1

11 -zn-i

... IC(unl)Iv1-zn+lt(u)-In-1z

n-1

tj+1

I C(u7+11)I v

zn-j+zn-j -zn+1

j=1

j=1

Therefore,

n-1

_yn_j

HC

j=1

n-1

IC(26j+11)Iv Zn-,i+zn-i-=n+l det Cn_1(t(u), N)

j=1

n-1 = det D(1)(t, u, z) H tj_+i ' j=1

where D(1) (t, u, z) is the following matrix

n-2 zn-1

Ilrv

t21V

I7fvn_2

zn-l-zn+l

IC(u21)Iv

t2Iz1v1C(2621)IZ1v-=n+1

...

Itn 11-1 I

...

ItnI

C(unl)Ivn-l-zn+l

,1IC(unl)Iv1-zn+l

Part I The general theory

46

It is easy to see that Jkv-1

det D(1) (t, u, z) < aiu2i + a2u3i +

-

n-1 1

= det D(2) (t, z)

+ an-lung >v du21

duns

-(zi-zn+1) qv

1-qv

i=1

where D(2)(t,z) is the following matrix Ivn-1

(1 (1 -

I.7rvtlt21I z1-zn )I7r n-2 t2 1vz1 v

(1- I7fvtn-1tnlly-1-zn)Itnlvn-1

2

V

...

v

(1 - I7rvtn-1tn 11 v1-z, ) Itn Iv1

1

Let vj be an (n - 1)-dimensional column vector whose i-entry is I7fv-itj Izn Let dj = 17rvtjt { 1I - Then, D(2)(t, z) = (v2 - d1v1, v3 - d2v2, ... vn - dn-lvn-1). Therefore, det D(2) (t, z) is equal to

, vn) - dl det(vl, v3,

det(v2,

, vn)

+ ... + (_1)n- 1d1 ... do-1 det(v1i ... , vn-1) = det

and

d1

( 1

Vi

V2

d1...dn-1

dld2 V3

vn

...

dl...d; = It11

/I

t,+1lv

Note that (n - 1)zn = nzn. It is easy to see that

tl '^, t2 zn-1... t,nz1 =t-Tcz(tl...tn)-nry1 Therefore, Wn,v(a, z) is equal to

t-TCZ(tl ... tn)

I1rv1 nzngv

z Ei
H i<)

where D(3)(t, z) is the following matrix I7rv-1t11vn

...

Itnii

vn

Itnlvl

By an easy computation, det D(3) (t, z) _ ( t 1.tn) n "1 I1v Iv zn det Cn(t, z).

1 - qv (z1-zi+l)

-(zi-z 1 - q,

2 Eisenstein series on GL(n) Since

-2 L (z'-z)+ 2z

47

-2 i<j
i<j
(z'-z4,

this proves (2.3.8). Q.E.D.

We have been using the assumption that the character < >v is of index zero. In general, if a = (av)v is the difference idele, < a, - >v is a character of index zero. So

for a finite place v, we define a function Qn,v(a, Z) of a E (k' )'-l and z as above by the formula Wn,v(ava, z) = O'n,v(a, Z) [J(1 - q,; (Z'-Zj+i)).

(2.3.12)

i<j

Definition (2.3.13) For a = (av)v E A', we define

on(a, z) = I Qn,v(a,,, z). vESRf

By definition, Wn,f (aa, z) =

1) . lli<j a (a z)j +

, an-1) E (kv )n-1 and aj E ,),' for all j. Then on (a, z) satisfies the following inductive relation

Lemma (2.3.14) Let a = (a1i n-1 Aj o'n,v(a, z) =

qv

C1(Zn-1-Zn)

... q, Cn-1(ZI-Zn) Y

j=1 cj=0 X

Cn-2-Cn-1

C1-C2

Proof. We choose t as in (2.3.6). Let

c = (cl... , cn-1), and t(C) _ (ivlt2 ... , jrCn-1tn). By (2.3.8), the right hand side is equal to n-1 Aj

q,)Cl(Zn-1-Zn)...(I,Cn-1(Zl-zn) t(c)

n_ zjir 1vz Ei<j
7li<j
j=1 cf-0

x detCn_1(t(c),-z), where vn_1 = TGL(n-1) Let (2.3.15)

n-1 ).j

B(t,z)

qvC1(Zn-1-Z.)...gvC,.-l(Zl-Zn)t(C)-vn-l z det Cn_ 1(t(c), -z) i=1 cj=o

Part I The general theory

48

The (i, j)-entry of Cn_1(t(c), z) is zn-i I-9-1 vcjti+ll z,.-i = qv-ciz,.-i n-9-1 ti+1l

xv'n

Also t(c)-Vn-'zqv C1(zn-1-zn) ... qv Cn-1(z1-zn) = t2 z^ ' ... tnzl(r(C1+...+Cn-1)zn

Let

qD(-ci(zn-i-zn)

t, z, c)

Then

n-1 Aj

B(t, z) _

n-j-1 ti+1lvzn ,l/

t2 =n-'

1
to z' det D(t, z, c).

j=1 c;-0

The order of the summation and the determinant can be changed. Since

aj

c,_ov cgi(zn-i-zn) -

1-

j+1)(zi-zn)

q'v (

1 - qv -(z'-z,)

n-1 B(t z) = t2 zn-' ... to 21 det D(2)(t, z)

-(zi-zn)' i-1 1 - qv

where D(2)(t, z) is as in the proof of (2.3.8). The rest of the proof is similar to the proof of (2.3.8). Q.E.D.

Definition (2.3.16) For r = (rl,

,

r,_1) E Rn-1, we define D(n, r) to be the

following set

x=

(xl,...

x,,) E Rn I x 1 +... +xn = 0, x i - xi+1 > ri for a = 1, ... n-1}.

Lemma (2.3.17) We fix r. Suppose a = (a1i there exist constants N1i

, an-1) E (ov \ {0})n-1. Then , Nn_1 < 0 independent of v such that

I0-n,v(a,z)I < IaiIN' ...

I(kn-1lNn-1

for Re(z) E D(n, r). Proof. We use induction on n. Let c1, , cn_1 be as in the proof of (2.3.14). There exist M1, , Mn_2 < 0 such that IQn-1,v(a27rvc1-c2 I

a 2 v 1f'1-czIvM1 ,

,

.

. .

, an-1 Vcn-2-cn-1 zl> ... zn_1

Mn-2 Ia n-1cn-z-Cn-1 v Iv

cn-2Iv 7rv But IalIv < I7r"I v v Ian-2 Iv < Ix v cl-C2Iv Mo = supi{Re(zi - zn), 1}. Then I

I7r V

cn-2-c,.-1Iv We define

<_ I Irv

C1(zn-l-zn) ... Ir Cn-1(z1-zn) Iv < Ia1... an-1Iv Mo V

49

2 Eisenstein series on GL(n)

Also the number of terms in (2.3.14) is bounded by lal

an_lly 1. Hence,

IQn v(a, z)I C lal ... an-llv Mo-llala2lM1 ... Ian-2am.-1IMR-2 Q.E.D. For infinite places, some integral formulas are known, for example [71]. However, we have a problem of uniformity. We illustrate the difficulty of infinite places by the example of GL(2). For details on special functions, the reader should see the classic book of Whittaker-Watson [82] (also see [18, pp. 50, 55]) Let be the K-Bessel function, i.e. (2.3.18)

2

forvEC,xE1l

f e-z(t+i)t"d
+

.

Let a = (a,) be the difference idele, and we assume that a = 1 for v E 9J1Q.

Proposition (2.3.19) (1)

W2,v(a, z)

-2lal"

7r

(2)

2 I(

2

)

K_(21rjaj,) for v E 9JII, 2

W2,,, (a, z) = (Z7r)-(2+a)r(z

+ 1) Kz

ja1,) for v E 9c.

(4-7r

Proof. Since the proof is similar, we only consider real places. Let v E 931n. We use the usual trick of multiplying the Gamma function as follows.

r (f---)

z)

=

_

f

tt e

2

1 + u2

d"tdu

Le_t(1+u2)+20tPtdu x2t_2

= vri fR+ e-t =7 2

t2dxt

e-I I«I(t+i)t2dxt.

laI2

fR+

Q.E.D.

This formula looks superficially nice, and we seem to have a good estimate concerning the growth with respect to a. However, we now have a Gamma function in the denominator. Because of Stirling's formula, F('+')-' is a function of exponential growth with respect to Im(z). Since we intend to consider contour integrals along vertical lines, this is very inconvenient. So this formula is not directly applicable for our purposes. On the other hand, in the cases n = 2, 3, it is possible to compute the Mellin transform of Wn,v(a,z) and actually get a polynomial growth estimate (see Bump [5] for example). But in order to generalize this method, one may have to prove an estimate similar to Stirling's formula for general hypergeometric functions, which is an open problem.

Part I The general theory

50

The asymptotic expansion of the K-Bessel function is known to be as follows. K,,(x)

=

- \ 2x)

(4v2 - 1)(4u2 -

2 e_x

r(

I 1 + L

(2r - 1)2)

r! 23rXr r=1

1

for x >> 0. However, we are going to prove that Wn,,,(a, z) has an estimate of polynomial growth with respect to Im(z). Since KZ (21r)aly) = 2ir

for v E MR,

2

Iakv 2 r(z

2

1)W2,,,(a, z)

has an estimate of exponential decay with respect to Im(v).

Therefore, either the convergence of the above asymptotic expansion is not uniform with respect to Im(v) or the error term is significant as Im(v) -i oo. So for the sake of a uniform estimate, we cannot use the asymptotic expansion either. Shintani avoided these difficulties in [64], and used the naive integration by parts.

This is the idea of our approach to handle Whittaker functions at infinite places. Since our idea is based on the use of integration by parts, it may be interesting to interpret our computation from the viewpoint of differential operators, for which we do not know if it is possible to do so. In (2.3.20)-(2.3.22), v E 9R..

In order to estimate Wn,,,(ca, z), it is easier to consider a slightly more general function. When v E 9A R (resp. v E 9Z) and a = 1, we denote W,,,, (a, z) by WW,R(a, z) (resp. Wn,c(a, z)).

Definition (2.3.20) Let a = (al,

, an_1) E (II8+)n-1. Let z, z be as in (2.3.2). For a non-negative integer j, we define Wn,j,R(a, z)

uiC(u1)Zn-1-zn+l ... c(un_1)zi-zn+l

fn-1

X Wn-1,R(a2C(u1)C(u2)-1,... an-1C(un-1)C(un)-1 zl ...

zn-l)

X e(alul + a2S(U1)U2 + ... + an-is(un-2)un-I)du1 ... dun_1. We defineWn,j,c(a, z) in the same way replacingui in the above formula byRe(ul)3, c(ui)zn-l-zn+1

etc. by

c(uj)2(zn_1-zn+1)

etc., and e(alul + ) by e(2Re(alul +

))

Obviously, 3

dai

Wn,n(a, z) = (27r)'Wn,9,R(a, z),

d3

i

daWc(a,

z) =

z).

(We consider the derivation considering al E R.) Consider functions of the form g(u) = f (u)c(u)a, where f (u) is a polynomial, and a is an integer. We define deg g = deg f - a. This is well defined. If deg g < 0, sup 1g(u)1 < oo.

Now we estimate Wn,j,R(a, z) by induction on n. For n = 2, we can obtain a precise result as in Shintani [64] as follows.

2 Eisenstein series on GL(n)

51

Lemma (2.3.21) Let l > 1 be an integer. Then W2,R,j (ai, z), W2,c,j (al, z) are entire functions, and for any such 1 satisfying Re(z1 - z2) + l > j for the real case and 2(Re(zl - z2) + 1) > j for the imaginary case, there exists a function cj,l(z) of polynomial growth such that IW2,j,R(a1,z)I :5 cj,l(z)all

IW2j,c(a1,z)I

cj,i(z)a121

Proof. In the real case, if f (u) = ujc(u)z+1 the l-th derivative of f (u) is bounded by a function of the form g(u)c(u)Ro(z)+1, where deg g < j - 1. Thus, one just has to use integration by parts l times. The imaginary case is similar. Q.E.D. Consider the domain {z I Re(z) E D(2, 0)}. If j = 0, by the remark after (1.2.1), the above estimate is true for any real number l > 1. Now we consider general n.

Proposition (2.3.22) Let r be as above. Then (1) Wn,j,R(a, z), Wn,j,c(a, z) are entire functions, (2) There exist Ml = Mi(r), , Mn_1 = Mn_1(r) E R, independent of j such that , ln_l are positive integers satisfying if 11,

11-12>M1+j, 12-13>M2i...ln_2-ln_1>Mn-2, 1n-1>Mn-1, then there exist estimates of the form a-,.-1 ,

Crj,i(z)a-l11

IWn,j,R(a+z)I 5

... an-1

-211

IWn,j,c(a, z)I <_ cr,j,I(z)al

where Re(z) E D(n, r), and cr,j,i(z) is a function of polynomial growth. Proof. We use induction on n. We first consider the real case. Consider the function , rn_1). Let z = Wn+1,j,n(a, z). Let r = (rl, , rn) E Rn and r' = (rl, (zl)...

, zn+l) E Cn+1, z1 + ... zn+1 = 0, and Re(z) = x = (xl, ... , xn+l) E

D(n + 1, r). Let ll be a positive integer, and 0 < m1, m2, m3

ll.

Let

0(a, u) Let zi = zi +

z

=

(a2C(ul)C(u2)-1,...

anC(un-1)e(un)-1).

and xi = Re(zi) for i = 1, 9(ul)C(ul)m(a2C(ul)C(u2)-1)m

, n.

/Consider a function of the form

Wn,m',R(,3(a, u), zl, ... , in)-

The derivative of the above function with respect to ul is 91(ul)C(ul)m,+l11(a2C(ul)C(u2)_1)m,Wn,m',n(,3(a,

+

92(u1)C(u1)m+l(a2C(ul)C(u2)-1)m'+1Wn

u),Rzi, ... , zn)

m,+1,R(N(a, u), zl, ... , zn),

where 91(ul) = 9'(ul)c(ul)-1 - m9(ul)u1C(ul) - m 9(ui)u1C(ul),

92(ul) = -27rfig(u1)u1c(u1).

Part I The general theory

52

Clearly deg gl, deg 92 < deg g. By( induction, the m2-th derivative of W0,n(/3(a, u), zl, ... , zn)

with respect to ul is bounded by a finite linear combination of functions of the form C(ul)m2 (a2C(u1)C(u2)-1)m4I Wn,m4,R()(a, u), 11, ... , zn) I,

where 0 < m4 < m2. Similarly, the m3-th derivative of e(a2s(ul)u2) with respect to ul is bounded by a finite linear combination of functions of the form where 0 < m5 < m3. We apply integration by parts with respect to e(alul) l1 times to Wn+l,j,n(a, Z). Then, by the above remark, Wn+l,;,R(a, z) is bounded by a finite linear combination of functions of the form c(ul)m3(a2C(u1)c(u2)-l)m5

a1 11

C(u1)Xn-X,.+1+1-i+11C(u2)Xn-1-s,.+1+1 ...C(un)x1-xn+1+1

I J

/

X (a2C(211)C(212)-1)m4+m5IWn,m4iQ$(6(a, u), x1, " ' , xn)Idu1 ... du n

where 0 < m4 + m5 < 11Since , zn) E D(n, r'), by induction, we can choose M2, , Mn indeSince Re(zl, pendent of m4 (and hence independent of ll, j) so that if l2i l3, , In are positive integers satisfying the condition

12-13>M2+m4, 13-14>M3, 1n-1-1n>Mn-1, In>Mn, then there exists an estimate of the form I

Wn,m4,n(13(a, u), xl, .

, .

, xn)I Cr',.7,1(z)(a2C(ul)C(u2)-1)-lz

... (anC(un-1)C(un)-1)-1

,

where cr,,j,l(z) is a function of polynomial growth (since m4 is an integer between 0 and 11 and cr',j,1(z) depends on ll, we do not have to write the dependency on m4). Then I Wn+lj,n(a, z)I is bounded by a finite linear combination of functions of the form / -11 -(12-m4-m5) -13 / a3 Cr',7,1(z)al a2 X

/

J. R X

an-ln

C(u1)"n-'n+1+1-9+11-12+m4+m5C(u2)xn-1-Xn+1+1+12-m4-m5-13 C(u3)Xn-2-Xn+1+1+13-14...C(un)x1-n+1+l+lndul.

dun,

r

where c'r, l(z) is a function of polynomial g owth.

By replacing M3i , Mn if necessary, we can assume that the integral over , un converges absolutely for x E D(n, r). The only condition that l2 has to satisfy is that l2 - 13 > M2 + m4. So l2 - 13 - m4 - m5 can be any integer greater than M2. u3,

2 Eisenstein series on GL(n)

53

Let 12 = l2 - m4 - m5. Then if 11- l2 - j, l2 -13 are sufficiently large, the integral over u1i U2 converges absolutely for x E D(n, r). Therefore, there exist M1, , Mn,

ln>M.,

such that ifll-12>M1+j, IWn+1j,R(Ct, z)I

c ,i,l(z)al

11 ...anln

where cr,j,1(z) is a function of polynomial growth. This proves (2.3.22) for n + 1.

Since a is real, Re(alul) = a1Re(ul) etc. So, in order to estimate Wn,j,c(a, z), we apply integration by parts to Re(ul), and the rest of the argument is similar to the real case. Q.E.D.

By the remark after (1.2.1), 11,

ln_1 can be arbitrary real numbers satisfying

11>>...»ln_1»0. Lemma (2.3.23) Let v E 9.R. Then Wn,v(a, Z) = Wn,v((-an-i, ... , -al), -TGZ). n n+1

Proof. If the diagonal part of the Iwasawa decomposition of n(u)TG for u E kv is a(u), the diagonal part of the Iwasawa decomposition of rGtn(u)-1 is (a(u)

2

So t(TGtn(u)-1)Z+P =

Let u E kv

'(_+1) 2

(a(u)-1)TG(Z+P)

t(n(u)TG)-"+P.

= t(n(u)TG)-TG(z+P) =

be the element such that n(ii) = TGtn(u)-1TG. Then

Wn,v(a, z) =

INk,

t(n

()TG)-TGZV(n(u))du

,_ni)

(n(u))du,

INk,

Wn,v((-an-1) ... , -al), -TGZ). This proves the lemma. Q.E.D.

Since -TGZ = (-zn, , -zl), the condition Re(z) E D(n, r) implies the condition Re(-TGz) E D(n, rn_1i , r1). Corollary (2.3.24)

(1) If 0 << l1 << l2 << estimate of the form

<< ln_1 or l1 >> l2 >>

IWn,.(a, z)I <-

>> ln_1 >> 0, there exists an

cr,t(z)IaiI-'1... Ian-11-1n-1

, ln_1 for Re(z) E D(n, r), where cr,j(z) is a function of polynomial growth (01, depend on r). (2) Suppose n = 2. Then, for any positive integer 1 satisfying l + r > 0, the estimate in (1) is true for Re(z) E D(2, r).

Part I The general theory

54

z), we can assume that a _

Proof. Since Wn,1,(a, z) =

z) or 1. Let t E Tke such that titti+l = ai for i = 1,

Wn v(a z) = t-C-P f

, n - 1. Since

t(tn(u)rG)z+Pbjv (1 ...,i)(n(u))du k

and the right hand side depends only on Itily,

, Itnly, we may assume that , an-1 E R+. Therefore, we can use (2.3.21)-(2.3.23).

al

Q.E.D.

Now, we can obtain an estimate of the global Whittaker function as follows. For , an_l) E (Ax )n-1 and t = an(tl, , tn) E TA, we define

a = (ai,

6(a, t) = (aitit2 1, ... , an_ltn_itnl) E (Ax )n-1.

(2.3.25)

Proposition (2.3.26) Let r = (ri, , rn_i) E R+ 1 (1) If h, , ln_l are real numbers satisfying 0 << l1 << 12 << ... << ln_1 or 11 >> 12 >>

>> ln_i >> 0, then there exists an estimate of the form n-1 er,l(z)Itlt21I-`1...Itn_1tn1I-

IWn(6(a,t),z)I i=1 o;Ekx

for all t E TA, Re(z) E D(n, r), and cj(z) is a function of polynomial growth. (2) Suppose n = 2. Then for any real number l > 1, IW2(6(a,t),z)I <- e1(z)It1t21I

«lEkx

where t ETA, Re(z) E D(2,r), and cl(z) is a function of polynomial growth. Proof. By (2.3.22),

IWn,o(6(a,t),z)I C cd(z)Ia1tit21I- 1 ...Ian-ltn-ltnlloon for some function c`(z) of polynomial growth. So bounded by

c(z)

n-1

t), z)I

1 -dl

Ialtltz III i<j ((v. - z + 1)I I

i=1

En-1 2n.Ekx IWn(6(a, t), z)I is

t

ai Ekx

3

I

Ian-ltn-ltn I0in_1 1

aaitit-' integral

1+...+Nn-11a1t1t21If 1... Ian-ltn-ltn 1It

17n(a6(a, t), z) I <- Iaf If

Since IS(zi - zj + 1)-1I is of logarithmic growth with respect to Im(z) for Re(z) E D(n, r), the above sum is bounded by n-1 laaitit2-1 f +N1 ... Iaan-ltn-ltn 1I f 1+1V.-1

cI (z) i=1

aiEkX

aait{t-1 integral

X Itit2 1I-h1 ... Itn_itn1

In-1,

2 Eisenstein series on GL(n)

55

for some function c'l (z) of polynomial growth.

Ifi +Ni>1,

l aaitit-1 l f +N; <) (li + Ni ). as EkX

a.itit-' integral

This proves (1).

For (2), we can be slightly more precise. Let id(atit21) C ok be the ideal determined by at1t21. Then 1 Q2(a8(a, t),

z)latlt2 'loo

aEk

<

N(C)-Re (z) lat1t21I It1t211-1 cCokideal

o:Ek

cJideal(ntj t21) ao: tlt21 integral

1`

< 1t1t211-1

N(c)-(Re(z)+i)N(d)-l

c,dC ok ideal

< ((Re(z) + l)((l). Q.E.D.

§2.4 The Fourier expansion of Eisenstein series on GL(n) In this section, we estimate the Fourier coefficients of EB(g°, Z). Consider

kn-1 such that ai

0 if and only if i E I. Let g° = k(g°)t(g°)n(u(g°)) be the

Iwasawa decomposition of g° as before. Then

t(t (9°)n(u)(n(u))du

EB(9°, z)a = a(-n(u(9°))) fN/Nk yEGk/Bk

1

='1/)a(-n(u(9°)))

rENk\Gk/Bk INA /N,k

t(

t(9°)n(u)T)z+(n(u))du.

The double coset Nk \ GklBk is represented by the set of all the permutation matrices. Let T be a permutation. If n(u) E NrA, t(t(g°)n(u)T)z+P = t(t(g°)T)z+P. So if there exists i E I such that T(i) < T(i+1), the above integral is zero. Therefore, we only consider T such that T(i + 1) < T(i) for all i c- I. For any such T,

I

t(t(g°)n(u)T)z+P,Oa(n(u))du

NA/N,k

=

f_

t(t(9°)n(u)T)z+PY/a(n(u))du

N,A

=

t(g0)TZ+P

fN,A

t(n(u)T)z+P,.l,b(a,t(go))(n(u))du. Y

Part I The general theory

56

First, we describe integrals of the above form in terms of the Whittaker functions. For that purpose, we prove a combinatorial lemma. Suppose that I = {il,jl + 1,...,jl + k1,j2,j2 + 1,...,j2 + k2i...

where j2 > ji + k1 +

jr, ...,fir

+ kr},

jr-i + kr_1 + 1.

Consider r such that T(i+ 1) < T(i) for all i E I. Let T1, be the longest element among permutations of I j,., , jm + km }, i.e. Tim is the product of transpositions (jm, jm + km), (gym + 1, jm + km - 1), .... Let Ti = H Ti., and T2 = Tl 1T. Lemma (2.4.1) l(T) = 1(T1) + l(T2). , {j1 + ki, ji + k1 + Proof. The permutation 'T changes the orders of { j1, j1 + 11, 1}, {j2, j2 + 1}, . So if jm < i < j < jm + km + 1, 1 < m < r, T(i) > T(j).

Therefore, L C I. Consider (i, j) E Tl 11,,. The pair (i, j) satisfies two conditions T1(i) > T1 (j) and

T(i) < T(j). If (i, j) E I, , T1(i) < T1(j), so this cannot happen. So Ir, nTl 1I,-2 = 0. If i < j, then (j, i) E I.,, . So j, < i < j < jm + km + 1 for some m. But by the assumption on T, T(j) < T(i), so this cannot happen either. This means that i > j, and (i, j) E I,. Hence, I- f rl 1Ir2 C I. So l(T) > 1(T1) + l(T2). But by (2.1.6), 1(T) < 1(T1) + 1(-r2). This proves the lemma. Q.E.D.

Definition (2.4.2) For a = (a1i we define

W, (a, Z) =

, an-1) E (Ax )n-1 and z = (z1,

f

JNrA

, zn) E Cn,

t(n(u)T)"Pba(n(u))du.

+ zn = 0 for convenience. We do not restrict ourselves to z satisfying z1 + ,1) to z and make it satisfy However, we can always add a scalar multiple of (1, the above condition. Note that this does not change the value of W, (a, z) or zi-zi+l Consider the decomposition r = Tint in (2.4.1). Then by a standard argument,

fl

(2.4.3) W, (a, z) =

t(n(ul)Tl)(n(u1))du1.

O(Zr2(i) - ZT2(i))

>9

fNA

T2 (i) < *2 (.i)

In terms of T, i>j

T2(i)
`b(z-2(i) - Z,2(7))

H O(Z-(i) -

Z1.(3)).

r1 (i)> -1 (7)

r(O<-U)

Clearly, (2.4.4)

f J _ t(n(ul)Tl)r2z+P'Wa(n(u1))du1 =rfj W , i-(a,T2z) Nr1 A

m=1

2 Eisenstein series on GL(n)

57

Example (2.4.5) Consider G = GL(9). Let a1i a2, a3i a5, a6, as ¢ 0. Then in the figure below, ui1 in * contributes q(zr(i) - zTU)).

0

L1

\

J 4 Wn(a,z). < ih as before. Let r be

Because of the choice of our measure, W.,,(a,z) _ IOk1

Let I= {i1,... , ih} C {1, , n-11, where i1 <

a permutation which satisfies the condition that r(ij + 1) < T(ip) for j = 1,

, h.

Let v be a permutation which satisfies the condition that if i < j, j I, then v-1(i) < v-1(j). The element v can be written in the form v = vh . . vl, where vj fixes i > ij, and v.-1 preserves the order of {1,

, ij - 1}.

Definition (2.4.6) We define the function EB,I,T,,,(g°, z) by the following sum t(9o-yv)TZ+aO«(n(-u(9°-yv))WT(6(a, t(9°'Yv)), z). °i Ekx if iEr yENv=o if for

Then by (2.1.9), EB(9°, Z) =

EB,I,T,v(9°, z)

if the right hand side converges absolutely and locally uniformly. We estimate EB,I,7,v(9°, z), and prove the required convergence also. Let T T1T2 be the decomposition as in (2.4.1). Let EB,I,7,v(9°, z) =

11

¢(zT(2) - zT(a))EB,I,T,v(9°, z)

By the remark after (2.4.5), L

WT(g°,z) _ I

lp(zT(i) - Z'

11 W,, - (9', 72 Z) m=1

The condition Re(zT2(i) - z72(i+l)) > 0 for all i E I, is equivalent to the condition Re(zr(i+l) - zT(i)) > 0 for all i E I.

Part I The general theory

58

Definition (2.4.7) Let 6 > 0. We consider I,,r = T1T2 as in (2.4.2). Let 6 > 0. We define Dr, DT,5, D1,,, D1,,,5 to be the following sets (1)-(4)

{x = (x1,... , xn) I x1 + "+x n = O, XT(i) - xr(j) > l for (ij) E IT 1, {x = (x1, ... , x,) I xl + ... + xn = 0, xr(i) - xr(j) > 1+6 for (i, j) E Ir,

(1) (2)

{x=(x1... , xn)

( 4)

+ xn = 0, xr(i+i) - xr(i) > 0 for i E I,

X1 +

x = (x1 ... xn)

(3)

1I,2

I

xr(i) - xr(j) > 1 for (i,j) E Tl xl + ... + xn = 0, xr(i+l) - xr(i) > b for i E I, xr(i) - xr(j) > 1 + b for (i, j) E T1

}

I

l

1Ir2

respectively.

By(2.3.24),if0<>.. >>lh»0, h

J

B,I,r,v(go, z)I< cl(z)

t(go,yv)rx+P fj I tii (go7'v)tii+1(go'Yv)-1I -1', j=1

L: 7EN:k

for x = Re(z) E Dj,r,6i where cj(z) is a function of polynomial growth. Let i i ei = (01 ... )0117 - 1)05 ...

0), eij = (0) ... ,0 ,0,... ,0,-1,0,... ,O).

We consider eij for i > j also. Clearly, eij = -eji. Then the right hand side of the above inequality is cl(z)

E

h t(t(go)'Yv)rx+P-Eii lie+j

which is equal to t(go)v(rx+P-E

cl(z) E Proposition (2.4.8) Let f = (fri

, fn) = - E ljeii. Then, for any M > 0,

there exist 0 << l1 << ... << lh satisfying the following two conditions.

(1) For any (i, j) E I,,, f (j) > M(2) There exist Aij > 0 for i > j and Aij > M for (i, j) E I, such that

v(f) = EAijeij. i>j

Proof. We use induction on h. Let v' = vh_1

v1. Let h-1

.f' _ (.fl, ... , fn) = -

ljeij j=1

2 Eisenstein series on GL(n)

59

Since l(v) = l(vh) + l(v'), I = Ivh l/ vh 1I, ,. If (i, j() E I,,,, v(vh l(i), vh l(j)) = (v (Z), V '(j)), V '(j) :!:- ih-1 < ih.

So adding -lheih to f' does not change these coordinates. Therefore, by induction, we can assume that M for (i, j) E vh 1I,,,. If (i, j) E vh(j) = ih. So we only have to take lh >> li, , lh_1. This proves (1). Note that for any T, CT-1(i),r-,(j) = Teij. Suppose vh1(ih) = ih. Then vh(i) = i for i < the Z > Zh, vh(Zh) = ih, and vh(i) = i- l for i = ih+1, , ih. By induction,

v '(fl) =

Bijeij, i>j

where Bij > 0, and if (i, j) E I,,,, Bij > M. v(f) = vh(v'(fl)) - v(lheih) = vh(v'(fl)) - vh(lheih), and vh(v'(fl)) is equal to

-/

Bijevh (i)vh(j)+

Bi j e h'

Bije,, h'(i)vh1(j) +

h'

(i)vh' (j).

-h '(i)
(j)

Since vh'I,,, C I,,, the condition on the coefficients is satisfied for pairs in the first term. We can ignore the second term, because vh1(i) > vh1(j). In the third SO vh1(i) = Z5, Zh < vh 1(j) ih. But term, (vh 1(j), vh 1(Z)) E -vh (eih) _ -(eih + ... + eih) = eih+12-h + ... + eah+lih. SO if lh >> lh-1,

-lhvh(eih)-

B2j

Cih+liheih+1ih+...+Cih+liheih+lih,

h

where Cih+lih,

Cih+lih > M. This proves (2). Q.E.D.

Lemma (2.4.9) There exists a linear function Aij (x) on {x = (x1, + xn = 0} for (i, j) E I such that X1 +

, xn) E C'

v(x) - x = E Aij(x)eij. (i,j)EI

Proof. Let v = Vh v1 and v' = vh-1 v1. By induction on h, there exists a linear function Aij(x) for (i, j) E I,,, such that v'(x) - x = E A?j(x)eij. (i,j)EI,,,

It is easy to see that v(x) - x =

Aij(x)evh'(i),hl(j) + vh(x) - X.

(i,j)EI,,

Part I The general theory

60

Since I, = Ivh I vh 1I,,,, we can ignore the first term. Suppose vh(th) = ih. Then vh (x) - x -xih eZh+lih + xtih+leih+2ih+1 + ... + xiheihih = xih (etihah - eihzh+l) + ... + xih-leihih-1 - xiheihih.

Since Ivh = {(ih, ih),

,

(ih, ih - 1)}, the lemma follows. Q.E.D.

These considerations and (2.2.6) show the following proposition.

Proposition (2.4.10) (1) There exist constants cij for a finite number of m's and (i,j) E Ir such that if D C DI,T,6 is a bounded set, then there exists a function cD,l(z) of polynomial growth such that if 0 << ll << ... « lh, 11

cD,1(z)t(90)v(TRe(z)+P-213eij)

IEB,I,T,v(9°,z)I

Iti(9°)tj(g°)-1Ic-,3

m (i,j)EI for Re(z) E D

(2) Let6>0. Ifv=1 and

or c6,1(z)t(g°)TRe(z)-FP-E `,i ei,

I EB,I,r,v (9°, z) I

for Re(z) E DI,T,6

The reason why we have to restrict ourselves to a bounded set D in (1) is that the choice of 1 in (2.4.8) depends on z. If v = 1, we do not have to use (2.4.8), and therefore we can get a slightly stronger statement (2). By (2.4.9) and replacing M in (2.4.8) if necessary, we get the following proposition. Proposition (2.4.11) Let D C DI,T be a bounded set. For any M > 0, there exists a function cD,M(z) of polynomial growth such that cD,M(z)t(90)rRe(z)+P 11

EB,I,T,v(9°, z)I

I ti(9°)tj(9°)-1I

A,j

i>j

for x E D, where Aij > 0, and if (i, j) E I,,, Aij > M. In particular, if g° is in the Siegel domain, Iti(g°)tj(g°)-ll is bounded. Therefore, we get the following proposition.

Proposition (2.4.12) In the same situation as in (2.4.10), there exists a function of polynomial growth cD(z) such that I EB,I,T,v(9

z)I C cD(z)t(g0)rRe(z)+P

for Re(z) E D, g° E KT°+52.

Note that we only have to choose one such M > 0 for (2.4.11), and the choice depends on D, so we only have to write the dependence of cD(z) on D. However, the function t(go)TRe(z)+P does not depend on D.

2 Eisenstein series on GL(n)

61

For later purposes, we prove an easy lemma.

Lemma (2.4.13) EB(TGt(90)-1TG Z) = EB(9o -TGZ) Proof. Similarly as in (2.2.15), EB(TGt(9°)-1TG,z) _

t(TGt9(9°)

TG'Y)z+P

-yEGk l Bk

_

(TGt(9'7-Gt-Y-1TG)TG)Z+P

^yEGk/Bk

l

t(9O7/)-TG(z+P)

7'EGk/Bk

= EB(90, -TGZ),

because -TGp = p. This proves the lemma. Q.E.D.

Chapter 3 The general program In this chapter, we define the zeta function and describe our general program to determine the principal part of the zeta function. One goal is to prove Shintani's lemma (3.4.31), (3.4.34) for GL(n). Using Shintani's lemma, we prove that certain distributions can be associated to some Morse stratum.

§3.1 The zeta function Let (G, V, Xv) be a prehomogeneous vector space. Let G' = Ker (Xv). Since G is connected and Xv is indivisible, G' is connected also. For practical purposes, we assume that there exists a one dimensional split torus To contained in the center

of G such that if a E To, it acts on V by multiplication by a and Xv(a) = ae for some e > 0. In other words, we assume that the constant eo in §0.3 is 1. We choose a maximal split torus T C G. Let T' = T n G. We define t = X*(T') ® IR, t* = X- (T') ® R. We can identify t with the Lie algebra of T+, and t*

with the dual space of t. Let tQ = X*(T') 0 Q, tc = X*(T') 0 C etc. Elements in tQ, tt are called rational elements. We choose a minimal parabolic subgroup T C P.

We define Goo ={gEGAI IX(9)I=1forallxEX*(G)}. Let G1 ={gEGAI IXv(9)I = 1}, TT1 = TA n Gli, and TA° = TA n G°O. Let T+ = T+ A

T+ nTA°.

Let w be a character of Gj/Gk. A principal quasi-character of Gj/Gk is a function of the form Ie(g)I8 where e is a rational character of G and s E C. If X is a principal

quasi-character of G1lGk, we extend it to GA so that it is trivial on To+. Let X = (X1,' , Xh) be principal quasi-characters. We define X(9) = f Xi(9) For later purposes, we introduce some notation.

Definition (3.1.1) Let L C Vk be a Gk-invariant subset. For 4D E Y(VA), we define ()L ('D, 9) = E -EL 4)(9x)

Definition (3.1.2) For a Gk-invariant subset L C Vks and a Schwartz-Bruhat function 4 E 9'(VA), we define (1)

(2)

ZL(D,w,X,s)= f A Ck w(9)X(9)IXv(9)IeL('b,9)d9, W, X, s) =

f i

w(9)X(9)IXv(9)I OL('P, 9)d9

vt91>1

if these integrals converge absolutely. If X* (G) is generated by one element, we do not have to consider X, and therefore, we write ZL((D, w, s) etc.

Our first task is to prove the convergence of the integrals in (3.1.2) for some subset L C Vks , a, be the set of simple roots of P. Let T, = It ETA I t«" Let al,

t«, > }, and T,, = T,1 n TAI, TO = TT n TA°. Also let T,+ = T, n T+, Tl+ = T,1 n T+, T,°+ =

T, nT+. Let p be half the sum of the positive weights. Let K be a special maximal compact subgroup. So GA = KPA. Let PA° = {p E PA I IX(p)I = 1 for x E X*(P)}. We take a compact subset 2 C PA°, and consider Cam. = KT,+S2, 61 = KT,J'+Sl. Then if q is sufficiently small and SZ is sufficiently large, 6, t surject to GAIGk, G1AlGk

3 The general program

63

respectively. There also exists a compact set 12 C GA such that 67 C S2T,,+, 671 C We call 67 (resp. 671) a Siegel domain for GA/Gk (resp. G,1j/Gk). W e choose a coordinate system x = ( x 1 , . . . , XN) whose coordinate vectors are

eigenvectors of T. Let 'yi E t* be the weights of xi. For x E Vk \ {0}, we define Ix = {i I xi # 0}. Let CC be the convex hull of the set {ryi i E I.J. I

Definition (3.1.3) A point x E Vk \ {0} is called k-stable if for any g E Gk, C9 contains a neighborhood of the origin. We denote the set of k-stable points by Vk. We define V.tk = Vks \Vk ('st' stands for `strictly semi-stable').

Proposition (3.1.4) Let L = Vk. (1) There exists a constant C = C(x) such that the integral (1) in (3.1.2) converges absolutely and locally uniformly if Re(s) > C. (2) The integral (2) in (3.1.2) converges absolutely and locally uniformly for all s. Proof. Let r be the dimension of V. We choose an isomorphism d : ][8+ -* T. Also we choose an isomorphism c : R+ -* To+ (To is as in the introduction). Clearly, T+ = T+To+

Definition (3.1.5) Let i be a positive integer. For p = (pi,

,

pi) E R+ and a

positive number M, we define rdi,M(p) = inf(µi'x ... p M), where we consider all the possible ±. Clearly, if M1 > M2, rdi,M1(p) < rdi,M2(p).

Lemma (3.1.6) There exists a finite number of positive numbers c1, that for any M > 1,

, ca such

a

Ova(4), kc(to)d(t1))I <<

to `Ncirdr,M(t1) i=1

forkES2,toEIIt+, and t1ER+. Proof. By (1.2.3), there exists a Schwartz-Bruhat function 0 < IF E 9(VA) such and toER+. thatI4)(kx)I _ W(x)forallkE52,xEVA. Let Since

kc(to)d(t1))I

Ov, (`1',c(to)d(t1))

for k E S2, we estimate Ovk (1P, c(to)d(t1)).

i E I} contains a neighborhood Let I C {1, , N} be a subset such that {ryi of the origin. We define I

Ot(`I'> c(to)d(t1)) =

E

`F(c(to)d(t1)x).

' Ekx for iEI =0 for i0l

By the definition of Vk, Ov: ('y, c(to)d(t1)) <

OI('ll, c(to)d(tl))

Part I The general theory

64

Let t1i , tr > 1. Since the convex hull of {ryi i E I} contains a neighborhood of the origin, there exist constants ei > 1 for i E I and fi > 1 for i = 1, , r such that r I

t1i

d(t1)>iEI ei'yi _ i-1

Then

r

(c(to)d(t1))2ie1 e "Y' = to EzEi e

ti

i=1

By (1.2.6), for any M > 1, r

OI(`I', c(to)d(t')) << t0 ME.EI et

f ti Mf`. i=1

The argument is similar for other cases. Therefore, there exist a finite number of positive numbers ci, , cb such that for any M > 1, b

E)I(1F,c(to)d(t1)) «I `t-M ;rdr,M(t1). jJO

Since there are finitely many possibilities for I, the lemma follows. Q.E.D.

We continue the proof of (3.1.4). We consider the same as in (3.1.6). Let a = Re(s). Since 15 C StT,+, Zvk (1), W, X, s) << f

to IX(d(t1))jevv (I',

c(to)d(tl))(tl)-2Pdxtodxtl

where d"to,dxtl are Haar measures on To+,T+ respectively. Consider c1, , ca in (3.1.5). If M >> 0, 1 X(d(t'))j(t1)-2Prdr,M(t1) is integrable 1, Ea 1 to-M°' is on T+. Note that this M depends on X. Moreover, if to integrable if Mci > a for all i. This proves (3.1.4)(2). Therefore, for (1), we only have to consider the subset {to < 1}. IX(d(t1))I(t1)-2prdr,M(t1) is integrable on T+. Then the integral We fix M so that over the set {to < 1} converges if a > Mci for all i. This proves (3.1.4)(2). Q.E.D.

We say that (G, V,Xv) is of complete type if for any X, there exists a constant C(X) such that Zv,,s (4), w, X, s) converges absolutely and locally uniformly for all 4), w, and Re(s) > C(X). Otherwise we say that it is of incomplete type. If (G, V, Xv) is of complete type, we write Zv (4),w, X, s) or simply Z(qD, w, X, s) for Zvba (4b, w, X, s) etc.

For the rest of this section, we consider groups of the form G/T, where G = GL(n1) x x GL(nf) and T is a (split) torus contained in the center of G. Let Gi = GL(ni) and Ti the subgroup of diagonal matrices. Let T = T1 x . . xT f. Then T is a maximal torus of G. Let Ni C Gi be the subgroup of lower triangular matrices with diagonal entries 1. Let N = N1 x . . x Nf. Then B = TN is a Borel subgroup

3 The general program

65

of G. Let K be the product of maximal compact subgroups of GL(ni)A which we defined in §1.1. The group GA has the Iwasawa decomposition GA = KTANA, and we write the Iwasawa decomposition of an element g E GA as g = k(g)t(g)n(u(g)), where t(g) = (ti(g), ... , t.f(g)), ti(g) = an: (tii(g), ... , tin; (g)) E (Ax )ni for all i, ( u( E A ` z -' for all i. Let pi, and u(g) u ui ( pf be half the sum of the positive weights of Gi and p = (pl, , p f). Clearly, T+ = T1+x . . xTf+. We define G°A = G°A x ... x GoA and TA° = TA f1 G°o etc. Let w1, , L of be characters of Ax /k x, and w = (wl,

, w f). We define

w(g°) = [I wi(det gi)

(3.1.7)

for g° = (g1, , gf) E G. Let 6#(w) = 5(w1) . .6(w f). Let dgi be the measure on GL(ni)°' which we defined in §1.1. Let dg° = fi dgi , A f) E R+, let c(A) _ Let cn: (Xi) be as in §1.1 for Ai E R+. For A = (A1, (cn, (A1), , cn f (Af)). By c, we identify R+ with a subgroup of GA. Any element of GA is of the form c(X)g° where g° E Goo. We define T+ = It E T+ I Xv(t) = 1}.

We choose a subgroup T+ C 1l so that T+ = T+/T+. Let g1 = tg° where t E T, g° E G. Let G1' = TG°O, and GA = R+ x G. We identify T°+ with R+. (A, g1), where A E l[8+. Let d x t be a Haar measure on T, and dg' = dxtdg°. Let We define dg = dxAdg1. If we choose dxt suitably, ZL(D, W, X, s) is equal to the following integral (3.1.8)

f

+xGA/Gk

Asw(gl)X(gl)OL('P, g)dg

We have a similar expression for ZL+(4), w, X, s) also. If X*(G/T) is generated by one element, G1' = Goo, and T+ is trivial. Therefore, in this case, we use the measure on G°o which we defined in §1.1. Also, we do not have to consider X. We define a bilinear form to define an appropriate Fourier transform. We identify

, nil. Let 'rG be the longest the Weyl group of Gi with permutations of {1, , XN) as element of the Weyl group of G. Consider the coordinate x = (x1i before. As we mentioned in the introduction, the weights of the contragredient representation V* and the weights of the representation g - tg-1 on V are the same. Therefore, there exists a bilinear form [ , ]v such that [gx, tgy]v = [x, y]v for all x, y E V. We define a bilinear form [ , ] v on V by the formula (3.1.9)

[x,y]v = [x,TGy]v

We define a Fourier transform 9v-D of 4D E 9'(V,) using [, ]v, i.e. (3.1.10)

< [x, y]v > dy.

(x) = 'VA

If there is no confusion, we write instead of vob also. (A-1,TGt(gl)-1TG). An easy consideraWe define an involution t on GA by g` = tion shows that [gx, gty]V = [x, y]v. One advantage of using this involution is that the Borel subgroup B maps to itself.

Part I The general theory

66

The determinant of GL(V) defines a rational character of G, which we call RV. Let rv be the principal quasi-character on Gl' defined by'cv(g') = Rv (g') We extend 'v to GA so that it is trivial on To+. The Fourier transform of the function 41(g1.) is for g1 E G. Let dk be the measure on K such that the volume of K is 1. For Ob E Y(VA) and w as above, we define (3.1.11)

Mv,.,b(x) =

JK

w(k)4P(kx)dk.

The operator Mv,, satisfies similar properties to those in Lemma 5.1 in [83] as follows.

Lemma (3.1.12) (1) For any k E K, Mv,,,4)(kx) = w(k)-1Mv,,,4(x). (2) Mv,.Mv,w = Mv,W (3) gvMv,W4) = Mv v4). Clearly,

ZL(4),w,X,s) = ZL(Mv,W'D,W,X,s)

Therefore, in this book, we assume that 4) = Mv,,d). In particular, if w is trivial, 4 is K-invariant.

§3.2 The Morse stratification In this section, we review invariant theory and equivariant Morse theory. We do not restrict ourselves to prehomogeneous representations in this section. We consider an arbitrary perfect field k. Let G be a split connected reductive group and V a representation of G both defined over k. This G does not correspond to the group G in §3.1. Instead, it corresponds to G' = Ker(Xv). We choose a maximal split torus T C G. We define t etc. as in §3.1. We fix a Weyl group invariant inner product (, ) on t*. We assume that if z, z' E t,, then (z, z') E Q. Let 11 11 be the metric defined by this inner product. We choose a split

Borel subgroup T C B and a Weyl chamber t* C t* so that the weights of the unipotent radical of B belong to t* by the conjugation b - tbt'1. We can identify t, t* by this inner product, and therefore, we consider 11 11, t_ for t also.

We recall the definition of stability over k. Let i : V \ {0} , P(V) be the natural projection map. Let k[V]Gk be the ring of polynomials invariant under the action of Gk. Suppose that f E k[V]Gk is a homogeneous polynomial. We define IP(V) f= 1r(x) I f (x) 0 0}. Definition (3.2.1) Let y E ll"(V)k. Then we define (1) y is semi-stable if there exists a homogeneous polynomial f E k[V] invariant under Gk such that y E P(V)f, (2) y is properly stable if there exists a homogeneous polynomial f E k[V] invariant under Gk such that y E P(V) f, all the orbits in 1P(V) f are closed, and the stabilizer of y in Gk is finite, (3) y is unstable if it is not semi-stable.

3 The general program

67

We use the notation 1P(V) (resp. lP(V)'oik) for the set of semi-stable (resp. properly stable) points. These are Gal(k/k)-invariant open sets of P(V)k. However, we will consider the set P(V)'oik in number theoretic situations only when P(V)E' _ P(V).(o)k.

The fundamental theorem of geometric invariant theory is the following HilbertMumford criterion of stability.

Theorem (3.2.2) (Mumford) Let y E P(V)k. Then (1) y is semi-stable if and only if it is semi-stable for all the non-trivial 1PS's of G, (2) y is properly stable if and only if it is properly stable for all the non-trivial 1PS's of G.

We can diagonalize the action of T because k is a perfect field. Let (xo, , XN) be a coordinate system of V whose coordinate vectors are eigenvectors of T. Let -yj E t* be the weight determined by the i-th coordinate. Consider the case when G = GL(1). Suppose that the action of GL(1) on the i-th coordinate is by c i for a E GL(1) and jl < ... <_ jN. For x c Vk \ {0}, we define II C {1, , N} as in §3.1. If f is a polynomial as in (3.2.1)(1), f is a summation of monomials whose factors do not entirely consist of xi's such that ji > 0. Therefore, if x is semi-stable, I, should contain i such that ji < 0. Moreover, if x is properly

stable, I,, should contain i1, i2 such that ji, < 0, jig > 0. For, if ji < 0 for all i E II, I., must contain i such that ji = 0 considering the action of a-1. Since y = lima-1-o ax should be in the orbit of x and the stabilizer of y is infinite, this is a contradiction. If 7r(x) is semi-stable or properly stable, the above property should be true for

all the non-trivial 1PS's of T and gx for all g E G. This implies that if 7r(x) is semi-stable (resp. properly stable), the convex hull of {7'i i E I9 ,} contains the I

origin (resp. a neighborhood of the origin) for all g E G. The definition of k-stable points in §3.1 was a number theoretic analogy of the above definition. In particular, if (G, V, X v) is a prehomogeneous vector space and dim G = dim V, the subset of P(V) which consists of positive dimensional stabilizers is a G-invariant subset and is equal to the set of unstable points. Since dimKer(Xv) = dimIF(V), all the semistable points are properly stable. This implies that VV g = V, . Therefore, (G, V) is of complete type. However, the reader should note that we are only using the trivial direction (only if part) of Theorem (3.2.2). The true value of the Hilbert-Mumford criterion of stability for our situation is the rationality of the Morse stratification. We recall the definition of the equivariant Morse stratification for the rest of this section. The following definition is due to Kirwan [35].

Definition (3.2.3) A point ,Q E t6 is called a minimal combination of weights if,3 is the closest point to the origin of the convex hull of a finite subset of {yi...... N}.

We denote the set of minimal combination of weights which lie in t* by 93. B is the index set of the stratification. Since B is a finite set, this means that the stratification is a finite stratification. Let ,Q E B. We describe the stratum Sp. The reader should note that Sp can be the empty set. W e change the ordering of {-y , .. , ryN} _ {'y , ... , ry'N} so that

Part I The general theory

68

We assume that n

((yl a), ... , (yN" a)) =

M0

n2

n

(ml, ... ml m2 ... m2 ... , mP, ... , MP)

where mo > 0, ml < ... < mP are coprime integers. Since /3 E 93, there exists some 1 < s < p such that o = 11/9112. Let e' be the coordinate vector which corresponds to yi'. Let Vb be the subspace spanned by {ei (y', /3) = I

We define

Zp=®b=sVb, WO =®b>8Vb, YO =Zp®Wp,

Zp={ir(x)I xEZp\{0}},Yp={7r(x,y)I xEZp\{0}, yEWp}, Mp = {g E G I Ad(g)/3 = 3}.

The group Mp is Stabp in [35].

Let pp : Yp -+ Zp be the projection map. Let vp be the indivisible rational character of Mp whose restriction to T is a positive multiple of /3. We define MQ = {g E MO I vp(g) = 1}. Since Mp is the Levi component of a parabolic subgroup of a connected split reductive group, it is connected. Therefore, Mp is connected also. _SS The group Ma acts on Zp linearly. Let be the set of semi-stable points with respect to this action. Let Pp be the standard parabolic subgroup of G whose Levi component is Mp and fixes the set Yp. Let Up be the unipotent radical of Pp. Since vp is rational, all these groups are split groups over k.

Let Ypk = p0 (Zak), and Spk = GkYak. We define Spk = -1(Spk) etc. Note

that if/3=0, YO =Z,3 =VandGo =G. The following inductive structure of the strata was proved by Kirwan [35] and Ness [48].

Proposition (3.2.4) (1) Vk \ {0} = HOE$ S. (2) Spk = Gk X p,3k Yak.

Since we are assuming that k is a perfect field, the following theorem follows from Kempf Theorem 4.2 [28] (also see Kirwan [35, pp. 150-156]).

Theorem (3.2.5) (Kempf) Suppose that x E Vk \ {0} and 7r(x) E Spk. Then there exists g E Gk such that gx E Yaks.

For a more detailed survey of the Morse stratification and its rationality, see [85]. We apply (3.2.4) and (3.2.5) to G' in §3.1. Suppose that 7r(x) for some x E Vk\{0} is unstable. Then there exists g E Gk such that gx E YR' for some /3 E 93. Clearly, points in YQk' do not satisfy the condition in (3.1.3). Therefore, Vk C V,". However,

Vk can be the empty set in general. So this is not the most precise notion. But it seems that in some cases, Vk gives the largest subset L for which (3.1.2) holds. We use the notion of the Morse stratification for the inductive computation of the principal part of the zeta function. For that purpose, we recall the notion of /3-sequences introduced by Kirwan [35, p. 73].

3 The general program

69

Definition (3.2.6) A sequence Z = (/31. , /3a) of non-zero elements of t is called a /3-sequence if for each integer j between 1 and a, (1) 3, is the closest point to 0 of the convex hull

Conv{yz -01 - .. - ,3j_1 (yi - Nk, ok) = 0 for 1 < k <j}, and I

(2) /3j lies in the unique Weyl chamber containing t_ of the subgroup n1
If a, a' are /3-sequences and a' is an extension of tl, we use the notation l -< ti'. We identify t with t* by the inner product ( , ), so we can consider /3-sequences as sequences of elements of t* also. We call a the length of Z and denote it by l(0). We also use the notation 1(T) for the length of Weyl group elements, but the meaning of the notation will be clear from the context. We consider 0 as a /3-sequence also and we define l(0) = 0.

Lemma (3.2.7) Let G be a split connected reductive group over k and x an indivisible rational character. Let G' = Ker(x). Let T C G be a maximal split torus. Then Gk = GkTk.

Proof. We only have to show that x is indivisible as a character of T also. Since all the characters of G are rational, we assume that k is algebraically closed. Let C be the connected center, and G = [G, G]. Then G = CG, G is semi-simple, and T = T fl G is a maximal torus of G. Suppose that x = XP on T. Since T is connected, this implies that X is trivial on T also. Since TIT = C/C fl G = GIG, we can extend k to a character of G. Regular elements are dense in G, so x = xP. Therefore, p = ±1. Q.E.D.

We go back to the previous situation and consider (G, V, Xv). For a /3-sequence -0, we define subgroups Ma, Ma, Pa, Ua C G and subspaces Yo, Za C V inductively as follows.

If c0= 0, we define M,=Pa=G, Ma=G', Ua={1}, and Yo =Za=V. If i) = (/3), we consider the Morse stratification of V with respect to G' (not G !). Let Pj = Pa, Ma = MQ be the groups which we defined earlier, and So = SO etc. Let

Ua be the unipotent radical of Pa. We define Pa = TP', Ma = TM,'. Then Ua is the unipotent radical of Pa also. Consider /3-sequences 0' = (/31. not consider Ma, etc. Suppose St,

, /3a+1) and _ (i31i ... , /3a). If Sa = 0, we do 0. Consider the representation of Ma on Za. We

consider the Morse stratification for this situation. Then it is parametrized by the /3a+i's for all the /3-sequences Z - Z"s as above. Let Pa, = P0.+1, Ma, = M0.+1 C Ma be the subgroups which we defined earlier. Let Pa, = TPP,, Ma = TM,, and Say = S0.+1 C Za etc. Let Z1,k be the set of k-stable points with respect to Maf1Ma . Let YY,k,Sa,k be the corresponding subsets of Zak. Let Zasstk = Zask \ Za,k. We define Ya',stk, Ss',stk similarly. We consider D"s such that Say 54 0. Let Ua, be the unipotent radical of P6,. By induction, Ma is connected.

Lemma (3.2.8) Mak =

MkTk.

Part I The general theory

70

Proof. We prove this lemma by induction on l(a). If l(a) = 0, this is (3.2.7). Suppose that a a', l(a') = l(a) + 1, and a' = ()31, . , 0a+1). Let v' be the indivisible rational character of Ma, fl Ma, which is a positive multiple of 0a+1 when restricted to Ma, fl Ma fl T. Since Ma, fl Ma is the Levi component of a parabolic subgroup Pa, fM' of Ma, it is connected. By definition, Ma, = Ker(v'). By (3.2.7),

(M5,nMM)k=MM,k(TflMs,flM.)k Let g E Ma,k. Since Ma'k C Mak, we can write g = g't for some g' E Mak, t E Tk. Then g' E (Ma, fl MM)k. Therefore, we can write g' = g"t' for some g" E MS,kIt ' E Tk. Hence, g = g"t't, proving the lemma. Q.E.D.

Now we consider the rationality of the stratification.

Proposition (3.2.9) Suppose that a - a' and l(a') = l(a) + 1. Then Sak ak x Pa,k

1"k

Proof. By (3.2.5), Mk x pa,,, Yak -+ Sak is surjective. Suppose that 91,92 E Mak, y1, Y2 E Y, ',k, and 9iyi = 92y2 We can write 91 = 9it1, 92 = 92% for some t1i t2 E Tk, g1 g2 E Mak. Since g'7r(t,y1) = g27r(t2y2), there exists p E (Pa, n MM)k C Pa,k such that g' = g2p by (3.2.4). Since 92pt,y1 = g2t2y2, pt1y1 = t2y2. Let p' = t2 1pt,. Then 91 = 92P',P'Yl = y2, and p' E P5,k. Q.E.D.

Equivariant Morse theory in algebraic situations like in this section was established around 1983. However, Kempf's paper appeared in Annals of Math. in 1978. Therefore, he proved the rationality of the Morse stratification before equivariant Morse theory was established. Of course, if (G, V,Xv) is a prehomogeneous vector space over an algebraically closed field, the equivariant Morse stratification is the decomposition into G-orbits. Kimura and Ozeki determined the orbit decomposition of prehomogeneous vector spaces over algebraically closed fields (see [31]-[33]). However, the rationality question was not considered there. The reader can probably see that the construction of the equivariant Morse stratification is algorithmic and it is possible to write a com-

puter program to determine the set B. It is a non-trivial task to determine which vectors in B give us non-empty strata. However, this is relatively easier than just trying to determine the orbit decomposition from scratch. The author carried out this task and obtained the list of B in some cases. One advantage of this approach is that one can determine the inductive structure of the strata So = G xpQ Yps using the vectors 3. In later parts of this book, we handle the orbit decompositions of prehomogeneous vector spaces from this viewpoint.

When we consider a group of the form G/T, where G is a product of GL(n)'s and T is a split torus contained in the center of G, there is an alternative way of determining Yag. Let Xv E X*(G/T) be as before, and G' = Ker(Xv). We choose a split connected reductive subgroup G" C G so that GL' surjects to G' with finite kernel. Then we construct Ma using G" instead of G. It is easy to see that Mak surjects to M5' with finite kernel contained in Tk. Since the notion of stability is rational over the ground field by (3.2.3) and Tk acts trivially on Vk, the stabilities of points in Yo with respect to Ma and Ma are the same.

3 The general program

71

§3.3 The paths We introduce the notion of paths and related materials in this section. We consider groups of the form GlT where G is a product of GL(n)'s and T is a split torus contained in the center of G. Consider (G/T, V, xv) as in §3.1.

Definition (3.3.1) (1) A path of length a is a pair p = (a,s), where 0 is a /3-sequence of length a, and s is a function from {1, , a} to 10, 1}. (2) We use the notation p -< p' if i -< W, and s' is an extension of s.

We define l(p) = l(1) and call it the length of p. We allow p = (0, 0) (which means that we do not consider the function s) as a path and define l(p) = 0 for convenience .

Let Ma, Sa etc. be as in §3.2. Let va be the longest element of the Weyl group of Ma. We define an involution of MSA by 6a(g) = va'g-lva. We define a bilinear form [ , ]a on Za using va as in §3.1. We define a Fourier transform 9a on Za using this bilinear form. Suppose that YD C V' C V are subspaces defined over k and iY E 9(VV). Then the restriction of T to S°(YSA) is denoted by R5WY. Any element y E YSA can be written as y = (yl, y2) where yl E ZSA, Y2 E WSA. We define RaW E .'(ZSA) by (3.3.2)

RsI(yi) = f

Ra'I'(yi,y2)dy2

Wan

for yl E ZSA.

For 4 E 9'(VA), and a path p = (D, s), we define 4kp inductively in the following manner. If l(zi) = 0, 4)p = I . If l(cz) = 1,

(P s(1)=0, (3.3.3) Cl

Suppose that p =

' s(1) = 1. s'), and l(p') = l(p) + 1 = a + 1. Then we

s) -< p'

define

(3.3.4)

Ro Dp

s'(a + 1) = 0,

. 'Ra4Pp

s'(a+l) = 1.

If l(p) = 1, 4)p E .O(VA). If p p', and 1(p') = l(p) + 1 = a + 1, (Pp, E .9(ZSA). We also define a homomorphism Bp from Ma to G inductively in the following manner.

Ifl(p)=1,0p(g)=g`. Ifp-
0p

s(a + 1) = 0,

l Bp0a

s(a + 1) = 1.

Bp

Definition (3.3.6) Pa# = M5U5# (resp. Pp# = MpUp#) is the parabolic subgroup of G which contains B and whose Levi component is Ma (resp. Mp).

Part I The general theory

72

Let Pp = Bp (PO), Mp = Bp (Me), and Up = Bp (U-0). Since Bp induces an automor-

phism oft which preserves t_, ep(P#) = Pp#. Let Be = B fl Ma, Bp = B fl Mp. Let No = N fl Ma, Np = N fl Mp. Let pe (resp. po#) be half the sum of the weights of Ue (resp. Ue#). We write elements of Uom, UU#m as no(ue), no#(uo#), where uei ue# represent elements of some affine spaces. Let due, due# be the measures on Uem, Ue#m such that the volumes of Uem/Uek, Ue#m/Ue#k are 1. If there is no

confusion, we write n(ue) instead of n(n). , /3a) be a /3-sequence. Let Me = Met x x Me f. Let Gl = TG°° , vea Gl fl Pp/Pek -i IL, be homomorphisms, which are positive multiples of /31 i , /3a when restricted to T+. We define Let 0 = (i31i

be as in §3.1. Let vdli (3.3.7)

Pam = {p E Gl fl Pam I vo1(p)

lea(p) = 1},

P°m={pEPom I IX(p)I=1for XEX*(P)}, MOm={gEGIflMem Ivol(9)= =voa(9)=1},

M°A={gEMAI IX(g)I=1for XEX*(M)}. If 0 = 0, we define P ,A = Mam = Gl etc.

Let Ste C BeA be a compact set, and i > 0. We define Tj,7 to be the set of t c TA fl M°A such that to > q for all the positive weights a of Be. Let T°,+ = Te°,,1 fl T+. Then for suitable S2, ii, (K fl Mem)TT,+S2e surjects to MmlMok. We call

this set a Siegel domain for M°m/Mek. It is known that there also exists a compact set S2o C Mm such that (K n Mm)TT,7+11 C 52eTT'+. , /3a. Let del(Ao1), , dea(Xaa) be 1PS's which are positive multiples of /31, Let AD = d11(Xe1) ... doa(Aea) for A11,

, Xea E

R. We define

AD = {)e} = R.

(3.3.8)

+Ra Then Xex = Then GI n Mm = ADMam. Let AD E Aei x E Zom, and it = µx. W e define homomorphisms e , , , , epa : Gl n Mem/Mek - R inductively Xa'+

in the following manner. Suppose that p < p', and l(p') = l(p) + 1 = a + 1. For i = 1, , a, we extend epi to AD, so that it is trivial on do'a+1(Ae'a+1). Then we define

epi =

(3.3.9)

and ep'a+l(A ') = 1,

epi

s'(a + 1) = 0,

e pi 1

s'(a + 1) = 1

,

A'oQ''+...+3 +1

By Definition (3.2.5), (/3a+1,13k) = 0 for i Let AD = , a. Therefore, A3'+' = 1. This implies that

(3.3.10)

epa+l(Ao')ep'a(Ao)-1

ep'a+1(da'a+1(Ao'a+1)) = {

s(a+ 1) = 0, s(a + 1) = 1.

ep'a+1(Ao')ep'a(Ao)

Definition (3.3.11) Let p = (cl, s) be a path of length a. We define Apo = {Xe E AD

epi(Ao)

1 fori = 1,... all

Apt = {Ae E Ae epi(A1) < 1 for i = 1, Ape = {A, E AD

epi (Ae) < 1 for i = 1,

, a- 1,epa > if,

, a-

1}.

3 The general program

73

We define Ap = T+ n MMA fl Z(MD)A, where Z(MD) is the center of M.D. Let A'a = A, A'. Let A'' i = Api A' for i = 0, 1, 2. We extend epi to Api so that epi is trivial on Ap for i = 1, , a. We choose a Haar measure dgD on GI n Ma so that if f (g1) is a K-invariant function on G1j/Pa#k, f(9ana#(ua#))t(9a)-2pa#d9adua#

ff(91)d91 = f a #nlPa#k

Let ep = (-1)#{1
(3.3.12)

We define wp = w if #{1 < i < a I s(i) = 1} is even and wp = w-1 otherwise. For w, 0 and 'F E 9'(ZDA), we define MD,,' E 9'(ZDA) by MD,WAF(x) = f

Kf1Maa

where dka is the Haar measure such that the volume of K n MDA is 1. Easy considerations show that if (b = Mv,W4), Mo,,,,, Ra(Pp = Ro Dp.

The spaces Za, Wa are invariant under Ma. This implies that the determinants of GL(ZD ), GL(WD) determine rational characters of Ma. Let Ra1(90 ), !02 (go) be the rational characters of Ma determined by Za, Wa respectively. This means that if tI E GL(ZD) is a scalar matrix, kal (tI) = tdlm Za etc. Let ,Cai (9a) = IRai (9a) (-1 for i = 1, 2. Let rla3(90) = Ica1(90)Ica2(9a)

Let go =gaga, where as E AD, 9a E MMA. If l(p) = 0, we define vp (g) _ op (X, 9) = X(g). We define up (go) = vp (X, go) for 1(p) > 0 inductively as follows.

Definition (3.3.13) (1) If l(p) = 1, we define X(9a)1ca2(9a)t(9a)-2pa

s(1) = 0,

p(9a) - ! l X(9a)- 1 kv(9a)- 1 Ka2(9a)t(9a) -2 P°

5(1)

= 1.

(2) Suppose that p -< p', and l(p') = 1(p) + 1 = a + 1. Then s(a + 1) = 0,

J op(9a')!a'2(9a')t(9a')-2"

p'(9a) = l ap(9a')

2P°'

z(a+1) = 1.

§3.4 Shintani's lemma for GL(n) In this section, we define and prove some growth properties of the smoothed Eisenstein series for groups which are products of GL(n)'s. x GL(nf) etc. be as in §3.2. We define Let G = GL(ni) x

tt* _ {zi = (zi1, ... zini) E Cn, I zi1 + ....+ zin = 0}, t°* = {zi E to* I zi7 < ziJ+l for all j}.

Part I The general theory

74

Let t°*=t°*x

and t°*=t°* x...xtf*-

Let EB;(gi, zi) be the Eisenstein series on GL(ni)°o. We define (3.4.1)

EB(9°, z) _ fl EB; (9i, zi)

Ifni=1,wedefine EB;(gi,zi)=1. We define a hermitian inner product on t°j by the formula (zi, z'i)i = zijz2j. Let C1i ,Cf > 0 be constants, and C = (Cl, ,Cf). We define a hermitian inner product (, )o on by (z, z')o = Ei Ci(zi, zy)i for z = (z1, ... , zf), z' = (zi, ... , zf). Let Ilzilo = (z, z)o. Let p = (pi, ... , pf) be as in §1.1.

let

(3.4.2)

Li(zi) = 2(zi, pi)i = E (zij, - zij2). jl <j2

If ni = 1, we define Li(zi) = 0. We define (3.4.3)

L(z) = E CiLi(zi) = 2(z, p)o, i

wo = L(p).

Lemma (3.4.4) Let r = (r1, . . . , rf) be a Weyl group element. Then if r # TG, L(-Tp) < L(p). Proof. The element TG is the only element of the Weyl group such that TGp = -pSo if r # rG, -rp 54 p. The hermitian form ( , )o is Weyl group invariant.

This means that -Tp is on the sphere of radius Ilpilo in t°*. Hence if -Tp # p,

L(-rp) < L(p). Q.E.D.

For a Weyl group element T = (ri i (3.4.5)

, r1), we define

Mr(Z) _ [JMTi(zi)

We remind the reader that MT,(zi) =

W(-ZiTi(jl) -'ZiT+(j2)).

11 j1 >j2

*;(jl)
Let O(z) be an entire function of z which is rapidly decreasing on any vertical strip. We always assume that V)(z) = V)(-TGz), i(p) 0- 0.

We define (3.4.6)

A(w; z) _

O(z)

w - L(z)

A, (w, z) = MM(z)A(w; z).

3 The general program

75

Let dzi = fj_1 d(zi,j - zi.7+i) and dz = 1i dzi. Definition (3.4.7) For g° E G°n, w E C, we define ,,1+...+n f-f

)

.,(g°,w, L) _ ( 27

f

EB(g°, z)A(w; z)dz

Re(z)=q

, qf), qj = (qil, qi7 ), and qi,j - qij+1 > 1 for all i, j. We call 8(g°, w, 3G) the smoothed Eisenstein series. It is defined for Re(w) > L(q). Since we can vary the choice of q, it is holomorphic for Re(w) > wo. If there is no confusion, we drop 0 and write 61(g°, w) also. We have used a weighting of factors in the above definition. This will simplify some computations in later parts of this book. By (2.4.13) and the assumption that Vi(-TGz) = zb(z), ?((g°)`, w) _ 8(g°, w). Let p be a path. We define a function a (go, w) of (g,, w) E (GA fl Man/-0k) X C where q = (ql,

inductively as follows. If l(p) = 0, we define ep (g°, w) = e(g°, w).

If l(p) = 1, (3.4.8)

-p(ga,w) =

s(1) = 0, e(gan(ua)`,w)dua s(1) = 1.

J fUDA/Uak ff(gon(uo),w)dua

l JUMIUak ll

If p-
-'P' (go" w)

fU,'AIU,,k 60p(go,n(ua,),w)dua,

s(1) = 0,

fUb,A/Ua,l -0p(ea(ga'n(ua,)),w)dua,

s(l)=L

Let (3.4.10)

ep'(ga'n(ua'),w) _

-p(ge'n(ua'),w) - 8p'(go,,w)

s(a+1) = 0,

tip(ea(ga'n(ua')),w)-fp (8 (ga'),w) s(a+1)=1.

The function ip(gon(ua),w) is the non-constant term with respect to Up. Let P = MU C G be a parabolic subgroup of G, where M is the Levi component and U is the unipotent radical. Then EU(g°, z) is, by definition, the constant term of EB(g°, z) with respect to U, i.e., EU(go, z) = f

(3.4.11)

EB(g°n(u), z)du, A /U,

where du is the measure on UA such that the volume of UA/Uk is 1.

Definition (3.4.12) For g° E G°n, w E C, we define 1

eU (go, w)

ni+...+nf-1

27f

where we use the same q as in (3.4.7).

Eu(g°, z)A(w; z)dz, JRe(z)=q

Part I The general theory

76

The function 'u(g°, w) is defined for Re(w) > L(q). Let Pa# be as in §3.3. Then &p (go, w) = &' U , # (Bp(go), w) for go E G O i fl M,A. Suppose that

jip-1) for all i, p (jio = 0, jiai = ni). Let 1 = (31, ... , J f), and Ii, = {jip-1 + , jt } for all i, p. Let r = (11, .. , Tf) be a Weyl group element. Consider the

1,

following condition for permutations.

Condition (3.4.13) If il, j2 E Jip for some i, p and ji < j2 then ri(ji) < Ti(j2). Let Tp be the longest element which satisfies Condition (3.4.13). If P = Pa# (resp. P = P,#), we use the notation Ta (resp. Tp) for Tp,. (resp. Tpn# ). Let r be a Weyl group element which satisfies Condition (3.4.13). For g° E G°o fl MA, we define

(3.4.14) Eu,T(g°, z) = MT(z)EBM(9°,TZ), 'ai+...+n.f-f

1

-'u,T (go' w)

Eu,T(9o, z)A(w; z)dz,

=

Re(z)=q

where q E DT and eu,T(g,w) is defined for Re(w) > L(q) By the usual consideration, (3.4.15)

Su(go,w) _

-'u,T(go,'w),

where T runs through permutations which satisfy Condition (3.4.13). If U = Up#, we also use the notation fpT(go,w) for Suppose l(p) = a. For i = 1, , a, let p = (Di,si) be the unique path of length i such that pi - p. We define two conditions for /3-sequences and paths. ou,#,T(Bp(go),w).

In (3.4.16)-(3.4.23), we assume that G1 = G°o (i.e. (G,V) is an irreducible representation).

Condition (3.4.16) (1) MaiA = M0NA for i = 11 (2) MoiA = M° A for' = 11

.

, a.

a - 1.

If p = (zz, s) is a path and 0 satisfies (1) or (2) of Condition (3.4.16), we say that p satisfies Condition (1) or (2) of Condition (3.4.16). x Ma f, where Mai C GL(ni), Suppose that Mo = Mal x

Mai = Mail x ... X Mtiai+l, and

Mail = GL(jail), Mail = GL(jai2 - jail),

Maai = GL(jaiai -jaiai-1), Maiai+l = GL(ni - jaiai ) for some 1 < jail < . . . < jaiai < ni - 1 for i = 1, , f . We allow some ai to be zero, and in that case, Mai = GL(ni). Of course, a ai. For convenience, let jaio = 0Jaiai+1 = ni.

3 The general program

77

In this situation, Mp is of the form Mp = Ml x

x Mp f, where Mpi C GL(ni),

Mpi = Mpil x ... x Mpiai+l, and

Mpil = GL(jpil), Mpi2 = GL(jpi2 - jpil ), Mpai = GL(jpiai -Mpiai-1), Mpiai+l = GL(ni - Mpiai )

for some 1 < jpil < < jpiai < ni - 1 for i = 1, , f . For convenience, let jpi0 = 0,3piai+l = ni Let A' be as in §3.3. We consider growth conditions for functions on the space A'piM°A/Mak for i = 1, 2, 3. Let r1iP = (rliPs)l<s<7'aip -jai,-, -1 E R7aip-jaip-'-1 ,

r1i = (rlip)l
Let r2 = (r21, ... , r2a) E Ra and r= (rl, r2).

tt

Definition (3.4.17) Suppose that AD E A' and t°° E M°A fl TA. Let

to = (tal j... , taf )5tai = ani (tail

C.. itaini )-

We define \/a12 = rja1 epi( Aa)r2i and arl

f ai+1

=

11 ( aij r-i+ltaijap-i+2 i=1 p=l p

)'rli Pjaip -jaip-1 )rlipl .. ( taijap-ltaijap 1

r) be the set of continuous functions f satisfying the condi-

Let tion

sup

(3.4.18)

If(A'k,ta)ItarlA''r2
a EA'p, ka Ena too ETaO+

fori=1,2,3. Consider elements in t°* of the form a = (al,

,

af), where

n-jpiai

ai = (ail, '

and ail,

' .

, aiai+l) ... , aiai+l )

> ail, '

, aiai+l E Rai+1 for all i. We consider such a satisfying the following

condition (3.4.19)

ail <

< aiai+l,

E(jpip - pip-i )aip = 0 for all i. P

Part I The general theory

78

Proposition (3.4.20) Let p be a path of length a satisfying Condition (3.4.16)(2). Then there exist a satisfying (3.4.19) and constants cl, , ca > 0 such that a

Op (Aa)a = ri epi(At )ci i=1

Proof. We prove this proposition by induction on l(p). Let p = (0, s) -< p' = (0', s') be paths such that p' satisfies Condition (3.4.16)(2), and l(p') = l(p) + 1 = a + 1. This implies that p satisfies Condition (3.4.16)(1). Consider ep'a+1 on M°A n MM,A. Bp, (M°A n MM,A) = M°A n Mp'A. ( 9, maps Mo to Ma.)

Suppose that Moip n Mo, = GL(lipl) x x GL(mip,ip). Let p = (pl,

GL(mipl) x

,

x GL(lipbip), and Mpip n Mp, _ pf), where lipbip

lipl

Mi = (Ail, ... , µ'iai+1) pip = aj,p -.lap-i ({A'ipl' ... µ2p1 ...

µ2p6 p

...

µipb p

and /Rips E } for all i p, s. We consider such p satisfying T7 yip' = 1 for all i +p lisµips Then any element of MIA n Mo'A is of the form pgo°, for some p and g.a, E Ma°,A. Let -y be an element in t°* of the form y = (y1i , y f), where 1ipl

lipbip

yi = (Yil, ... , yiai+l), yip = (Yipl, ... , yipl, ... yipbip, ... , yipbip and

(3.4.21)

yipl < ... < yipbip,

lips yips = 0 for all i, p. S

Also we consider an element y in t°* of the form y = (51, Mipi

, 5f ), where Mip.ip

yi = (yil ... i yiai+l)i yip = (yipl, ... , Yipl, ... , yipcip) ... I yipcip). and

(3.4.22)

yipl <

< yipcip>

mipsyips = 0 for all i, p.

Lemma (3.4.23) There exists y as above such that 9p,(p)7 = ep'a+l(A). Proof. By the definition of /3-sequences, ep'a+1(p) = p7 for some -y as in (3.4.21). Note that ppl = = p' = 1. Pp, = Bp, (Po,) is the unique subgroup of Mo which contains B n Mo and whose Levi component is Mo,. By B,', the positive eights with respect to Pa, correspond to the positive weights with respect to Pp,. Therefore, the existence of such a y follows. Q.E.D.

3 The general program

79

We continue the proof of (3.4.20). We first assume that p' also satisfies Condition (3.4.16)(1). Then AD, = Aa,. We can write AD, = da'a+i(7a'a+l)A identifying AD with fi l dai(Ao,i) Since 0,(A,) = AD 1, Bp (AD,) -_

I Op(Ai)Op'(do,a+l(X5'a+1))

l

l 0p(AD)- ep'(da'a+l(AD'a+1))

s'(a+1) =

0,

5 ' (a + 1) =

1.

By the above lemma (p = da,a+1(\a,a+l)), there exists y' satisfying the condition of the lemma for p'. This implies that 0p' (da'a+l (Aa'a+1))ry' = ep'a+l (da'a+l (Aa'a+l )) = ep'a+l (An' )ep,a (Aa,

)fl

By induction, we choose a E to* which satisfies (3.4.20) for p and Op(\a)a = , a. Let a = ma + ^y, where m is a l epi(AD)ci, where ci > 0 for i = 1,

fli

positive number. If s'(a + 1) = 0, epi(AD) = ep,i(AD), and if s'(a + 1) = 1, epi(A5) = ep,i(AD) Also Bp(Aa)5 = 1,Op,(da'a+i(A 'a+1))' = 1. Therefore, a

ep,a+1(Aa')mci.

Bp'(Aa') = ep'a+1(Aa')ep'a(Aa')+1 i=1

Hence, if m is large enough, & satisfies the condition of (3.4.20) for p'. This proves (3.4.20) for the case when p satisfies Condition (3.4.16)(1). We use the same argument again, and we obtain the case when p only satisfies Condition (3.4.16)(2). Q.E.D.

, nil for Let Ii C {1, , ni - 1} be a subset and Ti a permutation of {1, , f . Let vi be a permutation which satisfies the condition that jl < (Ti,... ,Tf), (I1,... ,If), T = j2, j2 Ii implies vi l(jl) < vi 1(j2). Let I = and v = (vl, , vf). Let I,, be as in §2.1. Let Dj,,Ti be as in (2.3.7). Let Dj,T = DIl Tl x ... x DIf,Tf. Let ai = (ail) ... ,aini-1) E kni-1 for i = 1,... f, ni(ni-i) for i = 1, , f . Let and a = (al, a f). Let ui = (uijl,j2 )jl>j2 E A 2 (gl,... (n(ui)) =< ailui2l + ... + aini-1unini-1 >. For go = , gf) E G°', we

i = 1,

define

(3.4.24)

(gi, zi) =

EBi,.i,,(gi,zi)=

J NiA/Nik EBi,«i (ginni (ui), zi) Y'.i (nni (ui))dui, t(ginni(ui)Ti)'i+,OiV).,(nni(ui))dui.

4

INcA

We assume that Ti (j + 1) < Ti (j) for j E Ii for all i, because the above integral is 0 otherwise. We define (3.4.25)

EB,«,T(go, z) = 11 EBi,oi,Ti (gi, zi),

i

0

EB,« (9, z) = 11 EBi,0i (gi, zi).

Part I The general theory

80

Also we define (3.4.26)

«(g°,

)nl+---+nf-f

1

JRe(z)=q --e, (go, w, ') )nl+...+nf-f 1 2ir

EB,«(9°, z)A(w; z)dz,

f

EB,«,T(9°, z)A(w; z)dz, e(z)=q

where we choose q as in (3.4.6) for the first integral and q E DT for the second integral. These functions are defined for Re(w) > L(q). Let (3.4.27)

i

aijEkx if jEIi °ij=o if jv!i

Wi,T,V (9°, w,

)

E E E E 1e ik i

aijEkx ifjEIi .ij=o if i97i

Since Di,T is contractible, the contour defining &«,T(g(Yi,

i

,T(9°('Y1,... ,ryf),W,0I

-YiEN

(g°, w, b) is well defined as long as we choose , ryf), w, b)'s so that q E DI,T. Of course,

l eI,T,v(9°, w,'01 < ii,T,v(9°, w, 0)

Unless the situation requires, we drop 0, and use the notation a (g°, w) etc. We estimate kj,,,v(g°, w). Let li be a function from Ii to the set of natural numbers. If Ii = {ii, ,jNi}, jl < < 3Ni, we define lip = li(jp). Let l = (ll, , l f). Let si,i, (l) be the element of the Lie algebra determined by (3.4.28)

ti''I' (c) _

I ti3 ijj+l l

1i(j)

jEIi

Let s I = s

and tsI(i) = rj ts`i(t). We fix T I. For s l (l) as above, we put q'(l) = rq - sj(l). Consider the following conditions.

E Dj T, l

Condition (3.4.29) (1) For any 6 > 0, there exists a function c6,1(z) of polynomial growth such that WT(6(a, g°), z)I <<

for {z I Re(z) E Di,T,6}. (2)M 9 (1)ivi(jf)

for (ii, j2) E Ivi.

-

q'(1)ivi(j2) >- M

c6,1(z)t(go)-s(`)

3 The general program

81

By (2.2.6), (2.4.8), (2.4.9), we get the following proposition.

Proposition (3.4.30) (1) There exists M > 0 independent of q, l such that if q, l satisfy the conditions (1), (2)M in (3.4.29) and 61 > 0, then ej,T,,,(g°,w) is bounded by a finite linear combination of functions of the form t(go)v(Tq+P-s1(d)) 11

'

11

Itij1 (9°)tij2 (g°)-'

IC:iv2

(.1i,.72)EI,,:

for L(q) + 61 < Re(w), where ci71.72 is a constant independent of q, l for all (j1, j2) E

Iv. (2) Moreover, if q E Di,,, I", (go' w) << t(g°)Tq+P

for g° in the Siegel domain and L(q) + 61 < Re(w). Note that in (1), we are not restricting ourselves to elements of the Siegel domain. Let

J

I tij1(9°)ti.92 (go)-' Ci3132 11 i (ji,72)EI,,.

This is a slowly increasing function on G°o independent of q, 1.

We have proved in (2.4.8) that there exists M1 (depending on q) such that if li, - li,+1 < M1 for all i, j then (3.4.29)(2)M is satisfied for a sufficiently large M. We are going to use (3.4.30) to estimate 9p (go, w), ip (g1, w) in this book. But our estimates are rather delicate, so the author would like to clarify the logic. We have estimates of the form (3.4.29)(1) where the range of l does not depend on q E DI,,. But for Condition (3.4.29)(2)M to be satisfied, the range of l depends on q E DI,T. When we use (3.4.30), we have two kinds of freedoms. One is the choice of 1, and the other is the choice of q. We can always fix q, and take l which satisfies (3.4.29)(1) as long as it satisfies (3.4.29)(2)M. But if we want to fix 1, and vary the choice of q, we have to be a little careful. For, if v2 # 1, the range of li could depend on q. Therefore, if vi l 1, we have to show that (3.4.29)(2)M does not depend on qj. On the other hand, if v = (1, , 1), since Condition (3.4.29)(2)M is empty, we can fix any l satisfying (3.4.29)(1) and not necessarily satisfying the condition that li9 - li7+l << 0 for all i, j, and vary the choice of q. So, for example, we can choose lij - lid+l >> 0 for all i,j. Also if some Ii consists of one element and vi = 1, we can choose an arbitrary real number lil > 1 by (2.3.24)(2). We go back to the previous situation where p is a path as before.

Theorem (3.4.31) Let CG = (9101 ...T.,)-l. Suppose that G1 = G°o and p is a path satisfying Condition (3.4.16)(2). (1) For any e > 0, there exist b = 6(e) > 0 and a finite number of points cl, t°* satisfying IIciIIo < c such that if M > L(p), I9p(go,w) - CGA(w;p)I <<

t(9a)c:

, cN E

Part I The general theory

82

on Aa(K fl Mom)To°,,7+Sta, and wo - 6 < Re(w) < M.

(2) Suppose that r = (r1, ,Tf), and r # r,. Then for any e > 0, there exist 6 = 6(e) > 0 and ri = (ril, rig) as in (3.4.17), such that rill, , ri2a > 0 for all i, Iriip8I < c for all i, p, s, and that if M > L(p), ep,T(A' ga, w) I < 1: A r2at(ga)T{1

on Apo(K fl MMA)Tb,,+fZ,, and L(p) - 6 < Re(w) < M. (3) There exists r = (rl,r2) as in (3.4.17) such that all the entries of ri (resp. r2 )

are negative (resp. positive), and that if wo < Ml < M2, 61p, (A' ga, w) I <<

Ao(Re(w)-

o)'2t(ga)(Re(w)-wo)T1

on Aa(K fl MDA)TT,7+f2o, and Ml < Re(w) < M2 for all r. , If) Let go = Aa ga Clearly, Bp (go) = Op ()4 )9p (go). ) . Consider I = (Il, such that jpip Ii for all i,p. Let o, = (Ol, ,o-f), v = (vl,... , vf) be elements of the Weyl group, where ai = fP=1 l 6ip, vi = nPl 11 vip, and o ip, vip are permutations of 3pip for all i,p and vip satisfies the condition in (2.1.10). Then, by the consideration in §2.1, Proof.

I,oT,v(0p(9D),w)

-'p,T(go,w) _ I,o,v

If (jl, j2) E

for some i, then j1i j2 E 3pip for some i,p. Also if go is in the a

Siegel domain, 0p (ga) is contained in some Siegel domain for Mp°A also. Therefore,

by (3.4.30), if q E DI,,T and 6 > 0, t,or,v(9p(go),W) <
for go in the Siegel domain and L(q) + 6 < Re(w). It is easy to see that Ei,,,,(ep (go), z) = 9p (A'

)ar=+PEI,o,,(ep (g), z)

Clearly 9p (A, )0 z+P = 9p (A, )T Z+P. We /choose an element ry in to* of the form 7 = (51, ... , 5f), where 3i = (Yip), lip = (Tip1, ... yI1pjpsp-jVsp-1 ), and

'Yip = aipl(-1,1, 0, ... 0) + ... + a?pJjp -.7pip-1-1(0, ... , 0, -1, 1), aip1, ..

, aipJp+P-.7v=r-1 -1 > 0, ryipl < ... < yiPJpip-9pip-1 .

Consider a in (3.4.20). Let h1, h2 > 0 be positive numbers and a = hla + h0Then if h1h21 >> 0, a is an interior point of t°*. If h1 is small, II&IIo can be arbitrarily small. (9p(ga),w) so that Re(z) = q We choose the contour in the definition of and q = (vr)-1(-p + ma), where m > 0 is a constant. Then Qrq + p = ma. Since -(ar)-'(p) is in the closure of DI,,, and & is an interior point of t°*, q E DI,ar.

3 The general program

83

Therefore,

eI,ar,v(Bp (ga ), w) << Op (Aa a

)m.h10Op

i mhlc

epi(At,

(ga°)mh2-1

0 nhz5 ep(ga)

i=1

By the above consideration, q E DI,-T as long as m > 0, and L(-(0'7-)-1(p)) < wo for all o, T. Clearly,

L((UT)-1(ma)) = mL((cr)-1(a)).

So, if 0 < c' < jL((orT)-1(&))j-1 for all a,T, we can choose m = c'(Re(w) - wo). Note that the choice of & does not depend on or. This proves (3). Suppose or # TG. Since L(-(UT)-1(p)) < wo, if m > 0 is sufficiently small, L(q) < wo. This proves (2). Note that if r Tp, aT # TG. Next, we consider the case aT = rG. By the above remark, r = Tp. i-1

Suppose Ii

0.

Let j E Ii. Let a' = (0 0, ai, 0,

, 0) E t°*, where

ai = (-(ni - j),

, -(ni - j), j, . . , j). Then if m' > 0 is a small number, p+m'a' is in the closure of DI,TC and L(p+m'a') < wo. If m'm-1 >> 0, then L(p + m'a' + TG(ma)) <wo and p+m'a'+TG(ma) E DI,TC. Let q = p+m'a'+TG(ma). Then TGq + p = m'TG(a') + mrG(a). Therefore, 11TGq + pilo can be arbitrarily small. Hence, we can choose q = p + m'a' + TG(ma) E DI,,, ,1). So Suppose that aT = TG and I = (0, , 0). This implies that v = (1,

eI ,rc,1(ga,w)=

)nl+...+nf-f

(2

fRe(z)=q

(z)t (O(g

))Gz+°A(w;z)dz.

Let 61,62 > 0. Let ql = (qll, "

, q1 f) E t°*, where qi1 = (giji) .. , glini ), Blip glip+l = 1 + 61 for all i,p except for i = p = 1, and q, 11 - g112 = 1- 62. We assume that 61i 62 are small and 6261 1 >> 0. Then L(q) < wo. Let q2 = (q21, , q2f) E to* . 61 for all i, p except for i = ) g2ip;) and Up g2ip+1 = 1 + such that q2i = (g2i1, p = 1 and q111 - q112 = 1. Let dz' = 1!(i,p)0(1,1) d(zip - zip+1). Then 6'I,TO,l(ga, w) is equal to ''7UU

nl+...+n/-.f

1

1

+

I

e(=)=q,

MTC(z)t(ep(ga))TC=+vA(w;z)dz

n1+...+nf-.f-1 Res

27f

IRe(z)=q2 =11-z12=1

[M,, (z)t(Bp (ga))TC=+PA(w; z)] dz'.

There exists 6 > 0 such that the first term is holomorphic for Re(w) > wo - 6. We continue this process, and eventually get CGA(w; p). This proves (1). Q.E.D.

Note that the right hand side of (3.4.31)(3) does not depend on M1i M2.

Part I The general theory

84

Definition (3.4.32) Let f, (w), f2(w) be meromorphic functions on a domain of the form {w E C I Re(w) > A}. We use the notation i - f2 if the following two conditions are satisfied.

(1) There exists a constant A' < w° such that i - f2 can be continued meromorphically to the domain {w E C I Re(w) > A'}. (2) The function f, (w) - f2(w) is holomorphic around w = w°. The following corollary follows from Theorem (3.4.31)(1).

Corollary (3.4.33) Suppose that G' = Go Aand p is a path satisfying Condition (3.4.16)(2). Suppose that f (ga) is a function on A'iM°ti/Mak, where i = 1, 2 or 3, and that there exist finitely many points cl, , c,n E t* such that If(ga)I << inf((X,t(ga))c`), x

and

I',i (KnM,A)Ta,, Sla

inf((Aat(ga))")dgo < co. i

Then

f

.f(ga)(g,w)dgCGA(w;P) f

Ani MaA/Mak

viMDAYMak

In other word, if we can prove that a function is integrable by estimating on the Siegel domain, then we can get rid of the smoothed Eisenstein series in the integral.

The statement of the next theorem does not depend on the path p, and we do not consider Condition (3.4.16) or the assumption that Gl = G.

Theorem (3.4.34) (Shintani's lemma for GL(n)) (1) For any e > 0, there exists 6 > 0 such that if M > wo,

f sup

g0 ga=kt,.n(u)E60 .e(w)<M

ni-1

I e(g°, w) - CGA(w; P)I If fl I tij (g°)tij+l (g°)-l i=1

I-E

< oo.

1

(2) There exists a constant c > 0 such that if M1 > M2 > w°i

f ni-1 sup

16' (g°, w) I If H I tij

(g°)tij+l(go)-l Ic(Re(w)-wo)

< 00.

gO=ktn(u)E60

Proof. The proof of this theorem is very similar to that of (3.4.31). The only place we used the assumption (3.4.16)(2) was the choice of a in the proof of (3.4.31). This time, we just choose & = h where y = (y1i , 5f), and

5 i =a1(-1,1,0,... 0) + ... + ai,,-1(0, ... , 0, -1,1), ail, ... aini-1 > 0, 5il < ... < 'Yini. The rest of the proof is similar to that of (3.4.31). Q.E.D.

3 The general program

85

,ni-1,

Note that ifcij is a constant for

f ni-1 f ni-1 1111 Itij(9°)tij+1(g°)-' cij(Re(w)-wo) «fJ [ i=1 j=1

Itij(9°)tij+1(g°)-1Ic(Re(w)-wo)

i=1 j=1

for go E 60, where c = infi j cij. The statement (1) of (3.4.34) implies that if f E C(G°°/Gk, r) for some r such

that rij > -j(ni - j) for all i, j, then

f

A/Gk

f (9°)8(9°, w)d9°

CGA(w;

p) J

f (9°)dg°. ,

The statement (2) of (3.4.34) implies that if f is a slowly increasing function,

fG/Gk

.f (9°)e(9°, w)d9°

is well defined, and becomes a holomorphic function in some right half plane. §3.5 The general process

In this section, we consider (G, V,Xv) such that (Me,ZZ) is a prehomogeneous vector space for all 0 = (/31, , /3a). (The character is the one which is a positive multiple of (da.) This condition is not always satisfied, for example, G = GL(n), V =

M(n, n) (the set of n x n matrices). In this section, we assume that (G, V) is an irreducible representation. Let 0, ?.' be /3-sequences such that i and 'Y2 E 9(Za'A). We define

a' and l(ti') = l(cl) + 1. Let t@1 E 9'(Z-OA),

OS,, (,P1,9D) = E 'I'1(9Dx),

(3.5.1)

xES,,k

E `y1(9ax), xE o%k

OZ,, ('Y2,9a') =

'F2(9a'x), xEZaik

for ga E G1 fl MS,t, ga, E GI fl MM'A. We also define

W1(9ax), Gs,,,st(`I'1,9a) E '1(95x),

Osa,('P1,9a)= XESa,k

xES,,,atk

E 'Fl(95'x), OY,,,st(W'1,9D') = E IF2(90'x), XEYY,k

OZa, ('1'2,95') _ E W2(9D'x), OZ,,,st(F2,9a') = xEZ,,,k

xEYa,,atk

L:

xEZo,,etk

'F2(9a'x)-

Part I The general theory

86

Let w = (wl,

, w f) E St(A < /k" )f be as before. In the following definition, we

consider the case where G = G. Definition (3.5.2) Let p = (Z,s) be a path, and T a Weyl group element which satisfies Condition (3.4.13) for Up#. For -P E .'(VA), w as above, and w E C, we define (1)

Ep,T (-D, w, w)

f

wp (ga )op (ga )E)z, (Ra'Dp, go ).p, (go, w)dga,

,,2 MaA /Mak

w, w)

(2)

f9p1 MDA/Mok -p,T+(oD, w, w)

(3)

wp (ga )Qp (ga jai (ga )E)z, (9a Ra gyp, Ba (ga )),?p,T (go, w)dga, `q,0 MaA/M k (4)

p,T#

w, w)

= Ra-bp(0)

f

wp(ga)op(go)Sp,T(go,w)dga,

AvoMaA/Mak

=p,T#(4),w,w)

(5)

_ 90Ra'Dp(0)

fno MA/Mok

wp(ga)o,p(ga)kai(ga)SpT(go,w)dga,

if these integrals are well defined for Re(w) >> 0. w, w, When we have to refer to the function z/i, we use the notation etc. We also define EP w, W)- st+(,D, W, w) etc. using '9z,. (Rasp, go) etc.

Clearly, w, w) = Cp +(4)1 w, w) + p,T,st+(,P, W, w) etc.

If the distributions in (3.5.2) are well defined, we define p ('D, w, w) _

Ep,T (4D, w, w)

etc. Equivalently, EP (-P, w, w) can be defined by replacing 6p,T(go,w) in (3.5.2) by .°p (go, w). It is easy to see that Ep,+(4), w, w) =

w, w) + Ep,st+(4D, w, w) etc.

also.

We still assume that G' = G. We consider a path p = (a, s) which satisfies Condition (3.4.16)(1), or satisfies Condition (3.4.16)(2) and there is no split torus

in the center of Ma which fixes Z,11. Let x E Z. This assumption implies that there is no split torus in the center of Ma, which fixes x, because Zak is a single Mad-orbit.

3 The general program

87

Let to, ta, ta, to be the Lie algebras of AD, A', A', T+ n M ,A respectively. Then

to=tf

ta.

Let x = (x,, , XN) E Zak. Since x is semi-stable, it is semi-stable with respect to the action of Aa also. Note that we are considering Aa as a group over R, and the notion of semi-stable points is not changed by field extensions. If p satisfies Condition (3.4.16)(1), Aa is trivial. The convex hull of {yiltii I xi # 0} contains the origin of to*. If p satisfies Condition (3.4.16)(2) and there is no split torus contained in the center of Ma which fixes ZD, the origin cannot be on the boundary of this convex hull, because that implies the existence of a split torus fixing x. Therefore, it contains a neighborhood of the origin of W.

Proposition (3.5.3) Let p = (0, s), ,r be as above. (1) The distributions Bp T w), and vp,T+(4), w, w) in (3.5.3) are w, w), well defined for Re(w) >> 0. w, w) are (2) Moreover if p satisfies Condition (3.4.16)(1), Bp,T# (4b, w, w), also well defined for Re(w) >> 0.

Proof. Let a = l(p). Then as acts on ZDA by multiplication by epa (AD). For go in the Siegel domain, we choose v(ga) E T+ so that t(ga)z = v(ga)z for any z E We choose 0 < IF E 9(ZDA) so that IRa4Pp(A gax)I << W(A v(ga)x) for g°° in the Siegel domain.

E)a,I(IF,A,v(ga)) _

(Aav(ga),x) sE ZDk

o ekx for.ef =O for i0I

We consider such I such that the convex hull of {yiIta®t$ i E I} contains the origin of td* ® to*, and the convex hull of {yilta i E I} contains a neighborhood of the origin of to*. We choose Io so that {x = (xi)i I xi = 0 for i V Io} = Z5. Then I

Ioz, (Ro bp, aaga)I << E E)a,I(IF, Av(ga)) ICIo

We fix I. We choose constants 0 < ci for i E I so that E, ciyiIt, to = 0. Since A5 acts on ZDA by multiplication by epa(A5), this implies that

(A u(ga))-Eie1

h'

= epa(A

)-Eiel°i,

Let C = EiEI ci. Clearly, C > 0. Let d = dimA5. We write elements of AD in the form as = A5Aali, where Aa E A5, 41) E A. Since the convex hull of {yilta i E I} contains a neighborhood of the origin of to*, as in the proof of (3.1.4), for any M > 0, there exist constants , h, i E I and a slowly increasing function fm (AD, v(ga)) such cM pi's for j = 1, I

that (3.5.4)

J

fM(As,v(ga))rdd,M(4

Part I The general theory

88

By (1.2.6), for any M1, M2 > 1, (eza (Rasp, )aga)I << inf(X'av(ga))-M1 E.er

CM2,j"Y'

< epa (,A')-CM1 fM2 (AD, v(ga ))rdd,M2 (dal) ).

We have the estimate (ge, w) I «A

(Re(w)-wo)r2

v(ga)(Re(w)-wo)ri

where r is as in (3.4.31)(3). We choose M2 large enough so that the function fM2 (AD, v(ga ))rdd,M2 (A(') )

is integrable with respect to A( 1). Then the resulting integral is bounded by a slowly increasing function f'(aa, v(go)) of AD, v(g°). Consider the function f '(X1 , v(ga ))epa

(Aa)-CMS Aa (Re(w)-wo)r2

v(ga)(Re(w)-wo)r1

Note that the functions epa() ), A' (Re(w)-wo)r2 depend only on AD.

If e, ()4) >_ 1, we choose Re(w) >> 0 so that the above function is integrable with respect to epi (A') for i = 1, , a - 1 and go in the Siegel domain. Then we take M1 >> 0 so that it is integrable with respect to epa (A'). If epa ()4) < 1, we fix M1 and take Re(w) >> 0. Since there are finitely many possibilities for I, this proves that =-p,T (4D) w, w), =-p,T+(4i, w, w) are well defined if Re(w) >> 0. We can estimate

aRaDp, Bp(Aaga°)) similarly except that the exponent of epa(A') is positive. Since we consider Apo for =p+(qD,w,w), the proof is similar. This proves (1). , epa) gives an We consider (2). In this case, AD = A', MMA = M°A. So (epi, isomorphism AD -* R. By (3.4.31)(3), for any slowly increasing function f(go) on ApoM,A/M'Ok, the integral Oza

JA0 Man /Mak

f (go )6p,T (ga, w)dga

converges absolutely for Re(w) >> 0. This proves (2). Q.E.D.

A similar proof to that in the above proposition shows the following.

Proposition (3.5.5) Let p = (cl, s) be as above, p -< p' = (O's') and l(p') = l(p) + 1. xi 0} Suppose that for any x = (xi) , XN) E S5'k, the convex hull of {ryiIta contains a neighborhood of the origin of to*. Then the integrals JA0 Man /Mak

f

wp (ga) kal (ga )o,p (go )esa, (go Ro Dp, 9a (ga ))ep (go, w)dga,

wp (ga )op (go )osa, (Ra gyp, ga )e, (ga, w)dgo

A'0 Man /Ma k

are well defined for Re(w) >> 0.

3 The general program

89

If p satisfies Condition (3.4.16)(1), since Aj is trivial, the condition in (3.5.5) is automatically satisfied for p'. Suppose that p satisfies Condition (3.4.16)(1). Let (3.5.6)

go) ='al(ga)

Osa, (JFaRa4'p, 00 (go)) a
t(a')=t(a)+1 E) sa, (RD D-b p, ga

a[a'

t(a')=t(a)+1

Jp(4',go) = J,(4), go) + tca1(go)-FaRa4p(0) - RD 41p (0).

If p = (0, 0), we use the notation J(4), g°), J'(4P, g°). By the Poisson summation formula, p (4', w, w) _ Ep+('D, w, w)

+

Lip (ga)op(ga)Oza(RDI)p,ga).p(go,w)dga A p0 MaA/Mak

P+ (D, L", W) + Cp+ (D, w, w) + =p# (4D, W, W) - Ep# (4), W, W)

+

JA V0 MaA /Mak

wp(go)op(go)Jp(-P, go)ep(ge,w)dgb.

By (3.5.3), (3.5.5), we get the following proposition.

Proposition (3.5.7) If p = (D, s) satisfies Condition (3.4.16)(1), then p

w, w) = up+ (p, w, w) + up+ (p, w, w) + p# ata'

t(a' )=1M+1 M+1

f

a

oMaA/Mak

fApoMaA/Mak

w, w) - Ep# (-t, w, w)

wp(go)Ka1(go)op(go)E)

wp (ga )rp (ga )es,, (Ra Dp, ga ).p (go, w)dgo

l(a')=t(a)+1

Let p be as above, p -< p', and l(p) = l(p)+1 = a+1. For go, E G°O f1 Ma,A, ua' E -01A, we define Ep'1(4', w, go,, ua') = wp (ga' )o p (ga' )OY,, (Ra bp, ga'na' (ua' )),

Ep'2('D, w,g5',ua') = wp(ga1)op(ga')ka1(ga')®Ya, (9aRaDp,ea(ga'na'(ua'))).

Definition (3.5.8) (1) If s'(a + 1) = 0,

fna Pa,A /P*'k

p, (4), w, w) is, by definition, Ep' 1(4), w, go,, ua' )f''p (ge' no, (ua' ), w)t(ga') -2p,' dga, dua,,

if this integral is well defined for Re(w) >> 0.

Part I The general theory

90

(2) If s'(a + 1) = 1, gyp' ((D, w, w) is, by definition,

f

poPa,A/Pack

Ep'2 O, W, go', u5' )fp (go, no, (u0' ),

w)t(go')-2Pp' dg0, duo,,

if this integral is well defined for Re(w) >> 0.

We consider paths in the following three classes. (1) The path p satisfies Condition (3.4.16)(1). (2) The path p satisfies Condition (3.4.16)(2) but not (1) and Ma contains a split torus in the center which acts trivially on Zo. (3) The path p satisfies Condition (3.4.16)(2) but not (1) and Ma does not contain a split torus in the center which acts trivially on Zo.

Let p' be a path such that p' = (-0', s'), and s'(a + 1) = 0. Then IA90 M,A/Mak

f

WP(go)ap(go)es,, (Ro4)p,g,)ep(go,w)dgo

ApoMA/P,k

Wp(ga)op(go)19ya, (RoDp,go)ep(go,w)dga Ep' 1(4), w, go,, uo' )e'p (ge' no' (u0, ),

w)t(ga')-2Pp' dg0' duo' .

Apo MaAnPa'A /Pa'k

The second step is because M5,,,, Ro(bp = Ra4Dp by the assumption in §3.1. Since

fa' A / U,' k rya' (RoDp,go,no,(uo,))duo' = k0'2(g5')ez,, (RD',Dp',go,), and Apo{do'a+1(Ao'a+1)} = Ap'2, the above integral is equal to

^p'(-D,W,w)+, p'(D,W,w). If p' belongs to the class (3), Ep, (4i, w, w) is well defined for Re(w) >> 0 by (3.5.2). Therefore, Ep, (1D, w, w) is well defined for Re(w) >> 0 also.

Similarly, let p' = (D', s'), and s'(a + 1) = 1. Then

f

Wp (ga)ka1(ga )op (go)Os, (go RoDp, O* (gs)).p (go, w)dga po MaA /Mak

_ =p, ($, w, w) + E"p, (4), w, w).

Therefore, p, (4k, w, w) is well defined for Re(w) >> 0 if p' belongs to the class (3). These considerations show the following proposition.

Proposition (3.5.9) Let p be a path in class (1). Then if 2p, (4i, w, w) is well defined for Re(w) >> 0 for all p', Ep=p(4),W,w) = Ep(p+(-D, W, W) + =p+((D,W,w) +

w) -

E Ep, (Gp, (-D, W, W) + E" (4), w, w)1 I a
3 The general program

91

This proposition is the basis of the iterated use of the Poisson summation formula. We now describe how we proceed.

We assume that (G/T, V, Xv) is an irreducible representation and of complete type for simplicity. This implies that G1 = G. For 4) E .'(V,1) and A E lam, we define 4)A(x) = 4)(,x).

We want to find points pi, , Pdv E C and distributions a((D, w), a2j (1b, w) for i = 1, , dv, j = 1, , dv,i with the following properties (3.5.10)

f

w(9°)J(, 9°)(9°w)d9° - a(, w)A(w, z), °e/Gk

dv dv,i

(3.5.11)

A-n' (log A)ia2j (4D, w),

a(,DA, w) i=1j=1=1

where A E R+. Clearly, 1

f A'-r; (log A)i d' A = (-1)1 j!(s -

pi)-1-1.

0

This implies that (3.5.12)

Zv(4), w, s) = Zv+(4), w, s) + Zv+(4, w-1, N - s) dv

dv,,

-13-1

+1'

giving a principal part formula. If (G, V, Xv) is of incomplete type, we need `adjusting terms' (see [86] or Chapter 4). In this book, the only such case is essentially the case of binary quadratic forms, so we do not discuss it here. For a general conjecture about prehomogeneous vector spaces of incomplete type, the reader should see [85]. In order to achieve (3.5.10) and (3.5.11), we use (3.5.9) repeatedly. If p' in (3.5.9) belongs to class (1) and Ma, is a torus or p' belongs to class (2), then we do not consider paths p' -< p". If p' belongs to class (1) and Ma, is a torus, 8p, (-D, w, w) is essentially a contour integral of a product of Dedekind zeta functions. Therefore, we can handle such distributions. If p' belongs to class (2), it is likely that we can show "p, (4), w, w), 8p' (,D, w, w) -' 0 as is the case in Parts III and IV. Suppose that p belongs to class (3). This implies that the representation of M, on Z° is reducible. We still do not have a general approach to handle all the reducible prehomogeneous vector spaces, but if the number of irreducible factors is two, we can use a technique to consider further paths. We discuss this feature of our program in Parts III and IV. So we are still obliged to show that Ep ((D, w, w) is well defined for Re(w) >> 0 for

p=

s) in classes (2), (3). We will handle these distributions when we consider individual cases. Of course, we want to ignore all the 8p (4), w, w)'s. However, it is too optimistic to expect that ,=p (4 , w, w) - 0 for all p as we will see in Part IV. For the space of pairs of ternary quadratic forms, the correct statement turns out to be Ep"p((D,w,w) - 0. p

Part I The general theory

92

Also it is too optimistic to expect that for all p, there exists a distribution by ('P, w) which satisfies the property (3.5.11) and

'p (4b, w, w) - by ((a, w)A(w; z)

This is because some cancellations have to be established between strata. Therefore, our process is quite delicate and heavily depends on the individual cases. The following proposition follows from (3.4.32).

Proposition (3.5.13) Consider a path p in class (1). Suppose that a

ap(AD) = rl epi(A,)"', i=1

and Xpi > O for i = 1,

, a - 1. Then

V+(4),w, W)

CGA(w; P) f

wp(go)ap(ga)'9z; (RoDp,9o)dgo,

Ant Man /Mak

p+(4p, w, w)

- CGA(w;P)

f

wp(ga)op(go)noI(go)19z; (goRa4,p,Oo(go))dgo

Ap0 Man /Mak

Next, we consider paths of positive length in class (2). We are still assuming that G°A = G. Therefore, we do not consider X. Suppose that p is a path which satisfies Condition (3.4.16)(2). Let Aa be as before. Let A2 be the connected component of Ker(A' -* GL(V)A) which contains 1. Then Aa = R+ for some c. We choose a subgroup Aa C Aa which is isomorphic to R+ for some d so that Aa = ADAD and A. 2 n A.3 = {1}. We write elements of Aa in the form $1) = A(2)A((3), where A(') E A, for i = 1, 2, 3. Let go = ADA(')g0 where go E G°A n

go E M°A.

a A2 We fix an identification

R+, and write x(02)

- (t(i) ... ) 11ac)

for tai), ... , 422 E R+. Let d" 42) = rli dx 4 . We choose an invariant measure dxaaz) on Aa for i = 1, 3 so that d- A(') = dx42)d-A(3), and dgo = Suppose that r is a Weyl group element which satisfies Condition (3.4.13) for M. Let 'yp,T,i(z) be a linear function for i = 1, , c such that (3.5.14)

0p(\(2))Tz = H(A((2))7p,r,i(z). i=1

We define (3.5.15)

LSpj7 = {z E * I -yp,T,i(z) = 0 for i = 1,

, c}.

3 The general program

93

Let dzI Ls,,r be the differential form such that dz = 11i=1 d'Yp,T,idzl Ls,,r

Let h E t* be the point such that 0,(

('))-Th

= op(A('))Bp(Aa2))P.

There exist numbers h1, , h, such that 0 (A (2)),h = (A(2))h; Consider I, v, v in the proof of (3.4.31). We consider the following condition.

Condition (3.5.16) For any T, a, there exists q E (LSp,T + h) n DoT such that if a E t* and IIaIIo is small, the function op(X043))Bp(A,43))T(9+«)+vep(t(9a))°T(q+«)+POZa (Ro(Dp, go)

is integrable on Ap2A3(K n MIA)T°q+S2o.

Note that OZ, (RD Dp, go) does not depend on A(2). The above condition implies that (LS,,, + h) n DoT # 0 for all 7-, a. In particular, (LSp,r + h) n DT # 0. Suppose that (3.5.16) is satisfied. We choose a subspace Hp,1. C Q so that tt = LSp,T ®Hp,r. Let z1 E (LSp,r + h), z2 E Hp,r. Let dzj be the differential form on LSp,T + h such that dz1(z-h) = dzILS,,.,. For Z2 E Hp,T, we define z2i = ryp,r,i(z2) for i = 1, c. We define dz2 = rji dz2i. Let Weput fp,T (Ao A(D3)1 901 w, Z17 Z2) Op(Ao$D3))T(Zl+z2)+PEBo,T(9p(9a),

= fp,r,I,a,v(A0A0

z1 + z2)A(w; zl + z2),

90, w, z1, z2) 9,(aoAD3))T(zl+z2)+PEB,I,°r,v(9p(ga),

z1 + z2)A(w; z1 + Z2).

Then we define the functions ep,T(A0 XD3)g , w, z2)+ 6Op,T,I,a,v(A0AD3)g , w, Z2)

by the integrals {

1

27r

(i)C

2 7r

-1

Jp,r(AoA(3),9a,w,z1,z2)dzl,

+h

Re(zl)=9

(3),

r+h.fp,T,I,a,v(AoAD90, w,zl,z2)dzl, Re(z1)=q

respectively, where we choose q in (LSP,T + h) n DT, (LSp,T + h) n DQr. The above functions are well defined if Re(w) >> 0 and IIRe(z2)IIo is small.

Proposition (3.5.17) Suppose that Condition (3.5.16) is satisfied for p. Let r be as before. Then (1) 8p (41, w, w), E`p,r(4D, w, w) are well defined for Re(w) >> 0, and

=p,T(',w,w T

Part I The general theory

94

(2) 'p,T ((D, w, w) is equal to ap(A Aa3))cp,T(Aa)a3)ga, w, 0)eza (Ra4)p, 9a)dxasdx4a3)d9a

JA,, A3 MO

-

Proof. We consider subsets of A22 like {A(2) I AO > 1 for all i}, {.X(2)

A(2)>1}etc.

I

A() 01 < -

Let Aa+ = {))(2) Aa2) >_ 1 for all i}. Since the argument is symmetric, we consider the set Aa+. We choose a E t* so that IIailo is small and I

(2))p.

0 (A((2))Ta _ (A(2))Pl ... (A DC 01

for some By assumption, .-n,T(AaA(')9a, w, z2)

ep,T,l,o,v(AaA(3)9a, w, z2),

_ I,o,v

and

Iep,,I,.,.(\ )43)ga,w,z2)I «9p(A where qa satisfies Condition (3.5.16).

It is easy to see that

(2)

))$p,T(9a,w) _

-

it v

i

T(AaA(3)9a,w,z2)fj(Aa2))zy'dz2.

c

Re(Z2)=a

Consider the function vp(AaA0 ))eza(R5-Dp,9a)e',T(AaA3)90, aw,

(3.5.18)

By Condition (3.5.16) and the choice of a, the order of the integration of the function (3.5.18) with respect to the set Aa+ and the contour {Re(z2) = a} can be changed. Therefore, p,T ((D, w, w) is well defined for all T, and (1) follows. The integral of the function (3.5.18) with respect to the set Aa+ and the contour

{Re(z2) = a} is equal to _1)c C

2

V

ap(AaAa ))oZ, (Rasp,

c

1

f i z2i

JRBc=2.>=P;

1

w, z2)

dz2.

The integral of (3.5.18) over the set {A(2) I A(2) < 11 A(2) > 1, .. , Aa2) > 1} etc. has

a similar expression.

Let e > 0 be a constant . If O(s) is holomorphic function on the set Is E C I -e < Re(s) < e} which is rapidly decreasing with respect to Im(s), fRe(8)=-

0(s)ds+J

28

e(s)=2

8

3 The general program

95

The statement (2) of the proposition follows by applying the above relation c times. Q.E.D.

When we apply our process later, we sometimes face strata Sp's such that there is a split torus in the center of Ma which fixes each point in Zp. In many cases (but not always), we can ignore such strata. This phenomenon is based on the following proposition.

Proposition (3.5.19) (The vanishing principle) Suppose that V)(P)

0,0(-7-GZ) = O(Z),

and Condition (3.5.16) is satisfied. Then if p

LSp,T + h, there exists a polynomial

P(z) such that P(p) 0 0, P(-TGZ) = P(z), and that if st' = ,b(z)P(z), then -p,T (D, w, w, 01) = 0.

Proof. We choose a linear function 1(z) so that 1(z) = 0 on LSp,,. + h and l(z) # 0. Let P(z) = l(z)l(-TGZ). Since -TGp = p, P(p) # 0 also. Since 1/)'(z) is identically zero on the contour, b(z) satisfies the condition of this proposition. Q.E.D.

Proposition (3.5.20) Let S#(w) = 6(w1) ... 8(w f). Then

fA/Gk

w(9°).'(9°, w)d9° = b#(w)A(w; P)

Proof. Let db be the measure on BA which is a product of measures we defined in § 1.1. for BA n GiA for i = 1, , f . Let BO = G°4 n BA. We choose a measure db° on BA so that

f

fBA/Bk

f(b)db=n1...nf JR1 f

f(c(A)b°)[JdxAidb°,

+BA/Bk

cn,(A1)...cnf(Af) for a1i... Af E Imo. where c(A) = Let pi = ani (I'i1' , ... , z:ini -1)' and µ = (p1i pi2µil1'

Boo -

r+1+...+nf-.t

i=1

µ f). We identify

x (Al /kx)nl+...+nf,

Then, by the choice of our and write b° = µt° where to E measure, db = P-2p fi , dxµijdxt°, where dxtO is the measure such that the volume of

(A1/kx)n,+...+nf

is 1.

Part I The general theory

96

By the Mellin inversion formula,

fG/Gk

w(g°)e(g°, w)dg°

n+.('JJ .+nf-f

_

w(g°) E t(g°'Y)z+°A(w; z)dg°dz )=4

1

JG/Gk

YEGk/Bk

fG/Bk

Re(z)=4

J e(z)=9 w(g°)t(g°7)z+PA(w; z)dg°dz

nl+...+nf-f

(2) 1

w(go)t(go)z+°A(w;z)dg°dz

7fY -1

1

= b#(w)

YEGkl Bk

n+...+nf-f

1

nl+...+n f-f

C2 7r

JR()=9

H µijj-+l-1 A ij

,

(

)

Hd x µzadz ij

= 6#(w)A(w; P) This proves the proposition. Q.E.D. Right now, (3.5.13) and (3.5.19) are the only general statements we can prove

without using properties of individual representations.

§3.6 The passing principle We have to consider various contour integrals in later chapters, and for that purpose, we prove a statement concerning a possible way to move the contour in this section. We first introduce some notation. Let p be a path, and T a Weyl group element which satisfies Condition (3.4.13) for Mp. We define ST = (STl, ... , STf )f STi =

(ST21,...

i ) ST2ni-1) E Cni -

by sril = ziT(n,) - ziT(n;-1),''' , Srini-1 = ZiT(2) - zir(1) Since the correspondence between z and sT is one-to-one, any function of z can be considered as a function of sT. For a function _f (z), let f (sT) be the corresponding function of s,.. So for example, we consider L(sr),AT(w;sr) etc. Let pT E be the element which corresponds to p. Let dsr be the differential form which corresponds to dz. Let H C C01+'.'+nf-f be a subspace of dimension d not necessarily going through the origin. We use the letter SH to express an element of H. We choose a differential form dsH on H. Let r(l),r(2) E H fl Let to be the line segment joining r(l), r(2), and D a domain containing lo. Let f(SH) be a meromorphic function which is at most of polynomial growth on any vertical strip contained in {SH I Re(SH) E D}. Let

r () w

a

=

1

27r

f (SH)V(SH) ds H fRe(sx)=r(i) W - L(SH)

3 The general program

97

fori=1,2. Suppose that there exists a polynomial P(sr) such that P(sH) f (sH) is holomorphic on {sH I Re(sH) E D fl H}.

Proposition (3.6.1) (The passing principle) Suppose P(pT) 54 0. Then there exists a polynomial P(z) such that P(-rGz) = P(z), P(p) 0, and if we define V (z) = P(z)V)(z), then E (f, r(1), w, 0') = =(.f, r(2), W, 01)

Proof. Let P(z) be the polynomial which corresponds to P(sT) by the above sub-

stitution. By assumption, P'(p) # 0. Let P"(z) = P'(-rGz). Since -rGp = p, P"(p)

0. Then P(z) = P'(z)P"(z) satisfies the condition of the proposition. Q.E.D.

Consider the following condition

0(Z)=0 (-rGZ), 1'(P):0. Consider the situation in §3.5. In later parts, we analyze the distributions

(3.6.2)

In this process, we have to study various contour integrals. What we can do by (3.6.1) is to replace the function O(z) if necessary to move the contour. By doing this, the condition (3.6.2) is not changed. Therefore, if for some L C Vks, we can prove that 8p (4), w, w) and

fA/Gk

w(91)OL(4),91)e(91,w)d91 ,., CGA(w;p)

ap(4),w), p

where ap w) is some distribution associated with the path p, then we can still conclude that a.($,w). w(g')OL($,g1)dg1 = JA/Gk

Caution (3.6.3) For the rest of this book, when we consider contour integrals of the form fRe(z)=q ' . dz or fRe(s,)=r ... ds r, we always consider w E C such that Re(w) > L(q) or Re(w) > L(r)

§3.7 Wright's principle In this section, we introduce a technique concerning the cancelation of higher order terms of distributions. We consider q,,,(s) for some n (see (1.1.2) for the definition). The use of the following proposition was suggest to the author by D. Wright after he read the first manuscript of [86] (it is used in [86]) Therefore, the author calls this proposition 'Wright's principle.'

Proposition (3.7.1) (Wright's principle) Let A E C, cl < 1 < c2, and e > 0 be constants. Let w E C be a complex variable. Let f (s) be a meromorphic function on the domain Is E C I cl - e < Re(s) < c2 + c}, having no poles along the lines Is I Re(s) = cl}, Is I Re(s) = c2}. Suppose that f (s) has a finite number of finite order poles and is rapidly decreasing with respect to the imaginary part of s. Also assume that there exists a constant A' such that 1

0' (s) .f (s)

2xRe(s)-C2 w - s - A ds - w -`q'1 - A

Part I The general theory

98

is meromorphic on a domain of the form {w I Re(w) > 1 + A - S} for some S > 0 and is holomorphic at w = 1 + A. Then f (s) is holomorphic at s = 1.

Proof. Suppose that f (s) has a pole of order j at s = 1. Since O (s) has a pole of order exactly one at s = 1, q5 (s) f (s) has a pole of order j + 1. Let 00

On (s)f (s) _

i--j-1

ai(s - 1)i

be the Laurent expansion at s = 1, where aj # 0. Then

f(s)

1

Le(s)_c2 w - 1 - Ads 1

f

-1

w-+h(w)+Eai(w-1-A)'

(S)Ads

JRe(8)=c1

i=j

for some rational function h(w) which does not have w - 1 - A in the denominator. Therefore, the proposition follows. Q.E.D.

We illustrate the use of this proposition by an easy example in the next section. Even though this proposition is very easy to prove, it does save a lot of labor for us in Chapters 12 and 13.

§3.8 Examples Before we start handling non-trivial cases, we consider two easy examples G =

GL(1) x GL(2), V = Sym3k2 and G = GL(2) x GL(2), V = k2 ® k2, applying what we have developed in this chapter. The first case was handled by Shintani in [64] and the proof in adelic language is in [83]. The prehomogeneous vector space G = GL(2) x GL(2), V = M(2, 2) has been studied by many people, but there does not seem to be any result which uses exactly the same formulation as ours. Cogdell [8] studied this case by the same method as ours (he used the group GL(2), but it is essentially the same as our case). However, his paper is written in classical language and he also chose a particular Schwartz-Bruhat function, so his analysis superficially looks very different from our analysis. Also the use of Wright's principle in the previous section simplifies the computation significantly. , wi) be a character of (Ax IV)', We first introduce a notation. Let w = (w1, and T E Y(A). For t = ( t1 , , ti) E Ai, we denote w(t) = i jj wj(tj). Let

s = (sl... si) E 0Definition (3.8.1) (1)

Oi(ww, t) = E T (tx). xE(kx )i

(2)

I tl sw(t)e(W, t)dxt.

Ei(W, w, s) = f Ax

/kx)i

3 The general program

99

If w is trivial, we drop w, and use the notation Ei(T, s) also. When i = 1, we define an analog of ZL+(-D, w, s) as follows. E1+(`I't w, s) =

I. p

/kx

ItI8w(t)ol(q1, t)dxt.

itI>1

If w is trivial, we may drop w and write E1+(,F, s) also. It is well known that E1+(WY, w, s) is an entire function, and

Ej('`,w,s)=E1+('I',w,s)+E1+(,Y,w-1,1-s)+S#(w)

(s(01

- s0))

where W is the standard Fourier transform of if. We refer to this formula as the `principal part formula for the standard L-function in one variable'.

Definition (3.8.2) For j = (jl,

so,i) E 0, we use , ji) E V j so = (so, 11 the notation Ei (j, ... ,ji)(1F, w, so) for the coefficient of 1!((s` - sl,0)ji in the Laurent expansion of Ei(W, w, s) at s = so. We also drop w in this notation when w is trivial. )

(a) The case G = GL(1) x GL(2), V = Sym3k2 Let T be the kernel of the homomorphism G -+ GL(V). We use the formulation in §3.1. Since dim G/T = dim V, Vk s = V,'. Therefore, this case is of complete type, and we use the notation Z((P, w, s), Z+ (4P, w, s) for the integrals defined in (3.1.8) for L = Vk 8.

In this case, we identify t* = to* with R so that a2(A-1, A)° = Aa for A E R. We

choose {a E R I a > 0} as ti. The Weyl group consists of two elements and the non-trivial element rG is the permutation (1, 2). We identify V with k4 by (xo, ... , x3) --+ fx(v1, v2) = x0v + xlv2 i v2 + x2v1v2 + x3v2. The weights of V are -3, -1,1,3. Clearly, 5Z \ {0} consists of two elements /31 = 3. Let 01 = (Nl),02 = ((l2). Let Pi3 = (01,51j),p2j = (D2,52j) be paths for j = 1, 2 such that sij(1) = 0 if j = 1 and sij(1) = 1 if j = 2. It is easy to see that Ma, = Mat = T and Walk = {(0) 0, x2, x3) I x2, x3 E k}, Y,,k = {(0, 0,x2, x3) I X2 E k", x3 E k},

Za1k = {(0,0,x2,0) I X2 E k}, Zalk = {(0,0,x2,0) I X2 E kX}, YD2k = Za2k = {(0, 0, 0,x3) I X3 E k}, ZO2k = {(O, 0, O, X3) I X3 E k"}.

Then by the general theory, Sark = Gk XBk YD k for i = 1, 2. We define a bilinear form [ , ] v by 1

1

[x, y]v = xoyo + 3x1y1 + 3x2y2 + X3y3

, y3). Then this bilinear form satisfies the property for x = (xo, , X3), y = (yo, [gx, tg-1]' = [x, y]', for all x, y. Let [x, y] v = [x, TGy]'. We use this bilinear form

as [,]vin§3.1.

Part I The general theory

100

We choose the constant in the definition of the smoothed Eisenstein series so

that C = C1 = 1. This implies that wo = 1. In this case, "rG = (1, (1, 2)) and CG = 921. We assume that (D E 9(VA) and MV,,I) = C By (3.5.3), (3.5.5), w, w), 8p,i w, w) are well defined for Re(w) >> 0 for i, j = 1, 2. Let - p;j J(4), g°) be as in (3.5.6). We define (3.8.3)

I0(4D,w) = f

w(g°)J((D,g°)dg°, p /C'k

I(4D, w, w) f

w(g°)J(, g°)8(g°, w)dg°.

Then by (3.4.34),

I(4i, w, w) . CGA(w; P)I

w).

By (3.5.9), (3.5.20), W, W)

EP (EP (-D, W, W) +

W, W))

P=P11,P12

+

EpEp (4), w, w) + CGA(w; P=pn,P22

By the remark after (3.4.7), "P12(4),w,w) =

P 2(4), w, w) =

Therefore, we only consider P11, 1121 Let d(A1) = (1, a2(AI 1, )1)), t ° = (t, a2(tl, t2)),

Where t,tl,t2 E A1, u E A, and to = d(A1)t°. We define d (.\1ttit2x2iAitt2x3), and on Ya2A = ZD2A by x3 -->

Lemma (3.8.4) ;p (4),W, w) - 0. Proof. By definition,

e(t°,w) - 400,W) = '{1},TG,1(t°,w). By (3.4.30), there exists a constant 6 > 0 such that e(t°, w) - 8N (t°, w) is holomorphic for Re(w) > wo - b and if M > wo, l >> 0,

eN(t°,w)I «A1. By (1.2.8), for any N > 1, IeYa1 (Re,, iD, t°n2(u))I << Al Nl sup(1, Al 3).

3 The general program

101

This implies that for any/ l >> 0, N > 1, Jey,1 (Ral -P, t°n2(u))I !e(t°, w) - 6°N(t°, w)JA2 «,i N+2 SUp(1, Al 3) Therefore, the integral defining Ep11(4P, w, w) converges absolutely for Re(w) > w° S.

Q.E.D.

We fix a Weyl group element T. Let sT = zr(2) - z.,(1). Then it is easy to see that sr-1 0 2 -3 -1 2 0 Tz+ (3.8.5) apll (t) = A1A1 = Al , apse (to) = A1, (t ) P = Al Easy computations show that 1w(t °)f (tt1t)d° = b#(w) f /kx f (q)dxq,

J 1/kx)3 A

f

A1 /kx )3

w(t 0)f (tt2)dxt ° =

6(w1)6(w2)

f

1 /kx

w1(q)f (q)dxq

for any measurable function f (q) on Al /k 1. Therefore, p11,T(-D, w, w) b#(w) 27r

/- fRe(s)=r>3 E1

P21,1 OD, w/,

_

,

2)AT

T )ds T,

w)

S(w1)Slw2) 3

(Rs s, al

1

21rv'----l fRe(sr)=r>2

E1(Ra2 w1 ST

+

1)AT(w; sT)dsT.

3

w, w) - 0 If T = 1, DT = IR and L(r) _ -r. Therefore, by choosing r >> 0, for i = 1, 2. Suppose T = TG. Then AT(w; sT) = ¢(sT)A(w; si). The point p corresponds to ) are holomorphic at sT = 1. sT = 1 . The functions E1(Ra1 sT - 2), E1(Ra2 C The passing principle tells us that by changing z// if necessary, l,T (4), w, w)

27r

(V

)=r y J 0
E1(Ral 4P, sT - 2)A, (w; sT)dsT

+ CGA(w; P)6#(w)E1(Ra1 4), -1) ,., CGA(w; p)b#(w)E1(Ra1 -D, -1).

Similarly, 3

Ep21,T (,D, w, w) - CGA(w; P)

b(wl)3(w2) E1(Ra2

w1, 3

These considerations prove the following formula I(-D, w, w) - CGA(w; P)b#(w) (E1(Ra1 &, -1) - E1(Ra11), -1)) 3

+CGA(w;P)b(w1)3 (w2) (E1(R2,wi1, 3) - E1(Ra2'P,w113) + CGA(w; P)T2b#(w)($(0) - 'D (0))

Part I The general theory

102

Therefore, (3.8.6)

,-1))

E1(Ra,

1°(,D, w) = 6# (w) S(w133b(w2)

+

(Ei(&2&wi1, 3)

- E1(Ra2-D,w1i 3)

+ X125#(w)*0) - co)). It is easy to see the following relations

E1(Ra1,P),, -1) = E1(Ral,P, -1), E1(Ra1 Via, -1) = A-4E1(Ral , -1),

El(Ra2,%,w1, 3) _ A-2El(Ra24)1w11

3),

El(Ra26),wi 1 , 3) = A ,P a(0)

= -P (0), 4a(0)

3), =A-4$(0).

Therefore, by integrating .\8I°(I)A, w) over the interval [0, 1], we get the following theorem.

Theorem (3.8.7) (Shintani) Suppose P = Mv,w w, s) =

El (R,, F,-1) s-4

a + 8(wl)S(w2)

Then

+ X2126#(w)

w, s) +

+b#(w)

.

4'(0) s-4 s /

($(0)

- E1(Ral-P,-1)'

E1(Ra2$,w1 1, 2)

3s-10

J

8

3 - E1(Ra2'D,w1, 3s-2

(b) The case G = GL(2) x GL(2), V = k2 ® k2 In this case, we have a canonical measure on the group GL(2)°' x GL(2)00. Let T be the kernel of the homomorphismG -+ GL(V). We consider (GIT,V) as before. A2))(a,6) = In this case, we identify t* = t°* with R2 so that (a2(A-', A,), a2(A2 1, A A2 We choose {(a, b) E R I a, b > 0} as t* . Any Weyl group element is of the form T = (T1i T2) where Tl, 7-2 are either 1 or (1, 2). For x E V, let xij be the (i, j)-entry. We use x = (xij) as the coordinate system of V. The weights of V are (-1, -1), (-1,1), (1, -1), (1, 1). Let ((al, bi), (a2, b2)) = ala2+b1b2. Then with the metric defined by this bilinear form, the weights of V can be identified with vertices of a square. The index set 'B\{0} consists of three elements ,l3 = (1,1), ,3' _ (1, 0),)3" _ (0,1). Let = (/3). It is easy to see that Ma = T and Yak = Zak = {x I x11 = X12 = X21 = 0}. Also, Zkg = {x E Zak I X22 E k" I. Let pi = (a, si) be a path for i = 1, 2 MQ,, MQ in §3.2 is SL(2) and Zp,, Zpu such that 51(1) = 0, 52(1) = 1. For can be identified with the standard representation of SL(2). So, Z7 k, Za ,k = 0. Therefore, there is only one unstable stratum.

3 The general program

103

We define the zeta function by the integral (3.1.8) for V,,S, and use the notation Z((D, w, s), Z+(,D, w, s) where w = (w1, w2) and w1i W2 are characters of Ax /kx However, since Vks = 0, the convergence of the integrals is not covered in (3.1.4). It easily follows from the following estimate (which is a consequence of (1.2.8)).

Lemma (3.8.8) Let 4) E 9(M(2, 2)A) and g° = (91, 92) E GL(2)°o x GL(2)0. Let gi = kia2(µitil, µi 2ti2)n2(ui) be the Iwasawa decomposition of Gi fori = 1, 2. Then for any N1i N2, N3 > 1, (µlµ2)-N1

OM(2,2)k (D, go) <<

SUP(1, µi lµ2) SUP(1, /-t1p2 1) SUP(1, µ1µ2) 1152)-N3

+ (11µ2 1)-N2(/ti

SUP(1, µ1µ2)

for g° in the Siegel domain.

Let [x, y]v = Ei,j xijyi,j for x = (xij), y = (yi3 ). Then [gx, tg-1y]v = [x, y]v for all x, y. We define [x, y]v = [x, rGy]v. Then by defining g` as before, [gx, g`y]v = [x, y] v for all x, y. We use this bilinear form as [ , ] v in §3.1. We choose the constants in the definition of the smoothed Eisenstein series so

that C = (1, 1). This implies that w° = 2. In this case, rG = ((1, 2), (1, 2)) and CG = Q2 2. We assume that 4D E ?(VA) and C By (3.5.5), 3p, (oD, w, w) is well defined for Re(w) >> 0 and E p, (4, w, w) _

rp, 4),w-1, w) as before. Let d(X1, \2) _ (a2(Al

1,

o

A1), a2(A2 1, A2)), t = (a2(41, t12), a2(t21, t22))

for al,.\2 E R+, tip E Al for i, j = 1, 2. Let to = d(.1i A2)t °. We define dxt o dxto in the usual manner. We define I°(1),w), I(4), w, w) by the formula (3.8.3) (the set Vks is different of course). Then I(l), w, w) ,,, CcA(w; p)I°(d), w),

and by (3.5.9), (3.5.20), 1(4), w, w) = =-P, (4), w, w) - Ep, (4b, w, w) + CGA(w;

The element to acts on Z-OA by X22 - A1A2ti2t22x22 Therefore, Ep2 (-D, w, w) = 0

unless w is trivial. We fix a Weyl group element r = (rl, r2). Let Si r= Zirl (2) - zir, (2) for i = 1, 2.

Let dsr = dsrldsr2. It is easy to see that (3.8.9)

av, (to) = A

2A2

d(A1,.2)rz+i = A1.,-1A2,2-1.

,

We make the change of variable µ = A1A2, A2 = A2 and q = t,2t22 Then `Sr1+l`8,2+1 = `5,-2-8,1 /sr1+1 1 2

2

Since

w(t JR+ xA1 /k x

°)µ3r1+1 p1(Ra

P, pt12t22)dxµdxt ° = 6#(w)El(Ro1P, sri + 1),

Part I The general theory

104

by the Mellin inversion formula, (D, W, w) =

f

8#(w) 27rv -1

sT1 + 1)A, (w; STI, STI)dsT1.

e(s,1)=r1>1

More precisely, we are using (3.5.17).

For T = (1, 1), ((1, 2), 1), (1, (1, 2)), L(r1i ri) = -2r1, 0, 0 respectively. We choose r1 = 1 + S for a small constant 6 > 0. Then L(r1 i r1) < wo = 2. So upl (4),W, w) " 0 unless r = TG. Suppose T = TG. Then AT(w; s,1, sT1) = (sT1)2A(w; sit, sil) Let

J(`), sit) = E1(Ra$, sil + 1) - E1(Ra-D, sit + 1). These considerations show that I(-D, w, w) ^ CGA(w; P)`3 6#(w)($(0) -.p(0))

+ 6#(W)

27r/

f

I/,

e(s,l)=r1>1

J((D, ST1)4'(ST1)2A(w; sil, sil)dsii.

Therefore, by Wright's principle, E1(RD$, 2) = E1(RDID, 2), and

I(,D,w,w) ^ CGA(w;P)9326#(W)($(0)-,D(0)) + CGA(w; P)6#(w) (E1,(1) (Ra &, 2) - E1,(1) (Ra D, 2))

.

This implies that IO(,p,W) = 932b#(W)+O)

-.p(0))

+ 6#(w) (E1,(1)(Ra$, 2) - E1,(1)(RA, 2))

.

The following relations are easy to verify. E1,(1)(Ra4)A, 2) = )1-2E1,(1)(RD D, 2) - A-2(log A)E1(Ra4), 2),

E1,(1)(RaCa, 2) = X-2Ei,(1)(Ra$, 2) +A -2 (log A)E1(RD$, 2). Therefore, by integrating a3I°(4a, w) over the interval [0, 1], we get the following theorem.

Theorem (3.8.10) (Weil-Cogdell) Suppose Mv,4,oP = 4. Then w, s) = Z+(4), w, s) +

+ b#( w)

-

w 1, 4 - s) + 2?28# (w)

2E1,(O)(ROP, 2)

(s-2)2

(0) - -(0) s-4 s

+ E1,(1)(RaD, 2) _ E1,(,)(Ra(D, 2)

s-2

s-2

E1,(o)(Ra-D,2) = 0. If for some v E 9A., 4)v E Co (V, ), then Therefore, the poles of the associated Dirichlet series are all simple. Cogdell [8] spent most of his labor trying to establish the cancellations of order two terms based on the equality 2) = E1(Ra$, 2) explicitly. However, we used Wright's principle, and were able to avoid this cancellation.

Part II The Siegel-Shintani case

Chapter 4 The zeta function for the space of quadratic forms We state the result of this chapter here.

Let G = GL(1) x GL(n), and Vn = Sym2kn, where n > 1. We consider the natural action of GL(n) on V. (if n = 1, a E GL(1) acts by multiplication by a2). We define the action of a c GL(1) on V,, by the ordinary multiplication by a. This defines an action of G on V,,. Let T C G be the kernel of the homomorphism G -+ GL(V). We consider the prehomogeneous vector space (G/T, VV) in this chapter. For a character w = (w1iw2) of GA/Gk, 4) E S°(VnA), and s E C, we consider the zeta function Zv, (4), w, s) which is, by definition, the integral (3.1.8) for L = Vim" if n 2 and for L = V2k if n = 2. We show the convergence of those integrals in §4.1, because it is not covered by (3.1.4). The case n = 2 is of incomplete type and requires an `adjusting term.' We define the adjusted zeta function ZV2,ad((D, w, s) in §4.2.

Let Sn,i C V n be the set of rank n - i forms. Then Sn,l, , Sn,n-1 are unstable strata. Let Yn,i = Zn,i C V. be the corresponding subspace for S,,,i. Then Yn,i Vn_i. Let Z.,.-2,0 C Zn,n-2 be the set of points of the form {(0, x1i 0)} by the above identification. We identify Zn,n-2,o with the one dimensional affine space.

For 4) E Y'(VA), Ri4> E 9'(Yn,iA) = 9'(Zn,iA) is the restriction of 4) to Y(Yn iA). For 4 E S°(VA) we define Rn,n-2 0$ E 9'(Zn,n_2,oA) by the formula

Rn,n-2,0(x1) = fR,n_2(O,xi,x2)dx2. If n = 3, we define +026(wl)6(w2)E1(R3,24),w1, 2)

F3(4',w,s) = Z

2s-3

s

+ 5(wi)b(w2)

+ 6#(w)

ZV2,ad,(0)(R3,14', (w1,1), 3)

s-3

3E1(R3,1,0ID, 2)

2(s - 3)2

E1,(1)(Rr3,1,0-D, 2)

+

2(s-3)

)

If n > 4, we define Fn (4), w, s) = Tnb#(w)

0)

+ E (n - i)93 8(w1)6(w2) IV _iRn,i4', (w1, 1), n 2 (2s - n(n - i) i#l,n-2 + Zln_2S(wi)S(w2) ZV,,ad(Rn,n-2I, (w1, 1), n)

s-n

)

Part II The Siegel-Shintani case

106

2 + (n - 1)6(w1)b(w2)

#()

+

Zy _1,(0)(Rn,14>, (wl, 1), n

n-1) 2

2s - n(n - 1) -2b- 1) + E1,(1)(R,_2 04>, n - 1)

2(s - n)2 (nE1(R4

)

2(s - n)

The notation Zv _1i(0)(Rn,14>, (w1i 1), n (n

1) will be defined in §4.4.

In this chapter, we prove the following theorem.

Theorem (4.0.1) Let n > 3. Suppose that M ,

Zv

('D'

w, s) = Zv

+(,D,

4> = 4>. Then

w, 3) + Zv +(4', w

1,

n(n + 1) 2

- s)

- Fn($,W 1, n(n + 1) - s) - Fn(4>,w, s). 2

Therefore, the adelic zeta function has double poles in general. However, it turns and the support of 4>v is contained in Vk',' for out (see (4.7.6)) that if 4> some v E fiIO, 6#(w)E1(R',n_2,04>, n - 1) = b#(w)E1(Rn,n_2,0 , n - 1) = 0.

We consider the case k = Q. The above remark implies that the poles of the associated Dirichlet series are all simple. Consider w = 1, and drop w from the notation Zv (4), w, s) etc. Let V C VAS be the set of forms with signature (i, n i). We choose a measure on (G/T)A so that the zeta function defined by (3.1.2) coincides with the integral in (3.1.8). Let µ,,,(x) etc. be as in §0.3. Consider such that 4P is the characteristic function of Vy and 4>, E Co (Vi). 4> Then it is easy to see that

Xi(4', dn) = f D(y.)dy. = $.(0) Since Z , (4), s) has the residue Z7n4) (0) = Tn(D,, (0) at s = dn, the Dirichlet series

xEGz\Vz

µco(x)bx,oo

1

o, x)

IA(x)I00

has a simple pole at s = do with residue Zn. This means that the Dirichlet series ltoo(x)bx,oo E xEGz\Vz' 000(x) IA(x)Ion 1

with residue -. Therefore, by the Tauberian theorem, has a simple pole at s = we have the following theorem.

4 The zeta function

107

Theorem (4.0.2) (Siegel [69]) Let n > 3. Then lioo(x)bx,oo Doo (x)

:EGz\vi°nvi Io(=)I-<X

ti

n

7.-X

For discussion on the error term estimate, the reader should see [65]. In order to state the principal part formula for the case n = 2, we need the `adjusting term.' For this reason, we state the principal part formula of the case n = 2 in §4.2. In this case, the filtering process was carried out by Datskovsky, and (0.1.2) is the asymptotic formula for GQ-equivalence classes. For this, the reader should see [9].

§4.1 The space of quadratic forms Let G, Vn be as in §4.0. Let T C G be the kernel of the homomorphism G -y GL(Vn). We consider Vn as the space of quadratic forms in n variables v = (Vi,.. , vn). Let do = n 2 1 . We identify Vn with the space of column vectors of dimension do by the following map

x = (x1m)1<ji<j2
Let To be the image of GL(1) in G. For (t, g) E GL(1) x GL(n), we define Xv. (t, g) _

,

tn(det g) 2 if n is odd, and X (t, g) = t 2 (det g) if n is even. We can consider Xv as a character of GIT. The character Xv is indivisible. It is easy to see that (G/T, Vn, Xv) is a prehomogeneous vector space and Vnk' consists of nondegenerate forms. For n = 2, V2k consists of forms without rational factors. We use the formulation in §3.3. Since X*(G/T) is generated by one element, G' = GA and t = to. If there is no confusion, we identify GL(n) with its image in G by the natural inclusion map. Let Yn,i = Zn,i be the subspace of Vn spanned by the coordinate vectors of xl,n's for 1, m > i. We identify Zn,i with Vn-i. Let Sn,i be the subset of Vn consisting of rank n - i forms. Let Zn ik be the set of rank n - i forms in Zn,ikLet Zn n_2k be the set of rank 2 forms without rational factors in Zn,n_2k. Let Zn,n_2,ok = Zn,n_2k \ Zn n-2k. Let Zn n-2 ok C Zin,n_2k be the set {(0, x1i 0)} when we identify Zn,n_2 = V2. Let Z';' a_2 ok = {(0, x11 0) ( x1 E kx}. We define Sn n-2 k = Gk ' Z, ,n-2k Let Pn,i C G be the maximal Sn,n-2,stk = Gk ' ZSnnparabolic subgroup whose Levi component is GL(1) x GL(i) x GL(n - i), where GL(i) x GL(n - i) is imbedded in GL(n) diagonally in that order. Easy considerations show the following lemma.

Lemma (4.1.1) iT (1) V n k \ {0} = Vnk TTTT 1.1 lli

1 Sn,ik (2) Sn,ik = Gk XPn.,ik Zn ik for all i.

We define a bilinear form [ , ] v as follows. n [x, Y1 V.

xiiyii + 2 E xijyij.

= i=1

i<j

Part II The Siegel-Shintani case

108

Then [gx, tg-ly]v = [x, Y1' for all x, y E Vn. This can be proved easily by reducing to the case n = 2. We define [x, y] n = [x, rrGy] n. We use this bilinear form as [ , ]v in §3.1. Let JFn, g = (A, g1), g` etc. be as in Chapter 3. Let E .Y(VnA). Then the Fourier transform of the function 4P(g ) is A-dn9n4)(9 ). If n = 2, we use the notation etc. for the integrals in (3.1.8) for L = V2k. The convergence of this integral for Re(s) >> 0 is covered in (3.1.4).

Definition (4.1.2) Let n > 3. For (D E 9'(VnA), w = (w1, w2) as in (3.1.8), and s E C, we define (1)

Zn(",w,s) = f

(2)

Zn+(4, ")IS) =

9w(gl)Ov k( 9)d9,

Ge/C'k

f

-

GA/Gk

sw(gl)Ov;°

9)d9

A>1

We prove that if n > 3, Zv, (4), w, s) is well defined for Re(s) > dn, and Zn+(4, w, s) is an entire function. Note that this does not follow from (3.1.4).

Lemma (4.1.3) (1) Suppose n _> 3. Then there exists r = (r1, , rn_1) E Rn-1 such that rj < j(n - j) for all j, and that for any N > dn, there exists an estimate of the form n-1 119

vnk ((D, Akt) I << A-N [J (tit-1

i=1

for (2) If n = 2, there exists a similar estimate as above for Ovvk (,D, Akt). Also for any

N>3,

«A-Nt1t21

lOvzstk(4),Akt)I

for A E R+, k E St, t = a2(tl, t2) E T°+.

Proof. Let V = Vn. We choose 0 <' E S°(VA) so that I-D(akt)I << T(At) for k E S2, t E T. We consider T+ as a subset of GL(n)°A. Let ylm E t' be the element determined by the coordinate xlm. , An_1 = to-ltn 1. Then Let t = an(tl, , tn) E T+, and Al = t1t2 1, n-1

1

t1 = Al n ... An-1

_L tj=alnn=1...Aj-1

-L ...71-1 Ain

n-1

1

to=Al^...an_1 Suppose tltm = fj A"-j. Then 2 n-j n

lmj =

n2 n

n

.9

l< m <j,

l<j<m, j
4 The zeta function

109

If x E VkS, the convex hull of {rytm xi,,,, 54 0} contains the origin (but not necessarily a neighborhood of the origin). For I C {(l, m) 1 1 < 1 < m < n}, we define VI = {x E Vk I xlm # 0 if and only if (1, m) E I}. We consider I such that the convex hull Convl of {71m 1 (1, m) E I} contains the origin. We define I

CI('I',9) =

'@(gx)

xEV,,nvl

Then Ovk=(w,At) <

OI(I,At).

We fix I and estimate (3I(qf, At). Since Convl contains the origin, we choose clm > 0 so that >(l m)EI Clm Ylm = 0. We define C = E(l,m)EI Clm > 0. By (1.2.6), for any M > 0,

e1(w, At) << X-CM II t-M E(1,m)EI C'-'Y'- 11 (l,m)EI

(Atttm)-1

(l,m)EI

=A-cm-#' 11

(titm)-1.

(l,m)EI

We consider the possibilities when

E Xtmj <- -j (n - j)

(4.1.4)

(l,m)EI

Suppose that (4.1.4) is satisfied. We first consider the case n > 3.

Case 1. j < 2

Since n-2j>Oandn-j>0, Xlmj > -

Xlmj

j(n-j)(n-j+1) n

(l,m)EI

If equality happens, then (1, m) E I for j < l < m, and (1, m) V I for l < m < j. But

j(n-j)(n- j +1) > -j(n-j),

n and equality happens only when j = 1. Therefore, (4.1.4) implies that j = 1. If j = 1, n - 2j > 0. So (1, m) I for l = 1 < m. Hence, the only possibility for (4.1.4) to happen is when j = 1, and I = {(l, m) 12 < l < m}.

Case 2.j>2. Since n - 2j < 0,

E

(l,m)EI

X1mj>-j(n-j)(n-j+1)+(n-2j)j(n-j)=_j(n-j)(j+1) n

n

If equality happens, then (1, m) E I if and only if m > j. But

j(n - j)(j + 1) > -j(n - j), n

n

Part II The Siegel-Shintani case

110

and equality happens only when j = n -1. Therefore, (4.1.4) implies that j = n -1

and 1={(l,n) I1=1,...,n}. We have proved that if I#{(1,m)12<1<m -j(n - j) (1,m)EI

for

We consider the case n = 2. Clearly, t2 = A,, t1t2 = 1, t2 = ail. Therefore, 11(i,m)EI tltm = A for some p< 1 if and only if I = {(1, 2), (2, 2)} or I = {(2, 2)},

If n > 3 and I = {(l, m) 1 2 < l < m < n} or {(l, n) l = 1, , n}, Convj does not contain the origin. So we do not have to consider such possibilities. Since I

#I < do and the choice of M > 0 is arbitrary, for any N > dn, we can choose M > 0 so that CM + #I = N. Let rj = infl(>(l m)EI Xlmj ). Then this r satisfies the condition of (1). If n = 2, and I = {(2, 2)}, then Convl does not contain the origin. However, Convj contains the origin for I = {(1, 2), (2, 2)}. But if x11 = 0, x E V2Stk. So we do not have to consider I = {(1, 2), (2, 2)} for the first statement of (2). Since lt2 3 = t2 2 = Al 1 the second statement of (2) follows also. 11(l,m)EI(tlt'm)_1 = ti Q.E.D.

Proposition (4.1.5) If n > 3, Zv. (4D, w, s) converges absolutely and locally uniformly for Re(s) > dn, and Zv +(4), w, s) is an entire function. Proof. Let t, Aj be as in the proof of (4.1.3). We only have to show the convergence of

ARe(8)Ov°k (41,

21t)t-2Pdx Adx

t.

It is easy to see that t-2" = rjj X, j(n-j). So if A > 1, we choose N >> Re(s) in (4.1.3), and this proves the convergence of Z , +(1), w, s) for all s. If A < 1, we choose N = do + 6, where S > 0. Then the integral over the set {A I A < 1} converges absolutely if Re(s) > do + S. Since S > 0 is arbitrary, this proves the proposition. Q.E.D.

Let C(G°A/Gk, r) for r = (rl, , rn_1) E Rn-1 be as in (3.4.17) (we consider the path (0, 0)). By considering (4.1.3) on GA, we get the following lemma. Lemma (4.1.6) If n > 3, Ovnk (4), g°) E C(G°A/Gk, r) for some r such that rj > -j(n - j) for all j. Also Ov2k (4), g°) E C(G°A/Gk, r) for some r > -1.

§4.2 The case n = 2 In this section, we consider the case n = 2. We use the formulation in §3.1. In this case, Vk' consists of forms without rational factors. Let V = V2 in this section. Let t* = t°*, t°,1° etc. be as in §3.8(a). We identify V with k3 by (SO,x1,x2) - fx(vl,v2) = x0v1 + xlvlv2 + x2v2.

4 The zeta function

111

The weights of V are -2,0,2. Clearly, '$ \ {0} consists of one element ,Q = 2. Let

a = (Q). Let pi = (D,si) be paths for i = 1,2 such that sl(1) = 0,52(1) = 1. It is easy to see that M, = T and Yak = Z,k = {(0, 0,x2) I X2 E k}, Zap = {(0, 0,x2) I X2 E k" E k}.

Then by the general theory, Sak = Gk XB, Z. Let Vstk be the set of rational forms with no double factors but with rational factors. Then Vk \ {0} = Vk 11 Vstk ll S1k.

Let H C G be the subgroup generated by T and (1, TG). We define Yok = {(0, xl, x2) I x1, x2 E k}, Yok' = {(0, xl, x2) I x1 E k", x2 E k},

Zol k = {(0,x1, 0) I xl E k}, Z'os = {(0, xl, 0) I xl E k" }.

Then it is easy to see that Vtk = Gk xHk Z'ok

Consider the function a(u) in §2.2.

Proposition (4.2.1) Let F C Af be a compact set, and L C k C k. a lattice. Then there exists a constant N > 0 such that

1 > a (u) > N fi (I xI x

vEmR

2 11 (IxIy + Iuly)-l'

+ Iuly) v

vE9Rc

for of E F, x E L \ {0}. Proof. Clearly, sup (1, i Iv) = Ixly 1 sup(lxl,,, Iulv). There exists a finite set S C Off such that if v E Of \ S, xthen x, uv E ov for x E L \ {0}, u f E F. So if v E fit f \ S, then Ixly, Jul, < 1. Clearly,

[J sup(lxly,

NO

vES

is bounded by a constant M > 0. So

fl sup(l, Ixly) <_ M 11 Ixly 1= MIxJ o. vE9R f

vE9R f

Therefore,

a(x)>M-'IxIO-01 f (1+IxI2V) VE9RR

= M-'

11 vE9RR

2

J

(1+IxIy)-1

vE9Rc

(IxIy + Iuly) 2 II (IxIy + July)-1 vE9Rc

Since the inequality 1 > a(x) is clear, this proves the proposition. Q.E.D.

Part II The Siegel-Shintani case

112

Proposition (4.2.2) Let Xv = {9° = (t,92) E G°A I ltl(92)1 >_

a(tl(92)-1t2(92)u(92))}.

Then for any measurable function f (g°) on G°A/Hk,

f ,°1/Hk.f(9°)d9°

= f

.f(9°)d9°,

XV/Tk

if the right hand side converges absolutely.

Proof. Note that the above condition does not depend on the choice of the Iwasawa decomposition of 92. We fix a place v E 'YROO. The subset {9° = (t, 92) E G°A ltl(92)1 = a(t1(g2)-1t2(g2)u(g2))} is a measure zero set, because if we fix u(g2), and Itl(92)Iv for all v c 991 \ {v}, there are finitely many possibilities for Itl(92)Iv. Consider a TG-orbit {g2ig2TG}. we can assume that ltl(92)1 > It1(g2TG)I. Then I

if 92 E GL(2)°A,

92TG =k(92)TGa(t2(92), tl(92))tn(u(92))

=k(92)TGn(tl(92)-1t2(92)u(92)) . a(t2(92),t1(92))

=k(92)TGklt(tn(tl(92)-lt2(92)u(92))) - n(ul) . a(t2(92),tl(92))

for some k1 E K fl G2A, ul E A. Therefore, Itl(92TG)I = It2(92)Ia(tl(92)-1t2(92)u(92))

By assumption, ltl(92)1

It2(92)Ia(tl(92)-lt2(92)u(92)), and It2(92)I =

Itl(92)I-1

So if g° = (t,g2) and ltl(92)1 > Iti(92TG)I, then g° E Xv. Hence, Xv surjects to G°A/Hk. It is easy to see that Xv is invariant under the right action of Tk. Since a(tl(g2)1t2(g2)u(g2))} is a measure zero set, this proves the {9° ltl(92)1 = I

proposition. Q.E.D.

Definition (4.2.3) Let v be a place of k. Let IF, T, be Schwartz-Bruhat functions on A2, kv respectively. For s, sl E C and a character w of A" /k", we define (1)

(2)

TV,v('yv,w,s,s1) = f

, xk

ItvIvw(tv)av(uv)814'v(tv,tvuv)dxtvduv,

ItIsw(t)a(u)s1 (t, tu)dxtdu,

TV (IF, w, s, s1) = J x xA

(3)

Tv+(`Y, w, s, sl) =

"x xA

I tI $w(t)a(u)s1 WY(t, tu)dx tdu,

i

(4)

T(111, w, sl) =

f

w(tl)a(u)311Y(tl, tlu)dxtldu. 1xA

4 The zeta function

113

If lY = ®1Y,,, TV (IF, w, s, s1) has an Euler product as follows

s,sl) = kl - 162t k1 f TV,v('I'v,w,s,s1) V

In (4.2.4)-(4.2.6), we drop the index v from tv, uv, because the situation is obvious.

Proposition (4.2.4) Let v E JJIf. Suppose that 'v is the characteristic function of ov and w is trivial on kv . Then 1-q(s+s1)

v

Tv,v N'V' w, s, sl) _

(s+31-1)).

qv

(1 - qv s)(1 Proof.

TV, ('F"' w, s, s1) =

Jo Itly-l sup(i, jut-l lv)-3ldxtdu 00

00

_

qv

a(s-l)

(1 - gv l)qv 6 sup(1, qv

(6-a))-s1

a=0 b=0 0o

_

qv

a(s-1)(1

a-1

00

a=0

b=0 as1

(gv(s1-1)

Cqv

E qv -(s-1)(j -

qv1)

qv1-l - 1

-

vq-1)

a=O =0

(q_a(s+s1-1)(qa(s1-1)

00

E

qv1-1

a=0 00

qv

as(gv1-l

- qv

1)

4

a=0 qvs1-1

-1

- qv v1-1

a(s+s1-1)(1

-

+

a- l 1 qv - qv

+ qv asl

- qv

1)

1-qvl - 1)

s)(gv1-1

1 - qv

1)

1

-1 - qv

(1 - qv

E qv b) 6=a

00

-

a31+b(s1-1) +

- qv 1)(E qv

(1 _ qv

(3+31-1))(gv1-1

- 1)

(s+s1)

(1- qv s)(1 - qv

(s+s1-1)

). Q.E.D.

Proposition (4.2.5) Let v E'JJI f. Then Tv,v(W , w, s, sl) is a rational function of qv 3, qv 31 for all v, holomorphic for Re(s) > 0, Re(s) + Re(si) > 1.

Proof. Choose an integer l so that Tv(x1,x2) is constant on the set {(x1ix2) E A2 x11y, Ix2Iv < qv a}. Then, w(t)ItIv lav(t)311I'v(t,u)dxtdu Cqv

= Cqv

l d tdu w(t)jtjv lav (u) t1

is

(s

1 - qv (1 - qv s)(1

-

-(s+s1 )

q,-(s+"'- 1)) k>1

w(t)dx t

Part II The Siegel-Shintani case

114

for some constant C. There exist open sets U1, , UN C k,, such that Ui C qv io,x, for some mi < 1, and that if (X1,X2) is in the support of Wv and'x21v > qv-1, then x2 E Ui for some i. By replacing U1i , UN if necessary, we choose an integer l' > l so that l1 is constant on the set It I Itl < qvl' } x Ui for i = 1, , N. Then u 31

iti.,
w(t)Itl$-'a (t) F,(t,u)dxtdu

gv `
N

_

Cigv

*ni(81-1)qv

qv (8+s1-1))-1 /

w(t)dxt

ov

i=1

for some constants C1,

, CN.

Clearly, (u,-)81

w(t)ItI , lav t

qlv(t,u)dxtdu

qv duly

is an entire function. Therefore, (at)Sl

/ti

_ w(t)ItI;-lav

'F,(t,u)dxtdu

sgv

qv `
is a rational function of q,-s, qv8', holomorphic for Re(s) + Re(sl) > 1. Similarly, the integral over the set {(t,u) E k,, x kv I qvi < ItJv, Jul,,, < q,-,'l is a rational function of qv 8, qv8', holomorphic for Re(s) > 0. Since the integral over is an entire function, this proves the the set {(t, u) E kv x kv q;' < ltlv, proposition. I

Q.E.D.

Proposition (4.2.6) Suppose v E 9Jt,,,. Then for any w, Tv,v(Tv, w, s, s1) is holomorphic for Re(s) > 0, Re(s) + Re(si) > 1, and can be continued meromorphically to the entire C2. Proof. First we consider the case v E 9J1R. Since w(t) = 1 or sign(t) for t E R>1, we may consider integrals of the form

fof_oot8(1 + u2)- w(t, tu)dxtdu. C'O

The above integral is equal to the integral

t8+s1-1(t2 + u2)-zWv(t, u)dxtdu o

FOO f°° r z

= J0 J

iz

re-l cos881-2 B1Yv(r cos 0, r sin 9)drdO.

4 The zeta function

115

We define 100

fl(Wv,Z1,z2)=

r2

Trz1COS129Wv(rcos0,rsinO)drdO,

J

2

too

2

f2(1Yv,z1,z2)=J

,rZlcosz20sin9AYv(rcos0,rsin9)drd9. 2

Then 1

fl(Wv, z1, z2)

(fl(a1'Fv, z1 + 1, z2 + 1) + f2(82Wv, z1 + 1, z2)),

zl + 1

f2zl, z2) = Z2 11

(f2(a1wv, z1 + 1, z2 + 1) - fl(a2wv, z1 + 1, z2 + 2)),

where a1i a2 are partial derivatives with respect to the two coordinates. The meromorphic continuation of Tv(Wv,w, 5, Si) for a real place v follows from the above formula. w(re2nV---1e) = e2""v----io. Next we consider an imaginary place. Suppose that

Let t = r cos 9e2i, u = r sin

9e2"1/'- 102. Then

d"tdu = 4r tan9drd9d¢1dq52.

Therefore, oo

T, (IF,., w, s, sl) = 4

p2

010 Jo

2n

p2"

F(r, e, 01, 02, s, sl)drd9d0ld02,

0

where 9e2"V/--1nk1

=r2s-l

F(r, 9, 01) 02, s, SO

Costs+2s1-3 9 sin

9e2"N---1"0',

x 11v(r cos

r cos

9e2"Vr-l"O2 ).

The meromorphic continuation of TT (tII0, w, s, Si) is similar to the real case, and it is holomorphic for Re(s) > 0, Re(s) + Re(si) > 1. Q.E.D.

Suppose IF = OT,. If w is a character of A' /kx, w is trivial for almost all v. Let 9)1oo C P C ))t be a finite set such that w is trivial on kv and 4f, is the characteristic function of the set ov for v 0 P. We define (k,P(z) = jjv0P(1-qv Z)-1, andTv,P(W,w,s,sl) = IOkI 2ttk111vEPTv(1I'v w s,51). By (4.2.4), TV (`I', w, s, Si) = Tv,P(`I') w> S, Si)

(k,P(3)Ck,P(8 + 81

(k,P(S + Si)

Definition (4.2.7) For SY, w, s, sl, and P as above, we define (1)

Tv(q', w, s) _

(2) Tv,P(IF,w, s) _

d

d dsl s1=0

TV (IF, L.), S, S 1), TV' (IF, U))

d dsl s1=0

Tv,P(W,

Si).

ds1 s1=o

TV (`I', w, Si),

Part II The Siegel-Shintani case

116

By (4.2.4)-(4.2.6), we get the following proposition.

Proposition (4.2.8) For any w, T('Y, w, s, s1) can be continued meromorphically to the entire C2 and is holomorphic for Re(s) > 1, Re(s) + Re(sl) > 2

Let b ° = t °(1, n2(uo)) E B00, and b° = d(\,)-b° E B. We define measures db °, db° on BA, BO in the usual manner (db° = dx Aldx t 0duo)

Proposition (4.2.9) Letw= (W1,W2) be a character of (Ax/kx)2. Then (1)

i/T,.

J1

asw(b °)ezo(4i, Ab 0)a(uo)sldx.db 0

f3A

converges absolutely and locally uniformly for all s, si E C, f1

(2)

J0

f

Xsw(b °)Oz,

°)a(uo)37d Adb °

0/ Tk

converges absolutely and locally uniformly for Re(s) > 2 + e, Re(si) > -e, where e > 0 is a constant, and w(b0)ezo(-D,b0)a(uo)sldb0

(3)

converges absolutely and locally uniformly for all s1 E C.

Proof. Let o, = Re(s), of = Re(si). There exist Schwartz-Bruhat functions 4)1i (D2 > 0 on A such that E) z,, (4), A0) << 1 41(AX1)4)2(AXluo) x1Ekx

Since the order of the integration and the summation can be changed, we can make the change of variable ul = Axlu0. Then (1)-(3) are bounded by constant multiples of the integrals

f Ilf J

'Dl(Ax1)(D2(ul)a(A-1xllul)"d"\dul,

Au-1

A

x1Ekx

E 4)1(AX1)4)2(u1)a(A-lxllul)6ldxAdul, x1Ekx

JAx1Ekx Y'

4)1(xlA2(ul)a(xllul)"Idol,

respectively. There exist a lattice L C k and a compact set F C A f such that 'D1(Ax1)4?2(ul) _

0 unless x E L, ul f E F. Since the finite part of J.ul is the same as the finite part of ul, by (4.2.1),

axlw + IA-lull)-1

a(A-1x11211) ? N II (Ix1w + IA-lull) vEmR

vEfilc

4 The zeta function

117

for some N > 0. If o1 > 0, then a(A-lxl lul)°1 < 1. If of < 0, since (Ixlly + IA-lu'lly) 2 < At k:Q av(Ax1)°lav(ul)o1 for

v E 9A98,

2°,

(Ixlly + IA-lulls)-°1 < A[k:Q[av(Ax1)°'av(ul)°1 for v E

,

we have the following inequality

aWixl lul)°1

A°' 11 av(Ax1)°1av(ul)°1. v EM

Therefore, the proposition follows from (1.2.6) (consider A = 1 for Q.E.D.

For later purposes, we fix some notation. Definition (4.2.10) For (D E VA, we define Rod E Y(YOA), R''

E 9'(ZZA) by

Rol)(x, y) = P(0, x1, x2), Ro,D(xl) = f(0,xi,x2)dx2. The proof of the following lemma is an easy consequence of the definition.

Lemma (4.2.11)

(1) F (2)

f

f

a sw(b o)ezb (41, Ab o)a(uo)sldx Adb o =

b(w1w21)7,(Ro(P,

A/Tk

w(bo)Ozo(41,bo)a(uo)sldbo = A/Tk

Definition (4.2.12) For 4), w as above, and s E C, we define Est(4)>w>s) = 6(w1w2 1)T1(Ro4),w1i

Let

1-s ). 2

00

Est(4),w, Z) = E Est,(i) (4),w,1)(z - 1) i=o

be the Taylor expansion around z = 1. Then (

-s`w12 2

1

_

Definition (4.2.13) ust(4', w, w) = f

w(go)Dv,st(4), go)8(go, w)dgo. A IGk

wl> s, sl),

Part II The Siegel-Shintani case

118

Then by (3.4.31),

,

t(4), w, w) is well defined for Re(w) >> 0. As in §3.8,

-pi (1), w, w) is well defined for Re(w) >> 0 for i = 1, 2, and

We define

J8(4', g°) = E ($((g°)`x) - -b(g°x)) , XEGk\Vk

w(g°)J8(,g°)dg°,

I°(-D,W) = f A/Gk

I(4), w, w) = f A/Gk

w(g°)J8(4), g°)e(go, w)dgo.

Then 1(4), W, w) ,,, CG (W; P)I°(1,W),

and by (3.5.21) and the argument of (3.5.9), I (d), w, w) = Epi ($, w-1, w) - Epl (4), W, w) + CGA(w; P)9J2S#(W)($(0) - 4)(0)) _1

Definition (4.2.14) Let 1b, w, s be as before. We define lS(W1W2 1)

ZV,ad(C W, 8) = Zv(D,W, s) -

TV(ItA,Wl, s)

2

We call ZV,ad(1b,W, s) the adjusted zeta function, and b("12 21)Tv(R°4), w1i s) the adjusting term. We devote the rest of this section to the proof of the following theorem.

Theorem (4.2.15) (Shintani) Suppose P = M ,,D. Then ZV,ad(C W, S) = ZV+(qb,W, S) + Zv+(4', w-1, 3 - s)

- 8(wl2 w2 1) (Tv+(Ro(b, wl, s) + Tv+(Ro$, wi 1, 3 - s)) (U3 (0)\ +28#(W) b(Wl)S(W2)

+

2

Wl 1, 1)

(s-2)2

6(W1)6(W2) 2

+ 1)

(s-1)2 (E',)(R,w1,

+

s-2

E1,(o)(R1) s-1

1, 1)

f .

Since the first four terms of (4.2.15) are entire functions and we proved the meromorphic continuation of Tv (R°4), wl, s) in (4.2.8), Zv (-D, w, s) can be continued meromorphically everywhere.

4 The zeta function

119

Corollary (4.2.16) The adjusted zeta function ZV,ad(4), w, s) satisfies a functional equation ZV,ad(', w, s) = ZV,ad(D, w _ 1,3-S).

We first consider

p, (4), w, w). The element to acts on ZSA by x2 -* )1b2tt2x2.

Let T be a Weyl group element. Let sT be as in §3.8 (a). It is easy to see that apl (to) = A2 (to)TZ+P = A '

'

Also if f (q) is a function on Al /kx,

f

w(t°)f(ttz)dxio

A1 /kX)3

= 6(co1)6(w2) f 1 /kX

wl(q).f(q)d"q.

Therefore,

w, 'w)

p1,7-

=

6(W2

1)6(W2) 2 27r

1

1

f

E1 (Ray w1

sT + 11

Re(s,-)=r>1

2

l AT(wjsT)dsT.

Ifrr=1,wecanchoose r>> 0andL(r)=-r<<0. So Bp,T(4D,w,w)-0. Next, we consider °st(-D, W, W).

Proposition (4.2.17)

w, w) - E

1

T

-1

J e(sr)=r>1

w, z)Ar(wi sT)dsT.

Proof. By the standard consideration,

w(g°)ev,(, g°)e(9°, w)dg°

=st(4), w, w) = f A/Gk

fGl/Hk JXV/Tk

w(g°)Oza

g°)e(go,

w)dgo

w(g°)Ozo (iD, g°)e(g°, w)dg° w(b°)Ozo (4), b°)e(b°, w)db°.

'XvnBA/Tk

Lemma (4.2.18) The integral fXy nBA/Tk

w(b°)Ozo (,D, b°)(oo(b°, w) - eN(t°, w))db°

is holomorphic for Re(w) > 0.

Proof. As in §3.8 (a), there exists a constant 6 > 0 such that e(t°, w) - &N (to, w) is holomorphic for Re(w) > w° - 6 and if M > w°, l >> 0, I .-(to, w) - eN (t°, w) I << All.

Part II The Siegel-Shintani case

120

Since tl,t2 E A' IV which is a compact set, by (1.2.3), there exist SchwartzBruhat functions 1D1i 4D2 > 0 on A such that

Oz., (4)fb1)e1(bofw) << E 4D1(xl)4'2(xluo)Ai x1Ekx

Then the above integral is bounded by a constant multiple of 2duo

JAx1Ekx E =

xlx

fi(xi)2(xiuo)a(uo)duo

f

=

2dul,

where ul = x1uo. Therefore, the convergence of the above integral follows from (4.2.9). This proves the lemma. Q.E.D.

If f (q) isf a function on A1/k<, wl(q)f(q)dxq.

f1/kx)aw(t0)f(tt1t2)dxt0 = 6(wl)6(w2) J1/kx

Also, `3T_ldx/1\1

I,>

^ Ck(up)

=

a(uo) 2 ST - 1

Therefore, by the above lemma and (4.2.11), 1

st((Df U), w) T

ST)dST.

ST -

27rV -1 fRe(sr)=*>1

This proves Proposition (4.2.17). Q.E.D.

It is easy to see that

A('wfs)ds,. ^ 0

1

27r

IRe(s4r> 1

sT

if r Let T = TG, and J(-D, w, sT) -

6(wl 21) (E1

(R&wif 2J - E1 (RwlT + 1 2

1

+ ESt(sfw )s ) ST-1

- F-St(-D, wf'ST) ST-1

4 The zeta function

121

By the above considerations, I0(oD, w, w) ^' CGA(w;

4D(o))

J(,D, w, s,)O(s,)A(w; sT )dsT. 1 J 27rV -1 Re(s,)=r>1

+

By Wright's principle, J(4P,w, sT) must be holomorphic at sT = 1. Since

El(IF,w1i

2E1,(-1)(W, w1,1)

sT + 1 2

sT - 1

)

+ E1,(o)(W,wl,1) +O(sT - 1),

we may conclude that w, w) ' CGA(w; 6(w1 6(w2)

( E 1,(°)( Ra

p)

(

w-1 ,

1

,

1

) - E 1,(0)( Ra

,

w1, 1 ))

w-1, 1) - Est,(1)(4), w,1)).

+ CGA(w;

Hence,

I0(4,w) _ 9326#(w)($(0) - 4)(0)) +

6(wl)6(W2)

1

2

,1) - E1,(o)(Ra41,wl,1))

+ Est,(1)(41, w-1, 1) - Est,(1)(It, W, 1).

It is easy to see that qla(o) _ (P(O), oa(O) = , wl,1) _ A-1E1,(o)(Ra1, w1i 1) - ))-1(1og A)E1,(_l)(R,P, w1,1), E1,(o)(RD

A-2E1,(o)(Rah,wl1,1)+A-2(logA)E1,(_1)(RD$,wi 1, 1). Also,

1

f

As,Da(O)dxA _

1)s4)a(t1)dXA

_

0

1

f AsEl,(o)(R, A,wl,1)d'A =

E1,(o)(RDP,wl,1)

+ E1(R5 , '1,

o

1

J

AsE 1,(0)( Ra a,

w-1 1 dx = El,(o)(RD$,w1 1, 1) ) 1

,

s-2

0 1

A8Est,(1)(I A, w,1)dx.A _ - 6(w12

1

f

0

1

A_-

21)

-

1),

_ E1,(-1)(Ra_,wl 1,1) (s - 2)2

(7'v(Ro41, wl, s) - 7'v+(RooD, w, s)),

6(wlw2 1) Tv+(Ro$, wi 1, 3 - s), 2

Part II The Siegel-Shintani case

122

because the order of the integration with respect to A and the differentiation with respect to sl can be changed by (4.2.9). This finishes the proof of (4.2.15).

§4.3 3-sequences In this section, we introduce some notations related to /3-sequences. The unstable strata of V. are Sn,,1, , S,,,,.-1, and Y,,,,i = Vn-i for all i. So we identify /3-sequences with subsets 0 _ 0 1 ,--- , ja} C {1, , n} where 1 < j1 < < ja < n. With this identification, YD = Vn-ia. If 0 and l(cl') = l(a) + 1, Pa, C Ma is a maximal parabolic subgroup. Therefore, all the ,3-sequences satisfy Condition (3.4.16) (1). Hence, Ep ((D, w, w) etc. are well defined if Re(w) >> 0. We choose da1(Aa1), , daa(Aaa) in the following manner da1(Aa1) =

da l Ail

a

In

in

-ji

\- n-i2) r.

da2(Aa2) =

'D 2

Iia-1

do,, (AD.)

'a

_

via-ia- In-ia

(n-ia)Iia-ia-1

-aa Let

AD = do (Aa1,...,Aaa) = dal (Aa1)...daa(Aaa),

AD = {A5 = d(Aal, ... , AD.) I Aal, ... , AD,, E R8 }.

Let go E Mam fl G. Then go can be written uniquely as go = Aaga°, where A5EAa,9a°EMA. We define (4.3.1)

aa1 = jl(j2 - j i )

( j a - j . -

-

ja),

t at = 2aj1(j2 jl) "' U. - j.-1), trla3 = trf51t352 = 2-a(n - j1) ... (n - ja). Let dx Aa = dx Aa1 ... dx AD,. We define an invariant measure dga on MDA fl G°°

by dga =,'ldxaadga. This measure satisfies the condition after (3.3.11). We define 'XJa = 331Qji2-il .. Tia-ia-1. Let S2a etc. be as in Chapter 3. If 0 = {i}, we also use the notation Fn,i instead of go. , ja} be a /3-sequence of length a > 0. Let jo = 0, ja+1 = n. We Let i) = {ji, define constants fai, hai in the following manner (4.3.2)

(n - 9i-1)(n - ji)

(n - ji)(n - ji + 1) fai =

2

,

hai =

2

for all i = 1, , a. Let Is = faa, ha = has We define a constant cp for each path p of positive length inductively in the following manner.

4 The zeta function

123

Suppose 1(p) = 1. Then (4.3.3)

cp

1i-za. 2

s(1)=1.

If p -< p' and 1(p') = 1(p) + 1, cp, =

(4.3.4)

j

2(ha ha,) cp

5'(a + 1) = 0,

2(ha+ha,) fa) cp

s-'(a + 1) = 1.

For w = (W1iW2) and a path p = (0,s), we define Sp(w) = 6(w2Db(w2). Let wp = (w1, 1) if #{i I s(i) = 1} is even, and wp = (wj 1, 1) otherwise. Let Ti be the set of p's such that Z = {1, , a -1, i} for some i # a, n - 2, and l(0) > 0. Let 312 be the set of p's such that 0 _ {1,... , a} for some a < n-3, , and

l(z) > 0. Let 3 be the set of p's such that 0 = { 1, ,a-1,n-2} where a < n-3. 933 is the empty set if n = 3. Let 3 be the set of p's such that a = {1, 2,- , n-2}.

§4.4 An inductive formulation We will formulate an inductive way of proving Theorem (4.0.1) in this section. We first have to introduce some notations. Let

Zv (4i, w, s) = E ai(s - so)

(4.4.1)

be the Laurent expansion of Z n (4i, w, s) at s = so. Then we define Zv,(_) (ID, w, so) = ai.

Of course, when we consider these values for a particular n, the meromorphic continuation of Zv. (4), w, s) has to be known. However, since we prove the principal part formula by induction on n, we do not logically depend on the meromorphic continuation proved in [60], [64]. We use a similar notation ZVZ,ad,(i) ('D, w, so) for the adjusted zeta function for n = 2. Suppose that 0 = {jl, , jQ} is a ,3-sequence such that ja = n - 2. Then we identify Za with V2.

Definition (4.4.2) (1) We define Z' ,o c Za to be the subspace which corresponds to Zo in §4.2 by the

above identification. (2) For T E Y(ZDA ), Ra,OW E 9(A), R°'Y E .'(A2) are functions which correspond to R'V2 ' OWY E 9'(A), ROW E Y(A2) in §4.2 by the above identification. We define Z'asok similarly.

Let J(4 D, g°) be as in (3.5.6). Let 9(g°, w) be the smoothed Eisenstein series which we defined in (3.4.7). We define (4.4.3)

IO(4),w) = f

GA/Gk

1(4), w, w) = f

w(g°)J(4,g°)dg°, w(g°)J(D, g°)e(g°, w)dg°.

A /Gk

Part II The Siegel-Shintani case

124

We remind the reader that 1

Zv (4b, w, s) = Zv+(,D, w, s) + Zv+($, w-1, do - s) + f A'I0(4>A, w)dx A. 0

Since

_ g°) - °v k ($, (g°)`), by (4.1.3), J(lk, g°) E C(G°°/Gk, r) for some r = (r1, , rn_1) such that rj > -j(n - j). Therefore, we can use Shintani's lemma for GL(n). The statement of J(-P, g°)

(3.4.34) implies that I(41, w, w) - CGA(w; p)I°(1, w).

By (3.5.9) and (3.5.20), we get the following proposition.

Proposition (4.4.4)

I

E

w, w) _

4)(0))

Ep,rp (4', w, w) + b#(w)A(w;

pE'a31Ual3U`a34

+E E

E Ep(2p#(4),w,w)-w,w))

pE`.az

Note that since all the paths satisfy Condition (3.4.16)(1), ^p($,w,w) is well defined for Re(w) >> 0 for all paths p.

Definition (4.4.5) (1) Let p E'Pi and l(p) = a. We define Ip(-P,w) = CpUabp(w)Zv

(Ra4)p,Lop, ha)

(2) Let P E'432 and 1(p) = a. We define 1p+ (41, w) = cpbp (w) (Zv a+(Ra,Dp, wp, .fo) + Zv-a+(ARo Dp, wp1, 0))

,

qP.fa0)

Ip# (4', w) = Cpb#(w)

Ipl (1)' w) = - c p'JJ n-a- 2 b#(w) + Cp93n-a-2b#(w) w)

Cp'un

(n -

a - 1) 2(n - a)2

E1,(1)(Ra, oRa,go4bp, n - a - 1) 2(n - a)

(n - a)E1(R0,,oRa,4)p,n - a - 1) a-2b#(w) 2(.fa - (n - a))2 E1,(1)(Ra,,oRa,-bp, n - a -

c

p'XTn_a_2b#(w)

2(fa-(n-a))

where a' _ { jl, , ja+1} is the unique ,3-sequence such that t and ja+1 = n - 2.

1)

,

-

a', l(0') = l(0) + 1

4 The zeta function

125

(3) Let P E X33 and l(p) = a. We define Ip (4), w) = cpJn-a_2bp (W)ZV2,ad(RaI)p, wp, n - a + 1).

(4) Let P E 4+4 and l(p) = a. We define Ip(4,w) = cpbp(w)ZV2,ad,(0)(Ra,Dp,wp,3).

Proposition (4.4.6) Suppose My ,,,4D = 4). Then

I°(1,w) =

4)(0)) + E 10(C w) pETi

+

2 n-2b#(w) ( Fl,(l)(Rn,n-2,0 , n - 1)

+

- E 1 (1)( Rn,n-2,0 n-1) ) ,

(Ip+(1), W) - Ip# (4, w) + Ip1(-D, L11) + Ip2(D, w)) PET2

+

Ip(,D,w)+ PET3

PET4

We devote §§4.5-4.7 to the proof of (4.4.6). For the rest of this section, we prove that (4.4.6) implies Theorem (4.0.1) by induction on n. The following lemma is the basis of cancellations of various distributions in this chapter.

Lemma (4.4.7) Let dx = fv dx be the ordinary measure on An, and dxt = H d" t, the ordinary measure on Ax. Suppose that I E Y(An) is invariant under the action of the standard maximal compact subgroup of GL(n)A. Then

f

4)(x)dx = Z (n) n

f

x

ItITD(t,0,... ,0)dxt.

Proof. By the choice of our measure, IA"'D(x)dx = I

kl

f

2

4b(xv)dxv, v

, 0)dx t.

fAX tIn(t, 0, ... , 0)dx t = it- l 1 f Itv l(ty, v

Let v E 9)l. It is easy to see that fk= (1 - q1)n v

Let fn(i) = (1 - qv 1)n

q(xii

it

qv

in

Then

00

fn(i) = (1 _ qv 1)n E

qvi-i2-..._in

i2 ,... in =i 00

+ (1 - qv 1)n

qv ii =i+1 inf(i2, ,in)=i

_ (1 - qv 1)q, ni + qv (i+l)fn-1(Z)

ii-i2-..._in

Part II The Siegel-Shintani case

126

By induction, fn(i) _ (1 - qv n)qv n. Therefore,

f,,.(i)4'(,rv 0 ... 0) _ (1 - qv n)

fk X

Itvly,P (t,,, 0, ... 0)dxt2,.

Next, we consider the real place. Let Dn be the unit ball in R. Then vol(Dn) _

it2F(2 +1)-1. So, ( xR)dxR =

fR ^

f

x+ + xR, 0,

,

0)dxR

R

r = nvol(Dn) J

(r, 0,

,

0)rndxr

= 2vol(Dn) fR ItRIR4)(tR,0,0)dxtR x 0, ... , 0)dxtR.

n r- 21(2) ftx

Finally, we consider the imaginary place. Note that dxc is two times the usual Lebesgue measure. As in the real case, Ixnclc, 0, ... , 0)dx,

-D(xc)dxc = f ( = 2nvol(D2n)2n f

r2n-P(r, 0,

(7r)

( ) fcx (27r)nln

,

0)dxr

tcInD(tc,0,... ,0)dxtc. I

Therefore,

(x)dx = Zk(n)1(27r)T2 IA"

9k

0)dxtv

J

Itln.p(t,0,... ,0)dxt,

Zk(n) fAx

because

(27r)T29tk.

Q.E.D.

Lemma (4.4.7) is essentially the relation between integrals with respect to the cartesian coordinate and the polar coordinate. Easy considerations show that (4.4.8) E1,(1)(Rn,n_2,o6a, n - 1) _

n_2 0

+

A-" 2

n - 1)

(log A)E1(R' ,n_2,A, n - 1),

El,(1)(Rn,n-2,o4)a, n - 1) _ A E1,(1)(R.,n_2,oob, n - 1) - A-n( logA)E1(R.,n_2,04, n - 1),

4 The zeta function

Zv (Rn,iCA, (W1 1, 1), n

_A

2

1+

2

(Rn,i,'),

Zv _i

127

n(n - i) 2

)

ZV _i (Rn,i-D, (WI

1, 1), n(n2

i)

),

n(n - i) ) (W 1, 1), 2

=A-

Zv, (Rn,i D, (Wl, 1), n(n2

2)

),

for i# 1,n-2.

Ifn=3,

(4.4.9) ZV2,ad,(o)(R3,1ia, (W1 1 1), 3) = A-3ZV2,ad,(O)(R3,1$, (W1 J),3) + )-3 (log A)9326# (W ).° 3,1R3,1$(0), Zv2,ad,(o)(R3,lDa, (W1, 1), 3) = A-3ZV2,ad,(O)(R3,1'D, (WI, 1), 3)

- A-3 (log A)'Z726# (W ).

3,1R3,1D(0).

Ifn>4, (4.4.10)

A-dn4(0), DA(0) = qD (0), (0) _ _1 ZV2,ad(Rn,n-4a, (W1 1), n)

n(n-,)

_

2

-1

ZV2iad(Rn,n_21, (Wl

1),n),

ZV2,ad(Rn,n-2'a, (w1, 1), n)

= A-"ZV2,ad(Rn,n-2', (01, 1), n), 1 n(n - 1)

Zn_,,(o)(Rn,l4)a, (Wl

-A-nZVn-,

,

1),

2

(0)(Rnl, (W1 _1, 1)) ,

)

n(n -1) 2

+ A-n(log A),Zn-lb#(W)9n,1Rn,1$(0), n(n - 1) 2 ) Zn-,,(o)(Rn,lca, (1)1,1), _ n(. -1) n(n - 1) 2 Z n-,,(0)(Rn,l I, (W1, 1), = 2

- A-

nc2 ,)

(log A)Tn-lb#(W)9n,1Rn,14D(0).

Lemma (4.4.11) If $ is K-invariant, (1)

Zn_2F_,1(Rn n-2 0'p, n - 1) =

(2)

21n_2E1(Rn,n-2,0$, n - 1) = 93n_19n,1Rn,1(D(0).

Proof. Note that 9n-2

'1(x12, ...

_1 = zk(1 1). We define `F1 E Y (An-1) by

,

x1n) =

JA 4)(x11, x12, ...

, x1n, 0, ... , 0 dx11.

Part II The Siegel-Shintani case

128

Let W2(x) = 1Y1(x, 0, , 0). By assumption, W1 is invariant under the action of the standard maximal compact subgroup of GL(n - 1)A. Easy considerations show

that E1(R71

n-

,9I'n,1Rn,1$(O)

=f

n - 1), W1(x12,"' ,xln)dx12...dxln.-

An-1

Then the first relation follows from (4.4.7). The second relation is similar. Q.E.D.

Definition (4.4.12) For n = 3, we define 3

{{/ J3(I',w) = Q36#(w)D+ (0)

2)

+ 6(W21)6(W2)ZV2,ad,(O)(R3,1(D, (w1, 1), 3)

+

#(w) E1,(1)(R3 1 0IP, 2).

For n > 4, we define fn(1D,W) = 9n8#(w)'D (0)

+

(n - i)1i8(wi)6(w2)

ZV

-i (Rn,iob, (w1,1),

n(no

i#1,n-2

+

Zn-26(wl)6(W2)ZV2,ad(Rn,n-2', (wl, 1), n)

+ (n - 1)S2 -1

(w2)

ZVn-1,(o)(Rn,lD, (w1,1), n(n2 1) )

+ Zn-26#(w) E1,(1)(Rn 2

n - 1).

Also we define (4.4.13)

1, 1(4,w)

E Ip(4''w)+ 1: Ip(F,w)+ E Ep1

PEP3

1(p)>1

1(p)>1 I a(1)=0

s(1)=0

PE'p4

s(1)=0

+ > (Ip+('D,w)-Ip#('D,w)+Ipl(1',w)+Ip2(4,w)), pEP2

s(1)=0

E Ip('D+w)+ pEP1 1(p)>1 s(1)=1

+

E

IEP3

1(p)>1

IpN,w)+ E Ip(4),w) pET4 s(1)=1

s(1)=1

(Ip+('D, W) - Ip#(D, L") + Ipl(4), W) + Ip2('D, w)) IEP2

s(1)=1

a) )

4 The zeta function

129

We consider three statements (An), (B.), (Cn). The statement An is as follows.

I°(-D,w) = fn($,w-1) - fn

(An)

The statement (B3) is as follows. I3,1(P, w) = -b(wi)b(W2)7'V2,ad,(0)(R3,1'P, (wi, 1), 3),

(B3)

I3,1(.p,w)

=6

1,

(wi 1), 3).

The statement (Bn) for n > 4 is as follows.

(Bn)

In,1(-D, w) _ -

In 1(4', w) -

(n - 1)b(wl)b(L02)

(n -

Zv, -,,(o)(Rn,l41, (w1,1), n(n2 1) ),

Z n-1,(o)(Rn,i

n(n2

1) )

, (w1 1, 1),

The statement (C3) is the empty statement. The statement (Cn) for n > 4 is the statement of Theorem (4.0.1) for Vn_1.

Proposition (4.4.14) The statements (An), (Bn), (Cn) are true for all n.

Proof. Clearly, (4.4.6) and the statement (Bn) imply the statement (An). So we only have to deduce the statements (Bn+l), (Cn+i) assuming the statements (An), (Bn), (Cn), and (4.4.6). Consider the case n = 3. In this case, T2i 33 are the empty set. Therefore, (B3) is Definition (4.4.5)(4). Now we consider the step from n to n + 1. Let 4 be as in Chapter 3. We multiply as to I°(4) a, w) in (An) and integrate it over A E [0, 1]. Then (Cn+1) follows by the relations (4.4.8), (4.4.9), (4.4.10). 1 Let 4) E Y(Vn+1,e), and W1 = Rn+1,1'D,'F2 = Rn+1,i . Let fn = n , hn,i = n n-i . By applying (Cn+1) to `I'1 and using (Bn), we get the following lemma. 2

2

Lemma (4.4.15) If n > 4, b(Wl)b(w2)Zn,(0)('1, (w1, 1), fn)

=

6(w2

flb(w2)(Zn +0I'1, w, fn) + Zn+(Wi, w, 0))

- 93n6#(w)

f

(0) n

+ E (n - i)

ib(wi)S(w2)

i961,n-2

(n - i)Zj6(wi)b(w2) i541,n-2

(wl 2hn

1

1), hn,i)

Zn (Rn,iW1) (w1, 1), hn,i) 2(fn - hn,i)

Part II The Siegel-Shintani case

130

(w1 1, 1), n)

+n-26(W1)6(W2) 2

- Tn_26(W1)b(W2)

+

In,1('F 1,W)

hn,n-2 ZV2,ad(Rn,n-2W1, (Wl, 1), n)

f, - hn,n-2

- In,1(W1,W) fn - hn,l

hn,l

+n_2/5#O W

-

nEl(Rn,n-2,o1, n - 1) + El,(1)(R',n-2,o1, n - 1) 2h 2

2hn,n_2

n ,n-2

-_2S #(w) (nE'(R2i,n 2(fn - hn,n-2)2

n,n-,oq1- 1)

+ E1,(1)(Rn-2,oW1, n - 1) 2(fn - hn,n-2))

)

/

For a path p = (a, s) for Vn, we associate a path p' = (D', s') for Vn+l so that

0'={1}U{i+1IiED},s'(1)=0,ands'(i+l)=s(i)fori=1,...,l(p). Any

path p' for Vn+1 such that 1(p') > 1 and s'(1) = 0 can be obtained in this way. By definition, cp' = - z cp.

Let pi = (0',s') be a path for Vn+l where 0' = 1}, s'(1) = 0. If we multiply (4.4.13) by - z , the sum of the first two lines and the last two lines is Ipi+(4),W) - Ip1#(4),W) + Ipi1(b,W) + Ipi2(D, W)'

The sum of the third and fourth lines is equal to

E

W),

p ,E'J1 1(p')=2 s(1)=0

where we consider paths for Vn+1. The sum of the fifth and the sixth lines is equal to Ip' (-P' L`')' p'E'V3

1(p')=2 a(1)=0

where we consider paths for Vn+1. Finally,

n In,l(Fl,w) 2

hn,l

n In,1('1,W) + 2 fn - hn,l

gives the rest of the terms in (Bn+l) for In+1,1(,D, w). This proves the step from (Bn) w) is similar using to (Bn+1) for n > 4 and In+i,i(-P, W). The argument for WY2. The step from (B3) to (B4) is similar using F3(1, w, s). This proves (4.4.14). Q.E.D.

Now we begin our analysis, and study the distributions ,=p ((P, w, w) in the next three sections.

4 The zeta function

131

§4.5 Paths in 131 We mainly consider paths in X31 in this section. We first determine o'p for an arbitrary path p. Lemma (4.5.1) Suppose up (AD) = fa 1 epi (Xa )x"' . Then

_

Xpi -

fori=

hai - hai+l

s(i + 1) = 0,

hai + hai+1 - fai

s(i + 1) = 1,

and Xpa =ha.

Proof. We prove this lemma by induction with respect to 1(p). Suppose that p = (-0,s) -< p' = (-0',s') and 1(p') = l(p) + 1 = a + 1. Let Xa' = do,(Xa'1, 1 Xa' a+l) The statement (3.3.10) says X2(ia.+1-ia)

(4.5.2)

D 'a+1

-

J l ea'a+1(Xa')ea'a(Xa')

ea'a+1(Xa')ea'a(Xa')-1

s'(a+l) = 0, s'(a + 1) = 1.

Therefore,

da'a+1(Xa'a+1)

(4.5.3)

2p _

(ea'a+1(Xa')ea'a(Xa,)-1)ha'

(ea'a+1(Xa')ea'a(Xa'))h°'

s'(a + 1) = 0, s'(a + 1) = 1.

In all the cases, Ya = Zo. So r,02(AD)

Suppose s'(a + 1) = 0. Then a-1 Qp'(An') = (IT ep'i(Xa')XCi)ep'a(Xa')ha i=1

(ea'a+l(Xo')ea'a(Xa')-l)h

, a - 1, Xp'a = ha - ha' , and Xp'a+l = ha' . Therefore, Xp'i = Xpi for i = 1, Suppose s'(a + 1) = 1. Then OD (AD,) = A,-,', and ep'a(Xa')-fa

ka1(ea(Xa')) = epa(Xa')fa =

Therefore, a-1

ep'i(A0')XO )ep'a(Xa')ha(ea'a+1(Xa')ea'a(Xa'))ha ep,a°.

Qp'(.X') _ i=1

Hence, Xp'i = Xpi for i = 1,

, a - 1, Xp'a = ha + ha' - fa, and Xp'a+l = ha' . Q.E.D.

, a - 1. Therefore, Xpi > 0 for all i. X14, hai = fai for i = 1, If P E 4'1e The following lemma is an easy consequence of (4.5.1). Lemma (4.5.4) Epaa3 f i=1 Xpil = ep The following lemma follows from (4.1.3).

Part II The Siegel-Shintani case

132

Lemma (4.5.5) Let p = (a,s) E 31 or T2 Suppose a = {1,

a - 1,i}.

Then

(n-i)(n-i+1)

for any N1i N2 >

2

op(Ab)IOza (Ra4)p, Xaga)l

(1)

a-1

<< fj

epi(Ab)Xn'inf(epa(A,)h.-Ni

epa(Aa)ha-N2)t(gO)r, l /

i=1

ap(Aa)ra1(Aa)IOza (Rasp, 00(.aga))j a-1 << 11 epi(Aa)Xeiepa(Aa)ha-d°-i+Nlt(go),.

(2)

i=1

for Aa E Ap2 and go E (K n MaA)ToO,+52a, where r = (r1,

rn_i_1) E R'-1,

rj < j(n - i - j) for all j, and t(g°)r is as in (3.4.17). Proposition (4.5.6) (1) Suppose p = (a,s) E X31. Then

ep=p(,D, w, w)

(2) Suppose P

E 132.

CGA(w;p)cp`2abp(w)Zv

(RsIp,wp,ha)

Then

Ep =p+(,D, w, w) "' CGA(w; P)cpZa6p (w)ZV.-is+(Ro Dp7 wp, ha), Ep

p+(-D, w, w) - C'GA(w; p)cpTabp (w)ZV.-a+(9a Ra 4D p, wp

1

0),

Ep=p# (11, w, w) - CGA(w; P)ep jp# (,D, w)'

Proof. Consider (1). We choose N1i N2 in (4.5.5) so that N1 >> 0 and N2 is close to (n-i)(2-i+1) Then ha - N1 < 0 and ha - N2 > 0. This is possible because ha > (n-i)(n-i+1) if p E'131. 2 This implies that the right hand side of (4.5.5)(1) is integrable on the Siegel domain. So we can apply (3.4.32), and ep-p(4),w,w)'"epC'GA(w;p) J 2

Mas/Mak

wp()()e(R0p,\ag°)dg

An easy consideration shows that the right hand side is equal to the right hand side of (1). The first two statements of (2) follow from (3.5.13). Consider the third statement of (2). Since ha > 0, ][li=1 epi(Ab)X"iepa(AD)ha is integrable on Apo also. Therefore, we can apply (3.4.32) to =-,# (-D, w, w). Thus, the proposition follows. Q.E.D.

§4.6 Paths in '133i'134 Suppose that p = (a, s) E '133 or '4i and l(p) = a.

Let a'-< a,and1(a')=a-1. ForWE9(ZZ'A),wedefine (4.6.1)

IF(gax) xEZ'°D°,Ok

4 The zeta function

133

Let p11 = (01, 511), p12 = (01,S12) be paths such that p - P11412, a1 = aU{n-1},

and 511(a + 1) = 0, 512(a + 1) = 1. Clearly, 93 = Ta, _ 93n.-a-1 By (3.5.9),

W,w)+ "p+(4),W, w)+ p#W,w)p#(4b) W, w)

W, W)

+ P12 (4), W, w) -'='Pii (-D, W, W).

Also

p+(D,W, w) =

(4), W, w) +

(D, W, w),

gyp+(4),W, w) _

(4), W, w) +

(
The following proposition follows from (3.5.13).

Proposition (4.6.2) (1)

Ep"p+(

(2)

fp,rp+(4,,W,w) ^' CGA(w>P)cpZabp(W)Zv2+(gaRa4Dp,Wn 1,3- ha).

We consider the contribution from Zastk. W e define Ma = f 1 Ma`, and Xa = Al x Man x Xv2. Let X.}. = {(uo, µ) E Ax R+ I t > a(uo)}. Let Ha = GL(1) x Ma x Hv2 and La = GL(1) x Ma x TGL(2), where TGL(2) is the set of diagonal matrices in GL(2), and Ma, TGL(2) are imbedded in GL(n) diagonally in that order. Then Zasstk = Mary xHak Z'ag0 .

Any element ga of Xa can be written in the form ga = (t, g'a, ba), where go E Man, ba = n2(uo)a2(it1, A-1t2), uo E A, µ E IlB+, and t, t1i t2 E A'. Let dba = duodX_µdxtldxt2i and d go = d"tdg°°dba. If f(go) is a measurable function on Ma°A/Hak, then (4.6.3)

fm.n/Hak

f(9°)dg° =

f

f(9a)d

Xa/Lak

if the right hand side converges absolutely. Let ga = Raga, and dga = dxAad g. Let Mp, Up, Up*, Tp etc. be as in Chapter 3. Also let Oz,.,t (RD Dp, go) be as in (3.5.1). By (4.6.3), Wp (9a )ap (Aa )E)za,st (Ra Dp, 9a).1p (9a, w)dx Aad9a

Apt Man/Mak

J

wp (9a )ap (Aa )Gz;,o (Ro-Dp, 9a )ep (9a, w)dx Aa d9a pi Man/Hak

fpl Xa /Lak

f

Wp(9a)op(Aa)Gz;.o(RIDp,9a).p(9a,w)

a,

Wp (9a )ap (AI)-a1(AD )Gza,,t (go Ra gyp, ea (9a ))-p (9a, w)dx Ao d9a

Ao Man/Mak

_

Jpo Man/Hak _

fpo Xa /Lak

g.

Part II The Siegel-Shintani case

134

The group Mp is either one of the following forms Case 1 GL(1) x GL(1)m x GL(n - a - 1) x GL(2) x Case 2 GL(1) x GL(1)m x GL(2) x GL(n - a - 1) x

GL(1)a-m-1 GL(1)a-m-1

Since it, E Pa1A, we can consider ga as an element of G°A n Pa1A. Let

9a = Aa(t,ga,a2(/Itl,µ 1121))

If we write ga = )talga°1 for aal E A,,,ga°1 E Ma A, then Aal = )adala+l(µ-1) Consider the function -'pll (ga, w) It is easy to see that n+m-a-1 a-m-1 ep(dala+l(µ-1)) =an( 1,... ,1,µ,Il-1,1,...

for Case 1 and m

n-m-2

Op(dala+l(/-1)) =an 1,...,1,µ,N-1,1,...,1 for Case 2.

Lemma (4.6.4) (1)

J

1Xa/Lak

wp (a)op

(AD )Oz, o (Rio 4p, ga )1p (go, w)d9a

^'0

fApj Xa /Lak (2)

_

wp () o p (A

(RD Dp, 9a )e 11,Ta11 (ga, w)d9a,

CJp

Apo Xa /Lak

Apo Xa /Lak

(gp, w)dga.

wp

Proof. We first compare ap(ga,w) with 0p11,'rpll (g0', w). By definition, -'p (ga, w) - -'P (go" w) = ip11(ga, w)

Lemma (4.6.5) For any e > 0 there exists S = S(e) > 0 and cl,

, Ch E t* = t°*

such that IIcjlb° < e for all i and for any l >> 0, M > w°i

for w° - 5 < Re(w) < M, and go in some Siegel domain.

Proof. Let I = In - m - a} for Case 1, and I = {m + 1} for Case 2. Then e,,', (op (ga ), w)

8pll (g-D, w) _ I,T

4 The zeta function

135

(see (3.4.27) etc. for the definition). Since v = 1 in this case, for any 6 > 0 and l >> 0, there exists a function ct,6(z) of polynomial growth such that I EB,I,T,1(9p(9a), z)I «Ct,6(z)t(Op(9a))TZ+P,-1.

Since I # 0, by the same argument as in the proof of (3.4.31), for any e > 0, we can choose q E DI, such that L(q) < wo, and IITq+PIIo < E. Therefore, there exist 6 = 6(e) and c E to* = t* such that IIcIIo < e and for any l >> 0, M > wo, 91,T,1(Bp(9a),'0) << t(ep(9a))cP-1

for wo - 6 < Re(w) < M. Note that there is no condition on p in this estimate (see the remark after (3.4.30)). We can write t(Bp(ga))c = t(ga))` for some c' E t*, and if the norm of c is small, the norm of c' is small also. Q.E.D.

- e'pi(9a,

Next, we consider the difference e,,, (ga, w) w). By definition, ep,li(90') = epi(Aa) for i = 1, a, and N'-Zepa(Aa) Therefore, (3.4.31)(2) implies that for any e > 0, there exist

6=6(e)>0,

andcEt*

such that ri < e for all i, IIcIIo < e, and if M > wo, a I&p11 (9a, w) - &pii,rp (9a> w)I <<

µ-r°+i t(9a )c

IIepi (Aa )r' i=1

for wo - 6 < Re(w) < M. Therefore, for any e > 0, there exists 6 = 6(e) > 0 such that if M > wo, ep (ga, w)-(go, w) is bounded by a finite linear combination of functions of the form a A-ra+1 t(9a)c

I'

epi(Aa)r i=1

for wo - 6 < Re(w) < M, and ga° in some Siegel domain, where c E t*, Irfl, IIcIIo < e

fori=1, ,a,andra+1>0.

We choose 0 < W E Y(A2) so that

Iez;,o(RaDp,9a)I << E T(epa(_a)x,epa(_a)xuo), XEkx

Ieza,j9-oRa,'p,ea(9a))I4 E W(epa(ta)-1x,epa(Aa)-1xuo) sEkx

These functions do not depend on ga. The function t(ga)' is integrable on the Siegel domain; therefore, we can ignore this factor. By (4.5.1), a

a

'i

-ra+1 i=1

-r.+1

Xv.+ri i=1

Part II The Siegel-Shintani case

136 Since ra+1 > 0,

µ-*a +ldxµ = 1_a(uo) - '11 2

µ>

ra+1

--(v.p)

Since #cai (7a) = epa (Aa)-3, multiplying i ,1(aa) affects only epa (.a ). Since ri can

be arbitrarily small, we can choose such ri's that Xpi + ri > 0 for i = 1, Let µi = epi (AD) for i = 1, , a. Then the integration with respect to µi, converges. So we only have to check that µaa+r°a(u0)_ -

1

iP(µx,Ixuo)d"µaduo

1

, a - 1. , µa_ 1

< oo,

xEkX 1 1

'F(N'a 1x, µa lx2lp)d"µa d260 < 00.

2

0

A

xEkEX

This follows from the fact that Tv2+(`P, w, s, Si) converges absolutely and locally uniformly for all s, s1. This proves (4.6.4). Q.E.D.

Let z = (z1,... , zn,) E C , z1 + ... + zn = 0. Let yl = za-m, y2 = za-m+1 for Case 1 and y1 = zn_m,_1, y2 = z,,-,n for Case 2. Let y = yl - y2. We define

T1=(a-m,a-m+1) for Case landT1=(n-m-1,n-m) for Case 2. In Case 1,

Tp(1) =n, ,,r,(m) =n-m+1,

Tp(m+1)=a-m+2,...,-rp(n+m-a-1)=n-m, Tp(n+m-a) =a - m, rp(n+m-a+1) =a-m+1, Tp(n+m-a+2)=a-m-1...... p(n)=1. In Case 2,

Tp(1) =n, ,Tp(m) =n-m+1, Tp(m+1) =n-m-1, Tp(m+2)=n-m,

Tp(m+3)a-m...... -(n+m-a+1)n-m-2, Tp(n+m-a+2) = a - m - 1, ,Tp(n) = 1. It is easy to see that Tp11 = T1Tp. Let U = U 11#, T = 7-p11. Then

MT(z) =

MTl

(Tpz)M" (z) = c(y)M1- (z).

Clearly, Bp(dal,a+l(µ-1))Tz+P = µl-y. Let I1 = {(i, j)

In-m+1<j<

n, 1< i < j}, 2= {(i,j) 1 1< i < a-m-a - m < j< n - m}, and I3={(i,j) I 1
n-m n-m m(Za-m. - zj) q (zi - zj) 11 O(za-m+1 - zj) H j=a-m+l j=a-m+2 (i,j)EIIUI2UI3

4 The zeta function

137

for Case 1, and

f

n-m-2

n-m-2

11 q5(zi - zn-m-1) [J q5(zi - zn-m)

O(zi - zj)

i=a-m

(i,j)EI1UI2UI3

i=a-m

for Case 2.

Let

9p(AD)rz+P = 9p(.a)T"z+P = ep1(Aa) i i(z) ... epa(Aa)7 (z)

Lemma (4.6.6) Suppose that ypi(z) =

cijzj+cio, where c, c0 are constants.

Then

(1) Ej cij + cio = 0, (2) caa_m = 1 for Case 1 and can-m-1 = z for Case 2.

Proof. In Case 1, n-a-1

m

a-m-1

-----------------

a

9p(d0a(ADa)) = an(

0.

a o- ia da

e

>

>

Therefore, 9p(daa(Aaa))T"z

-

AOa

(n-a-1)(Y1+Y22(za--+2+...+zn-,n

(n-a-1)Y+2((Y2-z-+2)+...+(Y2-zn-,n))

AOa

In Case 2, n-a-1

,m

Op (daa (Aaa)) = an(lr

' Aa (n_a_1)'

-

(n_a-1)'

A2

a-m-1

' 1 ... ,1

e

Aa

2

.

Therefore, (n-a-1)(Yl+Y2)+2(za-,n+...+zn---2) 0p(daa(ADa))T"z = A

_ So

a-1

ep(Aaa)T"z = fj ep(dai(Azi))T"zep(daa(Aaa))T"z, i=1

and the first factor is a product of powers of epi (.\a )'s for i = 1, (2) follows from (4.5.2). If z1i

, a - 1. Thus,

1, then Op(Asa)T"z+P = 9p(Aaa)TCz+P = 1. Thus, (1) follows. Q.E.D.

We define

Qp={zIzi-zi+l=lfori=a-m+2,,n-m-1}

Part II The Siegel-Shintani case

138

for Case 1, and

Qp ={zI zi-zi+1 for Case 2.

Let CO be a contour of the form {z E Qp I Re(z) = q}. Let

dzlcp=

1

l li=1 1 d,j Un-m dzj a

7'7a-m-1

lli=1

7'' n-1

dxi l In-m-2 dzi

Case 1, Case 2.

By the Mellin inversion formula, w2(det a)}1eS,T(ep(Aa(9a>ba)),w)dga a+1 MT(z)p1-IOp(Ao)T=+PA(w;z)dzlcp.

=6#(w2) (2v)

IC'

Let F0 1(4D, t, Aa, ba, z), Fp2 (D, t, Aa, b°°, z) be the following two functions ap(Aa)p1-IOp(Aa)7Z+P8Za.o (R5IDp, (Aot,1, bo)), ap (Aa)na1(Aa )pl-y9p (Aa )FZ+Ppza Al so

a

Ra,Dp, 9(t, 1, ba )).

let Ep,st+

7pa(z) + ha, 2 )

z)

&2(Y - 1) lli1 (7pi(z) + Xpi)

3 - 7pa(z) a-1

2p,st+(D, z) =

f

)

t a2 (y - 1) fi=1 (7pi (z) + Xpi )

, a. Then d" As = aa2 d" pl

Let pi = epi (AD) for i = 1,

ha,

1/kx XApi XX+X(Al/kx)2

dX Pa. Therefore,

w1(tt1t2)t1Fp1(-D, t, Xa, b°>

= 6#(w1)Ep,st+(41, z), wl(ttit2)}1Fp2('D, t, A1, ba, z)dxtd) Aadbo Al /kx xApo XX+X(Ah /kx)2

= 6#(wl)2p,st+(D' z). Hence,

=

1

a+1

6#(w) (2) J

z)MT(z)A(w; z)dzl cp ,

p

wp

w)dga

fA,,,Xa /Lak

=

1

_

a+1

b#(w) (2) f

p

Ep,st+('D,z)MT(z)A(w;z)dzlcp.

4 The zeta function

Let

[

hp(z) =

139

cb(zi - zi).

(ia)EI1u12uI3

In Case 1, we define a function fpo(y, za-m+i) by

fpo (y, za- +1) =z,_1=1 Res

.

Res

Res

za_m_1=1

Res hp(z)IQ,.

z1=1

In Case 2, we define a function fpo(y, zn-,n-2) by the same formula. Let fill(y) _ fpo(y,1) in both cases. Let 9tk

fp2(Y) =

Zk(y)

Zk(n - a) Zk(y + n - a)

Then Res z,_1=1

.. Res

. Res MT(z)IQ, = fpl(y) fp2(y) Res z,._m=1Resza_-+1=1z1=1

for Case 1, and

Res .. Res

Res

z._1=1

Res

za_,,_1=1

.. ResMT(z)IQ, = fpl(Y)fp2(Y) z1=1

for Case 2. Also Resy=1 fpl(y)fp2(y) = %DCG = 3n-a-1 n1

Let Qp = {z E Qp l zi = l fori = 1,... ,a-m-l,a-m+l,n-m,.. ,n-1} for

Case 1, and Qp = {z E Qp I zi = 1 for i = 1, , a-m-1, n-m-2, n-m, for Case 2. Any function on Q' can be considered as a function of y. Let 7i(y) = -Ypi(z)IQ,

,

E,8+ (D,y) _

z)IQp, Ep,at+(4), Y) =

, n-1}

z)IQp

Because of (4.6.6), -ypa(y) = y21

Lemma (4.6.7) 1a+1

1

Ep,st+('k,

z)MT(z)A(w; z)dzIcp

(1)

n

- 27r- J 1

(2i)

(2)

1

y)fp1(y)fp2(y) [A(w; z)] IQdy,

e(y)=1+b

a+1

f Ep,st+(4), z)MT(z)A(w; z)dzIc,

Le(y)1+S

,st+(

)fl(y)fp2(y) [A(w; z)]

dy,

where 6 > 0 is a small number.

Proof. Let Qpl = {z E Qp zl = 1}. Let Cpl be the contour of the form {z E Qpl Re(z) = q}. Let Cpl be the contour of the form {z E Qp I Re(z) = q} I

I

where gl-Q2=1-61,gi-qi+1=1 +62

foriV{1,a-m+2,,n-m-1}or

Part II The Siegel-Shintani case

140

{1, a - m, , n - m - 3}, and 61621 >> 0. Let dzI c,1 be the differential form on Cpl such that dzlcp = dzldzlcn1. Then i

(2) 1

a+1

fP

(2) f I (2) a

1

Res [Ep>st+(-D, z)MT(z)A(w; z)] dzI c,1

v/- 1

1

a+1

1

+

Ep,st+(iD, z)MT(z)A(w; z)dzIc,. 1

The second term is holomorphic for Re(w) > wo - 6 for some 6 > 0. If we continue this process, we get the lemma. Q.E.D.

These considerations show the following proposition.

Proposition (4.6.8) Suppose P ET3 U T4 Then (1)

Ep°p+(4),W,w) CcA(w;p)cpWabp(w)Zv2+(RaIPp,wp,h0) alb#(W)

+ Ep 27r

(2)

Re(y)=1+b

Ep st+(D, y)fpl(y)fp2(y) [A(w; Z)] IQ; dy,

ep=P+(4), W, W)

- CGA(w; p)cpZabp(W)Zv2+(gaRa4)p, Wp 1, 3 - ha)

+ EpValb#(W) 21N

1

JRe(y)=1+6

Ep,st+(4),y)fp1(y)fp2(y)[A(w;z)] IQ,dy,

where 6 > 0 is a small number. Next, we consider , Epl,

W, W)) Ep12 (p, W, W).

Lemma (4.6.9) Let T be as before (7 = Tp12 also). Then W, 211)

wp1. (ga1)aa1 (A01)9z11 (Ra1 p1. , go,)

t71a11 J

4v1.0 xMa1A/Mo1k

for i = 1, 2.

Proof. Suppose r # T. By (3.4.31)(2), for any e > 0, there exist

b=b(e)>0,

andcEt*

such that ri, llcllo < e for all i and if M > wo, << IS ,,,.(ga1,w)I

a+1

ri epl.j( j=1

1)r't(9a1)c

4 The zeta function

141

for wo - b < Re(w) < M, and gal in the Siegel domain. By definition, ha1 = 1. Since t1+T+1e1(Ro, fAX

I

p1i,

t)dxt

converges absolutely for i = 1, 2, the lemma follows. Q.E.D.

By (4.5.2), 16

Ao1a+1 = ep1ia+1(Xot)2eptta(Pat)

2 = ept2a+1(Asi)2

Therefore,

TZ+P

=

TZ+P

TZ+ P

ep (d,,.+,

a

_

epi(Ao)7Pi(z)XY-1 AY-11 a

= [Iep11i(Ao)'1pi(z)AY-i i=1

-eptta(Ao1)ryp(Z)-

' =i

2 ep11a+1(Ao1)

Y 2

a-1 Hep11i(Aot)'1pi(z) i=1

Also 0p12(Ao1)7Z+P

=

Bpeo(Xo)TZ+Pepeo(do1a+1(Aa1a+l))TZ+p

= ep(AD1)rz+POp(dala+l(ASla+1))7z+P a\\

_

fjepi (Ao)-'1pi (z)A5 i=1

as+l

a-1 =ep12a(AD,)'1111(z)+

1ep12a+1(Ao1)

2

ep12i(A01)'1pi(z) i=1

By definition, Xptia = ha define

- 1, Xpita+l = 1, Xp12a = ho - 2, Xp12a+1 = 1

We

bp(W)E1(Rot,DplltWpt y21) Ep11 (4)+ W, Z) =

Ep12 (fit Wt z) =

2to2(ho-1+ypa(z)- y21)I]'= (Ypi(z)+Xpi) 6p(W)F-1(Ro11)p12IWp 1' 2 ' 2%o2(ha - 2 + ypa(z) + y21)Ia=1(7pi(z) + Xpi)'

where we consider c.Jp, wp 1 as characters of AX /k < by restricting to elements of the

form (t,1). Let µj = ep1tij (ao1) for i = 1, 2, j = 1,

dxA, = ad-pi ... dXµa+1 = 2-1 a2

, a + 1. Then dxu,a+i

Part II The Siegel-Shintani case

142

Also , a1 = ,aa,1. Therefore, we get the following proposition.

Proposition (4.6.10) up1.

)a+11

1

w, w) - aa1

(.D,

1

w, z)MT(z)A(w; z)dzI c,

Ep,. n

for i = 1, 2. We define Ep w, y) = Ep1i (4, w, z) IQ, for i = 1, 2. We can get the following lemma as in (4.6.7) and the proof is left to the reader.

Lemma (4.6.11) =p1. (', w, w)

2

1 Re(y)=1+6

aV

Ep,, (-b, w, y)fp1(y)fp2(y) [A(w; Z)] I Q'p dy

for i = 1, 2, where b > 0 is a small number. The proof of the following proposition is similar to that of (4.5.6)(2), and is left to the reader.

Proposition (4.6.12) (1) Suppose P E 33 U X34. Then Ep p# ((D, w, w) - CGA(w; p)cp91a6#(w) TP(0) . (2) Suppose p E X33. Then

(0) epEp# ((D, w, w) ,r CGA(w; p)cpZab#(w) ha - 3

Now we are going to establish a cancellation of distributions. This cancellation basically comes from the case n = 2. In this process, we reconstruct the order two pole of the adjusted zeta function for the case n = 2. Let Ep1)

y) = Tv2+(Ra,oRa,Dp, Y 2

I

+ ha,

+Tv2+(Ra,o9aRa4)p, 3 -

2 Y)

y-1

bp(w)E1(Ra141P12,wp 1> 2 )

2(ha - 2 + y - 1)

1

2

- ha,

1-y 2

),

- bp(w)E1(Ra1Dp11,wp) 2(h, - 1)

b#(w)E1(Ra oRa-Dp, v21 +ha - 1)

y-1 +

En2)(.D,

y-1 y)

We consider the Laurent expansion of EP(3) (,D, w, y) at y = 1. Since Tv2+(Ra,oRa(bp, s, 0) = E1+(R',oRa4,p, s - 1) etc.,

4 The zeta function 1

Tv2+(Ra,oRa l)p,

2

143

+ ha, 0) + Tv2+(Ra,o " DRa(Dp, 3 - y 2

- E1(Ra oRa-Pp, y - 1 + ha - 1) + 2

2

R,,0Ra4'p(0)

+

1

- ha, 0)

Ra,ORa-917a4)p(0)

1

- 1-ha

2-

Clearly, 1

y1 2 + ha -1

2-y2ha

_

1

1

2(ha - 1)2 (y - 1) + O((y - 1)2),

ha - 1 2

lha+2(2 lha)2(y1)+O((y-1)2),

and

bp(w)E1(Ra1(-bp11,wp)1)

b#(w)El,(-1)(Ra1,Dp11,1)

2(ha - 1)

(ha - 1)(y - 1)

+

bp(w)E1,(o)(Ra1 -Dp11, wp) 1)

+ O(y

2(ha - 1) 1, 1L) 2(ha - 2 + y - 1)

- 1),

b#(w)E1,(-1)(Ra14)P12'1)

(ha - 2)(y - 1) _ 6#(w)E1,(-1)(RD1 '12,1) (ha - 2)2

+

by(w)E1,(o)(Ra14p12,WP 1, 1)

2(ha - 2)

+O(y - 1).

If b#(w) = 1, 4) is K-invariant. This implies that Ra,oRa-Pp(0) =

(Ra14Dp1111)1

Therefore, E(3) (4), w, y) is holomorphic at y = 1 and (4.6.13)

Ep3)(1),w, 1)

_ -b#(w) 2

ha) +Tv2+(Ra,o.3'iRs p, 3 - ha))

_ b#(w)E1,(-1)(Ra1Dp12'1) +

bp(w)E1,(o)(Ra1-Pp12,WP 1,1)

2(ha - 2)2

2(ha - 2)

_ b#(w)E1,(-1)(Ro14)p11' 1) _ 2(ha - 1)2

6p(w)E1,(o)(Ra14)p11Iwp,1)

2(ha - 1)

Let (4.6.14)

w, w) =

b#(w)E,st+(,p,

w) +

+Ep12(4,w,y)-Ep11(4,w,y) Also let °,p

8t (4),w,

w) be the following integral

b#(w)Ep,st+(.p, w)

Part II The Siegel-Shintani case

144

(4.6.15)

2 + ha - 1) fp1(y)fp2(y)[A(w;

E1(R

t71a36#(W)

Le(y)=l+ (y - 1) l1i_1 (1pi(y) + Xpi)

z)]I Q,, dy.

Then (4.6.16) tea

Y-1

(4)

JRe

aa3

Ep (W, W, W)fpl(y)fp2(y)[A(W;z)JIQ,,ay

Ep3)(

I

J

2irV -1 Re(y)=1+6

flid_-11(7pi(y)

y)

+Xpi)

fpl (y)fp2(y)[A(w; z)] I Q,, dy

+ p,st((D, W, W).

Since E(3) (4>, W, y) is holomorphic at y = 1, and Xpi > 0 for i = 1,

, a - 1,

(4.6.17) 2ir

^

Re(y)=1+6

7 llia=-11(7'pi(y)

+ Xpi)

fp1(y)fp2(y)[A(w;z)]IQ,,dy

C, Z' E (3) (4,' LL,'

+

Eptrla3

27rV '-1

f

dEp3)(i(,W,y)

fp1(y)fp2(y)[A(w;z)]jQ,,dy

e(y)=1-6 IJi=1 (7pi(y) + Xpi)

CpIjaE(3)(4p,W,1)A(w; z).

Note that Epaa3 f1a=1 Xpil = Cp. By (4.2.15), (4.6.8), (4.6.12), (4.6.17), we get the following proposition.

Proposition (4.6.18) (1) If P E f 33, Ep °v

w, w)

IiGA(w; p)Ip

w) + Ep =p,st

w, w)

(2) If P E 34, Epp ((D, w, w) N CGA(w; P)Ip (4),w) + Epp st (4>, w, w) + Ep ='p# (4D, W, w)'

§4.7 The cancellations In the last section, we established the cancellation of distributions based on the case n = 2. In this section, we will establish another kind of cancellation of distributions. This cancellation is established between -',1,,t (4), w, w) and °p2# w, w) for Pi E X33, P2 E X32 or P1, P2 E 134

; a - 1,nLet pi = (0l, s1), P2 = (D2,S2) be paths such that a1 = { 1 , 2}, 02 = {1, , a}, s1(i) = 52(i) for i = 1, , a - 1, and sl(a) + S2(a) = 1.

4 The zeta function

145

The correspondence between pl and p2 is one-to-one. Therefore, if pl is of Case 1 (resp. Case 2) in §4.6, then we say p2 is of Case 1 (resp. Case 2). If p2 is of Case 1, Mpg = GL(1) x GL(1)m x GL(n - a) x GL(1)' m. If p2 is of Case 2, GL(1)a-m-1 Mpg = GL(1) x GL(1)m+l x GL(n - a) x

Proposition (4.7.1) If a > 1, let p3 be the unique path such that p3 - P1, p2 and l(p3) = a - 1. (1) If a = 1, Epl

'1

(

/

WI w) +

W))

CGA(wiP)EpiOn26#(W)Z1(1)(R',,°Ral4)p,,n-1).

(2) If a > 1 and si(a) = 1, pi,st(''' W, w) + Ep2-p2#(, W,

Ep1

w) - CGA(wi P)Ipsl (I

W)

(3) If a > 1 and si (a) = 0, W, W) + 6P2;42# ('DI w, w)

Epl

CGA(wi P)Ip32(I, W)

We devote the rest of this section to the proof of this proposition. This will complete the proof of (4.4.6). Let I1, 12, I3 be the subsets of {1, definition,

, n -1} which we defined for pl in §4.6. By

n-m M1-,2 (z) =

H

O(zi - zj)

(i,j)EI1UI2UI3

11

Y'(za-m - zj )

j=a-m+1

for Case 1, and M'b2 (z) _

[

n-m-2

O(zi - zj) J O(zi - zn-m) i=a-m

(i,j)EI1U12UI3

for Case 2. Also by definition, Xp2a = hat = fat, and t021(A,2) = ep2a(AD2)-fat Therefore, opz(A )ND21(A12) = IIi=1

ep2i(A12)XU2i.

Lemma (4.7.2) "P2#(4', W, w)

aa21 f1,20 M,2A/M,2k W 2 (9°)a2P2 (A12)K121(A2 )2,r12 (0p2 (9a2 ), w)d A02 dg°2. Proof. Suppose r

rl,

r,2. By (3.4.31)(2), for any e > 0, there exist 6 = S(e) > 0

, ra+1 > 0, c E t* such that ri, IIcIIo < e for all i and if M > wo, a

Ip2,r (902 f w) I << [1 ep2j (A12 )T't(9az )c, j=1

Part II The Siegel-Shintani case

146

for w - S < Re(w) < M and ga°2 in the Siegel domain. Since /Caz (Aaz) = epza

(Aaz)-ho

ri + Xp2 > 0 for i = 1,

tr , aa2 (.a2 )Ka2 (A , 2 ) = 11a=1 e.2i (Aaz )Xe2 i . Since

, a - 1, and re > 0, the lemma follows. Q.E.D.

Let

Qp2 ={zIzi-zi+i=1fori=a-m+1,

,n-m-1}

for Case 1, and Qp2 for Case 2.

Let Cpl be a contour of the form {z E Qp2

I

Re(z) = q}. We use the same

definition of y1i y2i y as in §4.6. We define Q'2 similarly as in §4.6.

Lemma (4.7.3) Let a (Aaz)7-12 Z+P =

eazi(Aa2 )"12'(z).

epz

i=1

Then ryaza(z) =

(1-a2

on Qp,.

Proof. In Case 1, \

BP2(daza(AD2a))T2

Z+

\\(n-a)za-m-(za-m+1+...+Zn-m)-(n-a+2)(n-a)

P = ^a2a

, n - m - 1. Therefore,

On Qp2, z i = 1 for i = a - m + 1,

(n-a)za-m - (za-m+1

=(n-a)y+

n-a-1 i=1

zn-,t )

_

(n-a+1)(n-a) 2

- (n-a+1)(n-a) i=(n-a)y } (n-a-1)(na) 2 2

_ (n - a)(y - 1). Since Aa2a = epza (a02)ep2a_1(A02 )'1, we get the lemma for Case 1. Case 2 is similar. Q.E.D.

Zk (y)

fa21(y) = f i (y), fp22(y) = JJk(y + n - a)

fa22(y) =

w, z) =

Zk(91k

26#

a)fp12(y)-

a2 4'P2 (0)

(n - a)(y - 1) lli=1(lazi(z) + X020

4 The zeta function

We define Ep2#

w, y) = Epz#

147

w, Z) 1Q02 . By (4.7.3) and a similar computation

to that in §4.6, p2# (D, w, w) ^' 1aa23

(27rv/1

/ a Jc Ep2# (.Dw, z)MT2 (z) [A(w; z)dz] I c, 12

a0,3

,,

27 r V -i Re(y)=1+6

Ep2# (4, w, y)fpz l (y)fp22 (y) [A(w; z)JIQ' 2 dy.

Clearly, epl = -ep2. If a = 1, we define Elan.(4), Y)

E1(R1,oRa1 p1 >

y-1

y-1

2

+ n - a)

Zk (n - a) 9az Rax'PP2 (0)

y-1

9ik

If a > 1, we define y21

+n - a) E1(Ra1,ORa1,tp1, (Y - 1)(-Ynia-1(y) + Xp2a-1)

Zk(n-a) Nk

gal

R1e-bp2(0)

(y - 1)('Yp2a-1(y) +Xpza-1)

Lemma (4.7.4) We have the equality rypli (y) = ryp2i (y) for i = 1, (1) Yp1a-1(y) = 7p2a-1(y) - (n-a+2)(y-1) ifsi(a) = 0, (2) 'Ypia-1(y) ='Yp2a-1(y) +

(n-a2

, a - 2, and

if -,,(a) = 1.

Proof. If sl (a) = 0, Aa1a = epla(^al)2epla-1(\a1) 16

2,

1

'aza = ep2a (^a2) 2 ep2a-1(Aa2) 2

The lemma follows by the proofs of (4.6.5) and (4.7.3). The argument is similar for

the case sl(a) = 1. Q.E.D.

Sincen-aaaz3 ? = trlal3,

^

2

Ep1 ='p1,st

w, w) + Epz'='pz #

c''J, w )

a- E...' (-D' y) Ep1 9-0136#(w) f fp11(y)fp12(y)[A(w; z)] IQ,, dy. JRe(y)=1+b ii-1(Yp1i(y) + Xplti) (If a = 1, we do not consider the denominator in the above integral.) We consider Ecanc(4), y) only when w is trivial. This implies that 4) is K-invariant. Therefore, 4Pp, is invariant under the action of the standard maximal compact

subgroup of GL(n - a)A. So by (4.4.7), Ecanc(), y) is holomorphic at y = 1. We consider the value Ecanc(',1). If a = 1, Ecanc((P, 1) = 1 E1,(1)(Ra10OR514)p,, n - 1).

Part II The Siegel-Shintani case

148

Therefore, in this case, Ep,

..

past ('D, W, w) + Epa

pa# ('D, w, w)

P)Ep193n-26#(w)

CGA(w;

2

E 1 (1)(a1,O R' Ra , p1, n - 1).

Suppose a > 1. There are two possibilities for s1i52, namely, 51(a) = 0,52(a) = 1 or s1 (a) = 1,52(a) = 0. The following lemma is an easy consequence of (4.7.4).

Lemma (4.7.5) Suppose a > 1. Then (1) If si(a) = 0, (n - a + 1)E1(R'01,0R11,Dp1, n - a) Ecanc(4), 1) =

2 2Xpla-1

+

E1,1(Ra1,ORa1,Dp1 , n - a) 2Xp1a-i

(2) If si(a) = 1,

1) _ -

(n-a+1)E1(R'1,ORa,4)pi,n-a)

Clearly, aai3 =

2

+

E1,1(Ra1,OR51

2Xpja-1

2Xpla-1 1 30,,3

1,n-a)

and Ta, = `fin-a-1 Also a-2

a-2

-1

Xp,i =Ep3 aa33

Ep3trla13

i=1

Xp3i = Cp3.

i=1

Clearly, cp, = -Ep3 if si(a) = 0, and Ep, = Ep3 if sl(a) = 1. Therefore, Ep1 A,36#(W)

=

Ep1

E canc( ', y)

f e(y)=1+ lli=1 (1i(y) + Xp,i) 2

27r

'7a-2

Ya1393n-a-1b#(W)

fp11(y)fp12(y)[A(w; z)]IQ,l dy

Ecanc(4, 1)A(w; p)

l ii=1 Xp3i

+

Ep19'O136#(W) 27r

f

Re(y)=1-6 l1i=1 (rypli (y) + Xp1i) Re(y)=1-6

fp11(y)fp12(y)[A(w;z)]IQ,,dy

_1)o1(a)+l Cp3 Tn-a-16# (W) F'canc (-D, 1)A(w; p).

By definition, (n-a-}-1)(n -a)

Xp,a-1 =

n-a+1

= fa3 - (n - a + 1)

si(a) = 0,

2

sl(a) = 1.

Let a' = a-1. Then l(p3) = a', and n-a+l = n-a' etc. Therefore, this completes the proof of (4.7.2), and hence (4.4.6).

Remark (4.7.6) The order two terms of the poles of the zeta function are constant multiples of 6#(W)E1(R'n n_2

n - 1), 6#(W)E1(Rn n-2,o$, n - 1).

4 The zeta function

149

These distributions are zero if w is not trivial. If w is is trivial and 4D = Mv,,,,D,'D is K-invariant. Therefore, by (4.4.7),

E1(Rn,n-2,A)n - 1) = So, if

Zk(n - 1) k

- n,1R.,1ID (0).

and 4)v is supported on Vk'y for some v E 93

,

n - 1) _ -`Fn, 1Rn,1ID (0) = 0.

This implies that in the case k = Q, all the poles the associated Dirichlet series are simple as was proved by Shintani [65].

§4.8 The work of Siegel and Shintani Both Siegel and Shintani worked on the space of quadratic forms. We consider their work from our viewpoint. Even though I was an undergraduate student at the University of Tokyo and took some courses from Shintani, I was not personally acquainted with him nor was I aware of the theory of prehomogeneous vector spaces at that time. So the greater part of this section is my speculations based on papers by Siegel and Shintani. Siegel did at least two kinds of work on quadratic forms. One is the proof of what we call Siegel-Weil formula [67], and the other is the proof of the average density theorem for the equivalence classes of integral quadratic forms as in this chapter [69].

His argument in [69] was clever, and essentially determined the rightmost pole of the zeta function even though he used the argument based on his Siegel-Weil formula and did not use the zeta function explicitly. Shintani was apparently influenced by Siegel's paper and tried to interpret the content of Siegel's paper from the viewpoint

of prehomogeneous vector spaces. Here, we try to trace what Shintani might have thought in pursuing this goal. We assume that k = Q for simplicity. One difficulty of the space of quadratic forms comes from the fact that we have to adjust the zeta function for the case n = 2 as we saw in §4.2. If we use this formulation, the case n = 2 is more or less included in the general case. However, Siegel and Shintani handled the cases n = 2 and n > 3 quite differently. We used the smoothed Eisenstein series for all the cases, but Shintani used the ordinary Eisenstein series for the case n = 2. The function he considered was the following integral (4.8.1)

JGA/Gk IXv(

)8Ov, (, )E(g°, z)d,

where V is the space of binary quadratic forms, (t, g°) E A" x GL(2)11, and E(g°, z) is the Eisenstein series for the Borel subgroup of GL(2). This function can be considered as a two variable zeta function associated with the action of the Borel subgroup on V. As in the case of the smoothed Eisenstein series, the pole of (4.8.1) at z = p gives us the one variable zeta function. Shintani himself was aware of the approach in this book, but chose the above formulation. He might have chosen the

150

Part II The Siegel-Shintani case

ordinary Eisenstein series, maybe because he wanted to achieve the cancellation in §4.2 explicitly. Then why didn't he consider the generalization of the above function for the general case? Let us consider the following integral (4.8.2)

GA/Gk

jXv(7jsOvka(1b, g)E(g°, z)dg,

where V is the space of quadratic forms in n > 3 variables, g = (t, g°) E Ax x GL(n)°n, and E(g°, z) is the Eisenstein series for the Borel subgroup of GL(n). This would be the natural generalization of the integral (4.8.1). However, there is a slight difference here, i.e. we have to consider all the points in Vk". The effect of this is that the theta series Ovka (1), g°) is not rapidly decreasing with respect to the diagonal part of g° any more, whereas it is true for Ov,, (4), g°) for the case n = 2. In fact the function Ovk. (,D, g°) is barely integrable on G°n/Gk, and it is already non-trivial to show the convergence of the integral (4.8.2) in a certain domain. Moreover, we do not know if (4.8.2) can be continued meromorphically to a domain which contains p. This means that we cannot recover the one variable zeta function from (4.8.2). Shintani must have considered integrals of the form (4.8.2), and faced this difficulty. My guess is that this is the reason why he chose a different approach for the case n > 3. However, he could not compute the residue of the associated Dirichlet series for indefinite quadratic forms in general. So it was natural to look at Siegel's original approach and see how he handled this difficulty. Siegel's approach for n > 3 was quite different from the case n = 2 also. Roughly speaking we can summarize his argument as follows. First, he used his Mass formula to describe µ,,,(x) (see §0.3 for the definition) for this case. Then in order to estimate Iloo(x)bx,oo o CO

(4.8.3)

'

he considered integral quadratic forms with two different signatures (m, n-m), (m2, n - m + 2) such that the discriminant is less than or equal to Q

TIPg[s]. log p

= P

For 0 < S < N, let C,,,,(S) be the number of residue classes modulo Q which contain an integral quadratic form with the signature (m, n - m). Let D(S) be the number of modulo Q congruent classes such that detx = (-1)"-'S for x in such classes. Siegel related (4.8.3) with C,,,(S) by the Mass formula, and showed that Cm(S) + Cm_2(S) < D(S). By estimating D(S), he obtained an upper bound for Cm(S) + Cm_2(S). The lower bound is relatively easy, and this argument was the point of his argument. But why two different signatures (m, n-m), (m-2, n-m+2)? His argument was very clever, and I spent a long time trying to see from our viewpoint what he was trying to accomplish. My conclusion was that Siegel was trying to accomplish cancellations of divergent integrals one way or another. After investigating which distributions Ep (ot, w, w) may give such cancellations for some time, I reached the cancellations of distributions as in (4.6.18), (4.7.5). In a way, his

4 The zeta function

151

two different methods for n = 2 and n > 3 correspond to two different cancellations (4.6.18), (4.7.5) in this chapter. Siegel did not have a fully developed theory of Eisenstein series or adelic language to hand, and the fact that he was able to handle the case n > 3 is due to his genius. We now go back to Shintani's approach for the case n > 3. Recognizing the difficulty of using (4.8.2), Shintani used the approach essentially based on microlocal analysis. If we summarize his approach, we may say, that he considered the relative invariant polynomial as a hyperfunction, and tried to compute its Fourier transform. As I look back, he was unlucky in choosing this approach. Shintani was basically considering the associated Dirichlet series i (4), s) (see §0.3 for the definition). However, while the adelic zeta function has order two poles, the associated Dirichlet series only have simple poles. Nevertheless, the difficulty based on the order two poles of the adelic zeta function existed. In our approach, we indirectly reconstructed the order two part of the adelic zeta function in the inductive computation as in (4.7.5). But, since the associated Dirichlet series only have simple poles, Shintani was, in a sense, fighting against an invisible enemy. It is an interesting question to see whether it is possible to reconstruct the proof of Theorem (4.0.1) by micro-local analysis. Shintani passed away about three years before equivariant Morse theory was established. Nevertheless, he managed to produce all the essential ideas in this book in his single paper [64]. He had almost all the right tools except for the uniform estimate of Whittaker functions at infinite places. In this sense, there is no doubt that the content of this book is what Shintani would have done by himself. The purpose of this section is not to subordinate the contribution of Shintani concerning his work on the space of quadratic forms, but rather to point out how a true pioneer sometimes has to go through a tremendous struggle and yet may not achieve all his goals. And Shintani was a true pioneer of the subject.

Part III Preliminaries for the quartic case In the next three chapters, we consider the following three prehomogeneous vector spaces (1) G = GL(2) x GL(2), V = Sym2k2 ® k2, (2) G = GL(2) x GL(1)2, V = Sym2k2 ® k, (3) G = GL(2) x GL(1)2, V = Sym2k2 ® k2.

Case (3) was handled by F. Sato [55] in a slightly different formulation. These are rather easy cases, and are essentially the same as the space of binary quadratic forms. However, we need the principal part formula for the zeta function for these cases, because these representations appear as unstable strata for the prehomogeneous vector space G = GL(3) X GL(2), V = Sym2k3 ®k2 which we call the quartic case.

Coefficients of the Laurent expansions of the zeta functions for (2) and (3) appear in the Laurent expansion at the rightmost pole of the zeta function for the quartic case, and therefore they are particularly important. The principal part formulas which we are going to prove are (5.6.4), (6.3.11), (7.3.7).

Chapter 5 The case G=GL(2) x GL(2), V=Sym2k2 k2 §5.1 The space Sym2k2 0 k2 We consider the prehomogeneous vector space G = GL (2) x GL(2), V = Sym2k2®

k2 in this chapter. We will use a new technique concerning the choice of the constants in the definition of the smoothed Eisenstein series in this chapter. Generally speaking, if G is a product of more than one GL(n), it sometimes makes the computation easier to choose the constants in the definition of the smoothed Eisenstein series in a certain way. The reader can see the effect of this technique in the proofs of (5.3.9), (5.4.10), and (5.5.6). For the rest of this book, W is the space of binary quadratic forms in two variables v = (v1i v2). We can identify W with k3 by the map fx(v) = xovi + xlV1V2 + x2v2 _ (xo, x1, x2).

The vector space W is V2 in §4.2. Let V = Sym2k2 ® k2 = W ® W. Any element of V is of the form

f = (fl, f2), fl = fxl, f2 = fx2, where xl = (x10, x11, x12), x2 = (x20, x21, x22). We use x = (x1, X2) = (xlo, x111 x12, X20, X21, x22)

as the coordinate system of V. We define Gl = G2 = GL(2), and G = Gl x G2. The group G acts on V by the formula b) /lgl g)( f ) = (afglxl + b fgixz cfgixi + df9ixz) for g2 = (a \ dJ 2 fxl xz C

5 The case G = GL(2) x GL(2), V = Sym2k2 ®k2

153

Let T c G be the kernel of the homomorphism G -+ GL(V),and Xv(91,92) = (det gl)2 det 92. Then Xv can be considered as a character of G/T and it is indivisible.

Let

Let U = {x E V I pi(x) # 0 for some i}. Consider the map Vk D x - (p1(x),p2(x),p3(x)) E V2 Then G2k \ Uk = ]En, and the action of Glk can be identified with the action of

GL(2)k on ]P(Sym2k2). Therefore, this is a prehomogeneous vector space which is essentially the same as Sym2k2.

By the above consideration, Vk 0 0. Let G°,A etc. be as in §3.1. In this case, G' = GA, so t = V. We identify t with {(zl, z2) = (Z11, z12, z21, z22) E R4 z11 + z12 = z21 + Z22 = 0}. We define a Weyl group invariant inner product by (z, z') = Ei j zijzi'j for z = (zip), z' = (z2j). Let yij E t* be the weight of xij for all i, j. We use the notation yij (t) for the rational character of T determined by yij. This should not be confused with t7' which is a positive real number. We identify yij's with elements of R2 as follows.

Let f8 be the parametrizing set of the Morse stratification. Elements of B \ {0} are as follows. Cl 0

C2

Part III Preliminaries for the quartic case

154

C3

Let 01,

C4

, /34 be the closest points to the origin from the convex hulls C1i

, C4

respectively. We choose G" = SL(2) x SL(2) as G" in §3.1. Then Zpi etc. for i = 1, 2, 3 are as follows. Table (5.1.1)

RR

131 =

414)

4

4)

Zo

WO

X12, X21

X22

02 = (0)0i-2f 2) 03 = (-1, 1; -2, 2)

x20,X21,X22

/34 = (-1,1; 0, 0)

x12, x22

M1161

-

X22

SL(2) x {1} {(a2(t-1, t), a2(t2, t-2))} SL(2) x {1}

Let ai = ()3i) for i = 1, 2, 3. By the above table, Zaik = {x E Zp1k I x12, x21 E k" }. We identify Za2 with W and consider Za2k, Z'a2 ok etc. Since M,3 acts trivially

on Z,3, Zaik = Zaik \ {0}. The vector space Z-04 is a standard representation of Mao. Therefore, Za4k = 0. There is one /3-sequence of length 2 which is a4 = (/32,/33 - /32). Clearly, Za, is the subspace spanned by {e22}, and Z14k = Za4k \ {0}. It is easy to see that Vk is the set of x E Vk which are not conjugate to elements of the form (0, 0, x12, x20, 0, 0). This set corresponds to the set of forms without rational factors by the map U -* V. Let Yv,o (resp. Zv,o) be the subspace spanned be the subspace spanned by by {e12, e20, e21, e22} (resp. {e12, e20, e21}). Let {e12, e2o}. Let Z'vok = {x E Z' ,ok X12, X20 E k" }. Then VStk = GkZ'vok Therefore,

11 Vk \ {0} = Vk

3

Vstk

J__L

ll Spik. i=1

Let Za2,o C Z.O. (resp. Z-02,0) be the subspace spanned by {e21} (resp. {e21, e22}). Let Z'a2iok = Za2,ok \ {0}. Then Za2,stk = Ma2kZ'02,ok We write elements of TAO in the form (5.1.2)

t ° = (a2(tll,t12), a2(t21,t22)),

A = d(A1, A2) = (a2(A1 1, A1), a2(A2 1,A2)),

to = At °,

where A1i A2 E IIB+, tij E Al for i, j = 1, 2. We define

dxt° = fl dxtij, dx,A = dxA1dxA2, dxto = i,2

dxAdxt°.

5 The case G = GL(2) x GL(2), V = Sym2k2 0 k2

155

The inductive structures of VBik and Zt, ,stk can be described in the following manner. Lemma (5.1.3) Let Hat C G be the subgroup generated by T and (TGL(2),1). Let Hv C G be the subgroup generated by T and TG. Then (1) Vst,k = Gk XHvk Z'V,ok, (2) Za2,stk = Ma2k XHa2k Z'ja,ok.

The proof of the above lemma is easy and is left to the reader. Let w = (w1i w2) be as in §3.1. We use the alternative definition (3.1.8) of the zeta function using L = V,', and use the notation Zv (4i, w, s), w, s). Let [ , ]' be the bilinear form such that 1

1

2

2

[x, y] v = X1OY10 + X12Y12 + X20Y20 + X22Y22 + 1 X11y11 + 1 X21y21

for x = (x2j), y = (yjj) E V. Then it is easy to see that this bilinear form satisfies the property [gx, tg-ly]', = [x, y]'v. We define [x, y]v = [x, TGy]'I, and use this bilinear form as [, ]v in §3.1. Let (5.1.4)

b((g°)`x) -

Js(D,g°) = xEVk\V,

I°((D,w) = f

Cg°x), xEVk\Vk°

w(g°)J8(4',g°)dg°. A /Gk

For A E R+, let 4ba be as in §3.5. By the Poisson summation formula,

Zv(4', w, s) = Zv+(I,, w, s) +

w-1, 6 - s) + JR+ A8I°((>a, w)dx A.

We will study the third term in §§5.2-5.5. Throughout this chapter we assume that li = Mv,u,4) (see (3.1.11)). Let 8(g°, w) be the smoothed Eisenstein series which we introduced in §3.4. We choose the constants in (3.4.3) so that C1 = 1, C = C2 > 4. This choice of constants C1i C2 will have some effect later as we mentioned at the beginning of this chapter. We define (5.1.5)

I ((D, w, w) = J A/Gk

Since J8(4>, g°) = ev,. (,D, g°)

w(g°)J(, g°)e(g°, w, )dg°.

- ev,: ($, (g°)`), by (3.1.6), (3.4.34),

I(4, w, w) ,., CGA(w; p)I°(4), w).

We define (5.1.6)

V,st(4), W, w) = JCGOIGk w(g°) ev,k (4), g°)-O(g°, w)dg°;

=v,st(4), w, w) = JG/Gk w(g° )ev(&,

(g°)t)(g°, w)dg°.

Part III Preliminaries for the quartic case

156

The functions evik (,D, go), Cvtk (', (go)`) are slowly increasing by (1.2.6). Therefore, by (3.4.34), the above distributions are well defined for Re(w) >> 0. Since e((go)`, w) _ 6°(go, w), "v,st(4p, w, w) = Ev,st(4) , w-1, w).

Let 6#(w) be as in §3.6. By (3.5.9) and (3.5.20), (5.1.7)

I(-P,w,w) = 6#(w)A(w; p)+0) - -P(0)) +

ep(vp(1p,w,w)+Ep(4p,w,w)) p,l(p)=l

+ °v,st(4D, w, w) - EV'st(-D, w, w).

If p = (tI's), and 0 = tIi, p belongs to class (3) in §3.5. Therefore, °p (4b, w, w) is well defined for Re(w) >> 0. This implies that °,p (4), w, w) is well defined for Re(w) >> 0 also. If tI = a2, t3, °p (4, w, w) = 0, and Ep (I, w, w) is well defined for Re(w) >> 0 by (3.5.5). Therefore, all the distributions in (5.1.7) are well defined for Re(w) >> 0. If t = 04, tI satisfies Condition (3.4.16)(1), and therefore, ,3p(,P,w, w) is well defined for Re(w) >> 0 also.

Let i- be a Weyl group element as in (3.4.13). We use the same notation as in §3.6 in this chapter. So sT = (Sri, Sr2) E C2 where s,i = ZiT1(2) - ZiT1(1) for i = 1, 2.

We define ds, = ds,lds,2. For a similar reason to that above, Bp,T(4D,w,w) is well defined for Re(w) >> 0 in the same cases as 3p (,D, w, w)

§5.2 The adjusting term In this section, we define the adjusting term Tv(W, w, s, Si). This distribution is required in order to describe the contribution from V. Let ' E 9'(Zv,oA). Let q = (ql, q2) E (A1)2, uo E A, µ = (µl,µ2) E R+, dx q = dxgldxg2, and dxµ = dxµldxp2. We consider the function a(uo) in §2.2. Let wl, W2 be characters of Ax /kx, and w = (Wl, W2). We define w(q) = wl(gl)w2(g2). Let fv('F, It, q, uo, Sl, 82) = µ2l a(uo)32 ww(µlµ2 lql, µ1µ2q2, 2p1L2g2uo)

Note `2' in the third coordinate. Definition (5.2.1) For complex variables s, S1, 82 and q1, w as above, we define (1)

Tv(W,w,S,sl,S2) =JR2

x(Al)2xA

w(q)µifv('y,µ,q,uo,sl,32)dxpdxgduo,

(2) Tv+('I, w, S, S1, S2) = JR2 x(A1 )2 xA w(q)pifv('y, p, q, uo, S1, S2)dxidxgduo, µ1>1

(3)

TV' (12, W, Sl, S2) = f

+x(A')2xA

w(q).fv('I',1, µ2, q, uo, sl, s2)dx p2dx gduo.

Proposition (5.2.2) The distributionTv(1Y, w, s, sl, S2) can be continued meromorphically everywhere, and is holomorphic for Re 2 s1 > 1, Re si > 1, Re si + ti Re(s2) > 2. The functions (2), (3) are entire functions.

2

5 The case G = GL(2) x GL(2), V = Sym2k2 0 k2

157

Proof. The convergence of (2), (3) is easy, so we briefly discuss (1). Suppose that there exist Schwartz-Bruhat functions 'y1 E 9'(A), ty2 E 9'(A2), such that 'F (Y1, y2, 2y3) ='I'i(yi)'P 2(y2, ya). Then

TV(W,w,5,s1,52) = 2E1(IFI,wl,

s

2s1)Tw(W2,w2,

s

2s1,s2),

where the last factor is as in §4.2. So TV (W, w, s, s1 i 52) can be continued meromorphically everywhere. In general, we can assume that IF = and the finite part W f of T has a similar decomposition. So in order to consider the meromorphic continuation of TT(q', w, s, s1i $2), we can assume that Tv(W, w, s, s1i s2) has an Euler product and its finite part can be continued meromorphically everywhere and is holomorphic for Re 2 sl > 1, Re si > 1, Re sl + Re(s2) > 2. One can prove that the infinite part of Tv(W, w, s, s1i 82) can be continued meromorphically everysi Re sl > Re + Re(s2) > 1 where and is holomorphic for Re 2 sl > using the polar coordinate as in §4.2.

2

2

a

a

Q.E.D.

Definition (5.2.3) For 'P, w, s, s1i s2 as above, we define (1)

Tv (iY, w, s, 1 s )= d - ds2

TV ('y, w, s) s1) $2), 82=0

d

(2)

TV+('y, w, s, s1) = ds 2 TV' (IF, w, s1) =

(3)

TV+('y, w, s, s1, s2), 82=0

d

1

ds2 82=0

Definition (5.2.4) (1) Let 'y1 E Y(Zv,oA). Let p, q, no be as in the beginning of this section. We define eZv,o ('I'1, µ, q, u0) _ E 1y1(1!1µ2 181x, µ1µ'242y, 2µ1µ2g2yu0) x,yEk"

Note `2' in the third coordinate. (2) Let 'y2 E Y(Za2,oA). Fort' E Ax, uo E A, we define 'y2(t'x, t'xu0)

eZa2,0('y2i t', n0) _ xEkx

(3) For ty3 E 9'(Za2A) and gal E G°A fl Ma2A, we define ®z;2,o ('y3, 9a2) =

1:

'ya(9a2x.

xEZa2,ok

The following lemma is clear from the above definition.

Part III Preliminaries for the quartic case

158

Lemma (5.2.5) Letw1iw2 be characters of A' 10, andw = (w1,w2). Lets1is2,q, p be as in (5.2.1). Then for kk E So(Zv,oA), 1

Ty (T, w, 81,S2) fR+x(Al/kx)ZxA

w(q)l4a a(uo)82eZVo ('Y,1, i 2, q, uo)dxµ2dxgduo.

§5.3 Contributions from 01, 03 In this section, we consider paths p = (0, s), p' = (a, s') such that D = a1 or 03 w-1, w). So we only consider p. For 0-sequences a = a1, 03, Ma = Tk. Since l(0) = 1, we do not have to worry too much about Ape etc. for these 3-sequences. So, instead of the the general notation go E G°A n MaA, we just use t° in (5.1.2) to describe elements of G°A n MaA. When we consider A, Ax in the next three sections, we consider the standard measures on them. We start with some definitions. and s(1) = 0, s'(1) = 1. Since e(g°, w) _ 9((g°)`, w), -Ep, (4D, w, w) = SEp

Definition (5.3.1) Letw = (W1,W2) be as before. We define (1) bv,st(w) = b(wiw2 2), wv,st = (w2,w2), (2) Sai (w) = b(wlw2 1)6(w2), wai = (w2,w2), (3) ba2(w) = 6(w2), wag = (1,w1), (4) 602,-t GO) = bas(w) = 6D4 (w) = b#(w) = 6(w1)6(w2)

Definition (5.3.2) For 4)1 E .9(VA), 42 E .9(Z02A), we defineRv,o4)l E.(ZV,oA) and Ra2,A2 E 9'(Za2,OA) by Rv,o,D(x12, x20, x21) = f 4(0, 0, x12, x20, x21, x22)dx22, JA

(1)

R02,A(x21,X22) = -P(0,X21,X22)

(2)

Definition (5.3.3) For 4 E .9(VA), and s = (s1i s2) E C2, we define

s1-1 s1 + 282 - 1

bal (w)

2

2

,

2

where the first (resp. second) coordinate of Rol4D corresponds to x12 (resp. x21).

For s E C, we define Eal (4, w, s) = Eal (4), w, s,1). Let 00

(5.3.4)

Ear (1), w, s) = E Ea"(j) (4', w, So)(s - so)i i--2

be the Laurent expansion at s = so. For the rest of this chapter, r = (r1, r2) is a point in R2. First we consider the case 0 = 01.

Lemma (5.3.5)

(-ib, w, w) - 0.

5 The case G = GL(2) x GL(2), V = Sym2k2 ® k2

159

Proof. Let a = (al, a2) E V. For u = (ul, u2) E A2, we define

< an(u) > = < alul >< a2u2 > . Let

fl,a(t°) = f

®y,1 (4', t°n(u)) < au > du,

NA/N,

f2,a(w, t°) =

JNA/N,

9(t°n(u), w) < au > du.

Then by the Parseval formula,

fl,(t°)f2,-(w,t°)(t°)-2Pdxt°

'=p(-D+w, w) = L2/T,. w(t°) aEk2\{0}

Lemma (5.3.6) For any N1, N2, N3 > 1, Ifl,a(t°)I < A 2A-'(A2a21)-N1A2N2(A2 12)N3. aEk2\{0}

Proof. Let ui = u'l (x, u) = x21u1 + x12u2i u2 = u'2 (x, u) = X12U2. Then u1 X2_1 (ui - u2), u2 =x12 u2. Since A/k is compact, f1,a(t°) is equal to l

fA2/k2 x12,:21 Ekx

( 712(t°)x12,

721(t°)x2i, 722(t°)(x22 + ui)) < au > du

x22Ek

X12,

kx

A2/k2

Ra1D(712(t°)x12>721(t°)x21,722(1°)(x22 + ul)) x22E:k

x < au > du. Since

< au >=< aix21 u1 >< (a2x12 - alx21 )u2 >,

we only have to consider terms which satisfy the condition a2xr2 = a1x21. This implies that a1i a2 are both non-zero. _ Let 1) 1 be the partial Fourier transform of Rat 1' with respect to the third coordinate and the character < >. Then it is easy to see that fl,a(t°) = Al 2A2 1 E 411(712(t°)x12, 721(1°)ala2 'X12, 722(t°)-1a2x12) x12Ekx

Therefore, if we choose '2 E 9(A3) so that I'D1I < (D2i I fl,a(t°)I

Al 2'x2 103(D2, 712(t°), 721(t°), 722(t°)-1)

aEk2\{0}

Hence, the lemma follows from (1.2.6). Q.E.D.

Part III Preliminaries for the quartic case

160

Let M > 1 + C. Clearly, If2,«(w,to)I = i1,TG,l(t0,w)) aE(kx )2

where I = ({1}, {1}). Let 6 > 0. By (2.4.10), for any 11, l2 >> 0, <
Ilig jTG l(t0, Z)I

for Re(z) E DI,TG,6i where c1,6(z) is a function of polynomial growth. We can choose q E Dr,T,6 so that L(q) < wo. Therefore, there exists S' > 0 such that for any11,12>>0, Oi,TG,1(t0, w) << A A 12

forwo - S' < Re(w) < M. This implies that for any N1, N2, N3 > 1 and ll, 12 >> 0,

Ifl,a(t°)f2,a(w,t°)I «al'a2A 2a21(a1A21)-NIA N2 (A

2)N3.

«Ek2\{0}

Since the convex hull of {(2, -1)j(0j1)j(-2, -1)} contains a neighborhood of the origin of 1R2, the above function is bounded by a constant multiple of rd2,N(Al,.\2)

for any N > 0. This proves (5.3.5). Q.E.D.

Lemma (5.3.7) w)

_

(2)7r11

2

fRe rl>13,rr2>1

a

('p,

w, sT)'+T (w, sT)dsT.

Proof. Suppose that f (q) is a function of q = (ql, q2) e (A/ k' )2. Then J

(Al /k x )4

w(t0)f(t12t21,t11t12t22)dxto = sa(w) J

wo(q)f(q)dxq (Al /k x)2

after the change of variable Ql = t12t21, q2 = t11t12t22. Let t' = (ti, t2) _ (1q1,µ242) E (Ax)2 where p1,a2 E R+, gl)g2 E A1. Let dxt' = dxtidxt2. We make the change of variable µa = Al2A2i µ2 = A21. Then dxµldxµ2 = 2dxaldxA2i and Al = jtl 2 112 2, A2 = µ2

It is easy to see that Qp(to)(to)TZ+P =

T1-1

P1 2

2

112

Therefore,

sa2w)

f

W,sr)

(Ax /kx )2 (Ax

Hence, Ea(4),w,sr)AT(w;sr)dsr. _

Q.E.D.

1

1>3, r2>1

5 The case G = GL(2) x GL(2), V = Sym2k2 ®k2

161

Lemma (5.3.8) Cp,T ($, w, w) - 0 unless r = TG. Proof. We choose rl, r2 >> 0 for r = (1, 1), r1 fixed and r2 >> 0 for r = (TG1,(2),1),

and r2 >> 0 and r2 fixed for T = (1, TGL(2)) Then L(r) < w°. This proves the lemma. Q.E.D. If r = TG, then srl = Zii - Z12 and Sr2 = Z21 - Z22

Proposition (5.3.9) Let r = TG and s = sTl. Then (OD, w, W)) 2,7f

0-1

Re( >3

1

2

1)0(s)A(w; s, 1)ds.

rl

Proof. By (5.3.8), ED(-P, W,ST)AT(w;Sr)dsT

((D,W,w)

rl>3, r2>1 2

27r

+

f

Re(.

l>3,

ST)'+T(w;ST)` s, 1>>r=2r

>O

60

27

-1

Re(1>3r1

If rl is close to 3 and r2 is close to 0, then r1 + Cr2 < 1 + C, because C > 4. Therefore, we can ignore the first term. This proves (5.3.9). Q.E.D.

Next, we consider a path p = (0, s) such that 0 = Z. Proposition (5.3.10) By changing V) if necessary, 1z p(4D,W,w) - 0.

Proof. If f (q) is a function of q E X41 /k x ,

f(A1/kX)4

w(t °)f

(t12t22)dx t °

= 6# (w) f

f (q)dq

Consider the situation in (3.1.18). In this case, AD = {d(pi, pi) µl E ]f } and Aa = Aa = {d(p2i µ2 2) µ2 E III. }. We make the change of variable Al = Then dxA = 5dxµldxp2. Let r, sT be as before. It is easy to 91/p2, A2 = see that epl(A) = y5 and and ATZ-p = Misr1 Esr2-3/le2ry-2sry-1. Also LS,,, + h in I

1

(3.5.17) is {sr I sr1 = 2sT2 + 1}. Therefore, by (3.5.17), 'Ep,T (qD, w, w) is well defined for Re(w) >> 0 for all T, and

6a '-'(7,T

W1 w) =

2ir

(w)

JR(.2)3 r2>1

E1(Ra4l, sr2 + 1)Ar(w; 28T2 + 1, ST2)dsT2

Part III Preliminaries for the quartic case

162

Let pT be the point in C2 which corresponds to p by the substitution in §3.6.

Then it is easy to see that pT = (1, -1) for T = (TGL(2), 1), p,. = (-1, 1) for T = (1, TGL(2)), and pT = (1, 1) for T = TG. For these cases, pT is not on the line s,1 = 2sT2 + 1. Therefore, by (3.5.19), we can change t/i if necessary and assume that Bp,T (4), w, w) = 0. For r = (1,1), we choose r1 = 2r2 + 1 >> 0. Then LT (r) = -r1 - Cr2 < w°. Therefore, p T ((D, w, w) - 0 for this case also. This proves the proposition. Q.E.D.

§5.4 Contributions from D2,04 In this section, we consider paths which start with /32. We do not consider paths

p = (cz,s) such that s(1) = 1 for a similar reason as in §5.3. Let p2 = (02i52) be

a path such that 52(1) = 0. Let Ni = 04,s4i) be a path for i = 1,2 such that 541,(1) = 0 and 541(2) = 0,$42(2) = 1.

We use the notation Eat (Ii, w, s), Ea2,ad(T, w, s)

for the zeta function, the adjusted zeta function for the space Za respectively. Also we use the notation Ta(Ra2,OW1w1s) etc. for the adjusting term etc. We define Ea2,ad,(j) (W, W, s°) similarly as in (5.3.4).

Since D2,04 satisfy the condition (3.4.16) (1), (5.4.1)

°p2 (D, w, w) =

w, w) + Ep2+(4, w, w)

p2#(4?,w,w)=p2#(4)1W1W) + yP42 (4?, w, w) - up41 (cb, w, w).

Also w),

P2+('P1 w, w) _ 12+(4D, w, w) +

w, w) =

w, w)

w, w).

Let f be a function on Za2A. Then J(A1/kx)z

= 6(w2)

w2(t21t22)w1(det91)f(t2291x)d"t21d
f

w1(detgl)f(t2291x)d"t22,

where gl E G1A, X E Za2A. It is easy to see that ape (a2(A1 1, Therefore, by (3.5.13), (5.4.2)

-1),1) = all

(R, 2),

ypz+(,D, w, w) ^ CGA(w; PS2 wED2+D2 I, wp2

.02+((k> w, w) - CGA(w; P)ba2 (w)Ea2+(-Fa2 Ra2 "P, wpzl, 1)'

Let gal = (n2(u°), 1)t°. We define X-02 = {kga2 I k E K, Al < a(u°)-2

5 The case G = GL(2) x GL(2), V = Sym2k2 0 k2

163

As in the case W = Sym2k2, Xa2/Tk is the fundamental domain for G°A/Ha2k. Let dg-02 = dxt°duo. Then p2,st+(4p1 W, w) = IX02 naA/z.k W(9-02 )E)Z0'2 (Ra2'P+9a2)A2 282 ( 2, 049021

1-'2 nA1aA/Tk <1

W (9az

w)dg

(Ra2'Pl 0.02 (9D2

Lemma (5.4.3)

f

(1)

^ (2)

a2 nBA/T,,

J J

2

(Ra2-D, 902)A22eN(t°, w)d9a2 a2 nBA/Tk

!C2 nBA/Tk ,.

W(a2)®z2(Ra21)) 2g (2,w)d

a2

w(2(Ra2 , Ba2 (2 ))A2 S,2 (21

fBA/Tk

W(9,2)eZa2

Proof. Let I = ({1}, 0). Then I8p2 (902 1 W) - -N (t°, W) I << E 41-,1(9021

w).

T

Let M > wo. By (3.4.30), for any e > 0, there exist 6 = 6(e) > 0 and c E t* such that Ilcllo < e and for any l >> 0, 0

(t)c)"1

4,73 (W121 w) <<

forwo-5
.+xAx /kx xA

Ai It'I`

where c' is some constant, and l' can be arbitrarily large. Since T0e+(`f'1 s, s1) converges absolutely for all s, i, the above integral converges absolutely. Q.E.D.

Let r, sT be as before. We define (5.4.4)

Ep2,st+((D, sr) =

T0e+(R02,0R024" 1 + Sr2, 12T1) Tae+(Ra2 °a2 Ra2 41, 2 - 9T2,

Sr) _

,

STl -1

Sr1-1

1

2 2

Part III Preliminaries for the quartic case

164

Lemma (5.4.5) (1)

J

nBA/Tk X1>1

(2) f 1

st(W) dal (2)

2

Ep2,st+(d),

sr)AT(w; ST)dsT.

R`(°*)=* 1>1, r2>1

fX02 nBA/zk W(ga2 )0Zaz.o (RaAf ea2 (go. ))A21eN,r(t0f w)d-jaz X1<1 2

daz,st(W)

Ep2,St+(4),ST)Ar(w;Sr)dST.

rI>', '2>1

In particular, these integrals are well defined.

Proof. Suppose that f (q) is a function of q E (Al /k")2. Then

f f

W(t°)f(tiit12t22)d"t° =

A'/kX)4

d2,st(W)

f

l /kx

Therefore, (1) is equal to da2,st(w)

6z02,0(. 52,0R524) t',UO).2eN,T(to

w)dx.2dxt'duo,

R+XAX /kX XA

where t' = )(2t11t12t22 Since (t°)'=+P = jt'js.2+1A1r1-1 the lemma follows for (1). The proof of (2) is similar. Q.E.D.

Lemma (5.4.6) If r 54 TG, 2 1

(1)

C27r / 1

(2)

C27rv/---l

f

E2,5+(, ST)XT(wj Sr)dST ^ 0,

rl>1, r2>1

)f 2

Re(ar)=* 1>1, r2>1

Ep2,st+(4)1 sr)AT(wisT)dsT .v 0.

Proof. Since Tae+( , S, Si) is an entire function for IF = Ra2,°Raz 4)> Ra2,oga2 Ra2 4)f we can choose rl, r2 >> 0 for T = 1, r1 fixed and r2 >> 0 for T = (TGL(2), 1), and r1 >> 0 and r2 fixed for T = (1, TGL(2)). Then L(r) < wo. This proves the lemma. Q.E.D.

By moving the contour in the above integrals so that r2 < 1, we get the following proposition.

Proposition (5.4.7) Let r = rG, and s = srl. Then (1)

(2)

WI w)

W, w)

da2,st (W ) 27f

R`(°)=*1 Ep2,st+(1)1 Sf 1WS)A(wi s, 1)ds, rl>1

2,P2,st+(, ()

27r

fRe(.)1 rl>1

L'p2

s, 1)¢(s)A(w; s, 1)ds.

5 The case G = GL(2) x GL(2), V = Sym2k2 0 k2

165

Next, we consider P41, P42. Let r, sT be as before. We consider the domain DT (2.4.7) for r = Re(sT) also.

Lemma (5.4.8) (1)

w7 w)

`-'p41 ,T

_

Qaa4(w)

E1(Ra4"Pp417

J

R`(")=r

-ST1 + 2ST2 + 1

R.("r)-r rEDr, -1>3-2r2

ST1 + 2ST2 - 3

rEDr, r1<2r2+1

(2)

p42,T ('D,

w7 w)

_

s.22t)..,

A,(w; ST)ds.r.

Pba4 (w) 27r

1/-

AT (wj ST)dST.

Proof. Consider sT such that r = Re(sr) satisfies the conditions in (1) and (2). It is easy to see that Ap412TA° = {t° 1 A2 < 1}, and Ap422TA = {t° 1\2 > 1}. Easy computations show that 1

0

2

2

0

2

O`p41 (t) = A1A27 O'p42 (t ) = AlA27 (t0)Tz+P = As"-'As 2-1,

1-3,2

A1f

Suppose that f (q) is a function of q E Ax IV. Then

(ti2t22)dxt° = ba4 (w) f/kx f(q)dxq fA1/kx)4 (

after the change of variable q = t212t22. Let t' = pq where µ E R+. We make the change of variable p = A A2, A2 = A2. Then dx pdx A2 = 2dx A1dx A2i and

A1=µ2A22. So o,P41

(t°)(t°)TZ+P

=p

',.1+2a,.2+1

1}a

2r1 A2

2

°P41 (t°)Ba2 (to)Z+P

= µ

l+s 2

-s

+2s

-3

A2

Since 1

2 JR2 xA1 /kx

,I1+ar, \ fA 2 ^2

a2<1

+1_ 2

2

OZ04 (R a41)p41 Y

II

` q)dJC ^2dx dx

q

)

-srl + 2ST2 + 1 1

fR+XA1/kx f'I 1+er7 A2 e1,+2ar2-3 OZ04 2

2

` 'I x x (Ra44)p42,pq)dA2d

/a] q

a2 >1

E1(Ra4'DP42'

Sr1+1 2

)

ST1 + 2ST2 - 3

the lemma follows. Q.E.D.

Lemma (5.4.9) If T TG, p4i,T ('D, w,w) - 0 for' = 1, 2. Proof. If T = 1 or (TGL(2),1), we fix r1 and choose r2 >> 0 for P41, P42. Then L(r) < wo and r1 < 2r2 + 1, r1 > 3 - 2r2. If r1, r2 are close to 1, then L(r) < wo

Part III Preliminaries for the quartic case

166

and rl < 2r2 + 1 for T = (1, TGL(2)) and P41 If rl >> 0, then rl > 3 - 2r2, and L(r) < wo for T = (1, TGL(2)) and P42. This proves the lemma. Q.E.D.

Proposition (5.4.10) Let r =To, and s = sTl. Then r (1)

1
3-s

A 1
s -1

a4 (w)

T

"V42 (`r, w, w)

(2)

21r

2) cb(s)A(w; s, 1)ds,

I/'

eva4 21r -1 J

w, w)

)

El(Ra44Dp42''

1v

,

¢(s)A(w; s,1)ds.

Proof. P41

- fr1=2, r2=i Re(ar)=r

+

Pty

AT(w, sT)dST

i)

2ST212

rl =r2=2

yy

E1(R04iDP41' 9 2AT(w;sT)dsT !l -STl + 2S-r2 + 1 Jn1(Ra4

(w) 1
ON (w)

27r J

p41,

(s)A(w; s,1)ds

3-s

2e(.>=rl

E1(R04

0(s)A(w; s,1)ds,

3-s

R<*i<3

because L(2, 42) = 2 + AC < 1 + C (C > 4). This proves the first statement. 4 El(R041)P41,-2-)AT(w;s,)dsT

ti

$T1 + 2ST2 - 3

JRe(sr)-(2,4)

+

Paa4(w)

2

J 2aV-1 L-(.)=j 1
S- 1

Pba4 (w) /' 27rVr--1 1
1

)

)

0(s)A(w; s, 1)ds

q,(s)A(w; s, 1)ds

for the same reason. Q.E.D.

Proposition (5.4.11) p2#((D, w, w) - A(w; P) Pb#(w)1b(0)

(1)

2

'p2# ((D, w, w) " -A(w; P)96#(w)`7'02 Rat Y (0).

(2)

Proof. Let 7-2 be a permutation of {1, 2}. By the Mellin inversion formula, _

2

(27rV

+) I

1A/G1kXT2A/T2k

b#(w) f

a2<1

A(1,T2)(w; -1, s) e

rl>2

s-2

ds.

w(91,t2)A21.I{l}xN2,(1,T2)((91,t2),w)d91dxt2

5 The case G = GL(2) x GL(2), V = Sym2k2 ®k2

167

If T2 # TGL(2), we can choose r >> 0, and ignore this function. 11 72 = TGL(2), 1

27r-,/

A(,,,2) (w;

1f

-1,

s-2

*1>2

s) ds

- -Pb#(w)A('w; p).

Similarly, 2

(27r) fGO1A/G1kXT2A/T2k w(gl,t2)A2r'11}xN2,(1,T2)((g1,t2),w)dgldxt2 a2S1

= b#(w) 2ir

JRl >1

A(1T2)(w;-1,s)ds

8+1

Bb#(w)A(w; p) 2

This proves (5.4.11). Q.E.D.

§5.5 The contribution from Vsik In this section, we consider the contribution from V. Let b = (n2(uo),1)t°(1, n2(u2)), db = ))2 dxt°duodu2,

where t° E TT°, uo, u2 E A. We define

Xv={kbI kEK,A1
Proposition (5.5.1) Let T =TG, and s = sT1. Then v,st(

w w)

TV' (Ry,o(D, wy,st,1, Pbv,st (w) f J R<(<>=r1 s-1 2ir

2 0(s)A(w; s, 1)ds.

*1>1

Proof. By (5.1.3), =V,st(w, 4p, w) = f

A/Gk

w(g°)Ov(, g°)6(g°, w)dg°

JA/Hv k

w(g°)Ozc o ('b, g°)e(g°, w)dg°

JX1 fBA/Tk w(b)Ozv

0

(41,

w)db.

Lemma (5.5.2) Let c1i c2 be any constants such that c1 +2c2 > 0. Then the integral JAil x v nBA/Tk

A22Ozv.u (4, b)db

Part III Preliminaries for the quartic case

168 converges absolutely.

Proof. Let µ = a12A2, and IF = Rz,,,o4D. It is easy to see that

f

Ozv (4), b)du2 = .112A-'6zvo (91, 1, µ, q, uo),

where q = (qi, q2), qi = t12t2l, q2 = ti1t22 Therefore, the above integral is equal to A11-2Ac2+1 eZV

fRx(A1/kx)2xA a':5

o (`1',1, µ, q, uo)dx aid x µdx gduo

-(UO)-1

1 +2C2 µC2+10

=

1

zv,o (',1,µ,q,u0)dxA1dx/tdxqdu o

R+x(A1 /kx )2 xA

a(-0)-1

a1

- I2 x(A1/kx)2xA _

C2+1

µ

< _ 1+2c2

a(uo) Cl +2c2

2

pzv,u('I',1,µ,q,uo)dx µd xt°duo,

which converges absolutely.

Q.E.D.

Lemma (5.5.3)

f

(b)O, (, b)e(b, w)db v nBA/Tk

f

w(b)Czo (, b)eN(t°, w)db.

XV n BA/Tk

Proof. Let M > w°. By (3.4.30), there exists a slowly increasing function h(A1, A2) and a constant 6 > 0 such that for any 11, l2 >> 0, 1-1(b, w)

- -IN (t', w) I << h(\1, A2) sup(A

,

a2, A A2 )

for w° - 6 < Re(w) < M. Therefore this lemma follows from (5.5.2). Q.E.D.

Suppose that f (q) is a function of q E Ax /kx. Then

J

A1 /k x )4

f(ti2t21,ti1t22)dxt°=6v,st(w)f A1 /kx )2 wv,st(q)f(q)dxq

2 q2 = ti1t12 We make the change of variable after the change of variable q1 = t2 µ = A-2A2, Al = Al. Then dx µdx Al = dx Aldx A2, and A2 = .1iµ Clearly,

IA ez; ,o

(n2(uo),1)t°(1, n2(u1))duo = A 1

2

A2

1ezv,O (I',1, µ, q, uo).

Therefore, 8v,gt('D, w, w)

^ 6v,st(w) f

R+x(A1/kx)2XA A1<

a(up)-1

wv,st(q)6zv,o('P,1,µ,q,uo)µ.'N(t°,w)dxaldxµdxgduo.

5 The case G = GL(2) x GL(2), V = Sym2k2 0 k2

Let r, sr be as before. We define

169

Ev,St,r (Z , w, w) by the following integral

(5.5.4) bv,st (w) R+x(A1/kx)2xA

wv,st (9)ezvo (`I',1, p, q, uo),a.N,T(to, w)dxa1dx µdxgduo,

al< a(u0 )-1

if this integral is well defined. Let (5.5.5)

Ev,st(p,w,sT) = bv,st(w)

TV1 (RyoD, wy,st, Sr21

-srl-23.2+31 2

Sr1 + 2Sr2 -3

It is easy to see that (to)rt+a = Ai,1+2sr2-3µ3r2-1 Therefore, we get the following proposition.

Proposition (5.5.6) The distribution E'vt,T(4>, w, w) is well defined for all r and

vst,r('p, w, w) - (

2

Re (ar>=r

27r

rE Dr, rl>3-2r2

If r # rG, we can choose r1 or r2 >> 0, and v,st,r(4), w, w) r 0. Let r = TG. Then C27r

+

2

-i

f TV' (Rv,o4, wv,st,1, 123)

Pby,st(w) 21r

1

S-1

Re(e)=r1

rl>1

¢(s)A(w, s, 1)ds.

Since C > 4, 2 + s C < 1 + C. Therefore, we can ignore the first term. This completes the proof of (5.5.1). Q.E.D.

§5.6 The principal part formula In this section, we prove the principal part formula for the adjusted zeta funcThis implies that

tion. Throughout this section, we assume that 4) = MD,u,, Ra 4) = RD 4) for all the path p. We define

J1(4),w,S) =

bal(es) 2

E2(Ral-D,wal,

s-1 s+1 2

2

+ ba2,st (w) T0e+(Ra2,oR02 'b, 2,

1 2

s-

Ta2+(R02,0 g-02 Ra2 -P,1,

+ ba2,st (w)

+ba3(w)

12 s)

s-1

(El(Rs4 42 s-1

s-3 1_8)

1

+

8)

bv,st(w)Tv(Ry,oP, wv,st,1,

s-1

2

Part III Preliminaries for the quartic case

170

J2(1b, w) = Sae (w) (Eaz+(Raz , w02, 2) + Eat+(-97a2 Ra2

wa2, 1))

(0)

Then, by (5.3.9), (5.3.10), (5.4.2), (5.4.7), (5.4.10), (5.4.11), (5.5.1),

w, w) -

0

27rV /-l IR.(j )1''1

(Ji( &, w-1, s) - J1(), w,

s,1)ds

+ CGA(w; p)(J2($, w-1) - J2(4,, w)).

We define

J3(P, w) = 80 1(w)E0 1,(o) (C w, 1) Sa2,st (w) 2

(Ta2+(Ra2,oRa2 1D, 2) + Tae+(Ra2,oRa2 gal , l))

ba42w)

+ _

El,(1)(R04(Pp42,1) sa4 (w) 2

(El,(-1)(Ra44PD41, 1) + El,(o)(Ra4(I)p41, 1))

by,st(w) 11V 2

wv,st, l)

Tv k

l))

+ Sae (w)(Ea22+(Ra2'P) wag) 2) +

-'2728#(w)

(2o

) + 24)(0)) +

Then, by Wright's principle, Jl w-1, s) - Jl (4), w, s) must be holomorphic at s = 1, and the following proposition follows.

Proposition (5.6.1) I°((b,w) = J3(&,w-1) - J3(-D, w). Let IF = Ra2 41). Then, by the principal part formula (4.2.15), Ea2,ad,(o) (

)wag, 2) =Eat+('1') L')02)2) + Eat+0, wa2', 1)

8(1) 2

(Ta2+(Raz,o'y, 2) + Tae+(Ra2,o'I',1))

- 9326(w1) 8(w1) 2

(o2wo) +

F(0))

1)(Ra44)p4111) + El,(o)(Ra44Dp41,1))

Therefore,

8a42w)

J3(,b, w) = 8a 1(w)Ea1,(o)(11, w,1) +

+8a 2 (w)E 0 2,ad, (o)(Raz

E1,(1)(Ra44)p42, 1) +X28# (w) -P (0)

, waz ,

2)-8v,st(w)Tv(Rv,o-P,wv,st,1).

The following relations are easy to prove, and the proof is left to the reader.

5 The case G = GL(2) x GL(2), V = Sym2k2 ®k2

171

Lemma (5.6.2) W,1)

(1)

=

w,1)

2

-,\-2(log.1)Eal,(-1)(,b,w,1) + A-2EO,,(o)(4b,w,1). w-1, 1) =

(2)

1A-4(logA)2E0,,(-2)*

w-1, 1)

+ A-4(1og.A)Ea1,(-1)(4',w-1, 1) 1).

2A-2(log

(3)

E02,ad,(o)(R021'a,w02,2) =

A)2E02,(-2) (Ra24P,w02,2)

- A-2(log A)Ea2,(-1) (Ra2

W0212)

+ ^-2Ea2,ad,(o) (Ra2 `l', W021 2).

-4(log )2Ea2,ad,(-2) (Ra2 $, wa2 2)

E02,ad,(o) (Ra2 'a, wa21, 2) =

(4)

+2 \-4(log A)F'a2,ad,(-1) (Ra2 $, wa21 2) + A-4Ea2,ad,(o) 0, w0-21, 2).

E1,(1)(Ra4")Ap42,1) = IA-2(logA)2E1,(-1)(Ra4Dp42, 1)

(5)

+ A-2(1ogA)El,(o)(R04')p42,1)

+A-

2E1,(1)(R04'p42,1).

E1,(1)(Ra4Cp42,1) = 2a-4(1ogX)2E1,(-1)(Ra3-9702$,1)

(6)

+-4E1(1)(Ra3

1).

Since

-6(2wl)E1,(-1)(Ra4''p42,

2) _ Ea2,ad,(-1)

w02, 2) =

6(wl) 2

1),

E1,(o)(Ra4 'bp42,1),

etc. and 604 (w) = E(wl)6a2 (w), 6a,2w) 602 (w)Ea2,ad,(o) (Ra2

(w)A-ZE02,ad,(o) 602

, w0212)

+

E1,(1) (Ra4 Ap42 ,1}

(Ra2 (I), W02,2)+ 6a42w) A -2E1,(l)(Ra4 DP42,1), 1

602 (w)Ea2,ad,(o)(6'\p42,w02 ,2) +

= 6a2 (w)A-4Ea2,ad,(o)(R 12

f

6a4 (w) 2

w-1) +

,T

El,(1)(R04Y'Ap42) 1)

6042w}A-4E1,(1)(R04p42,1).

Part III Preliminaries for the quartic case

172

Definition (5.6.3) wv,st, s - 1,1).

w, s) -

Zv,ad(4D, w, s) =

We call Zv,ad(4, w, s) the adjusted zeta function, and 6"2 " )Tv (Rv,o4), wv,st, 81, 1) the adjusting term. It is easy to see that 1

f AsTv(Rv,o(Pa, wv,st,1)d"A = Tv(Rv,o-P, wv,st, s - 1,1) 0

- Tv+(Rv,o I, wv,st, s - 1, 1),

_

1

wvgt,1)d"A = Tv+(Rv,o$,wv,gt, 5 - s,1). 10

We define w) = bat (w)Ea2,ad,(o) (Ra2 C w02 2) +

aa (w) 2

E1,(1) (Ra, P42' 1).

Then, by the above considerations, Zv,ad(4,w,s) satisfies the following principal part formula.

Theorem (5.6.4) Zv,ad(4, w, s) = Zv+(4D, w, s) + Zv+($, w-1, 6 - s)

- 6v5(w) 2 dy,st(w) 2

wv,st, s - 1,1)

Tv+(Rvo1P, wvst) 5-8 ,1)

+ Z2s#$(0) 2 (w)

s-6

- 4'(0)

+ E($, W-1)

s

s-4

3

+

(-1)i+1 Ea1,(-j+1) ( w-1) E (s - 4)j j-1

- E('D, w) s-2

_ Ea1,(-j+1) (4, w) (s - 2)j

The following functional equation follows from the above formula.

Corollary (5.6.5) Zv,ad(4, w, s) = ZV,ad($, w-1, 6 - s). Also it is easy to see that Ea1,(-2)

w) = 26#(w)E2,(-1,-1)(Ro, ID, 0, 1), (C w) = 601(w) (E2,(-1,o) (Ra1(p, wa1, 0,1) + E2,(-o,-1) (Ra14 , wa1, 0, 1)),

Ea1,(o) ('P, w)

=dal (w) (E21(o10)(Ra1 2

+

da1(w)

Lao, 107 1) + E2,(-1,1) (Ra1(), wa1, 0, 1)

wa1, 0,1)

Chapter 6 The case G=GL(2) x GL(1)2, V=Sym2 k2® k §6.1 Reducible prehomogeneous vector spaces with two irreducible factors We consider two prehomogeneous vector spaces G = GL(2) x GL(1)2, V = Sym2k2 ® k,Sym2k2 ® k2 in this chapter and the next chapter. In both these cases, V is of the form V = V1 ® V2 for some irreducible representations V1, V2. Therefore, we consider such representations in general. We assume that Vks ¢ 0. We choose T+ as in §3.1. If we restrict the weights ryi to T+, the convex hull should contain a neighborhood of the origin. This can happen only if dim T+ < 1. If dimT+ = 0, G1 = G°A, and we considered such cases. Therefore, we assume that

dim T+ = 1. We choose an isomorphism a : R+ - T+. Then any element of G1 is of the form gl = a(A1)g°, where Al E R+, go E G. We define kv1, icv2 to be the rational characters defined by the determinants of GL(V1), GL(V2) respectively. Let rcv. (gl) = Iicv (gl)I-1 for i = 1, 2. We assume that a(X1) acts on V1, V2 by multiplication by A', Ai for some a > 0, b < 0. This means that a dim VI

iy1 (a(A1)) = Al

,

Icv2 (a(A1)) =

Ai dim V2

For i = 1, 2, let [, ]v, be a bilinear form on V such that [gxi, 9Lyi]v; = [xi, yi]V, for all x, y, where g` = TGtg-'TG as before. Let [ , ]v be a bilinear form on V defined by the formula [(xl, x2), (yl, y2)]v = [xl, y1Jv1 + [x2, y2]v2.. For P E S°(VA),

let Ry'I E .Y(V A) be the restriction for i = 1, 2. For P E 9'(VlA) (resp..So(V2A)), we define a Fourier transform 9v1 4) (resp.." v1 4)) by [ , ] v1 (resp. [ , ] v2 ). We also define partial Fourier transforms gv1 CF, gv24) for AD E .O(VA). Since the restriction

to Vl or V2 gives the same result, we use the same notation. If g' E GA,

(6.1.1) 6v,--(4),g1) = icv(9')ev,,. (-q'v4', (9l)`) + iv(91)9v4)(0)

+ E (iv(g1)es, (-q'V4, (g')`) - E s

- D(0) g1))

SO0V1,V2

+E

(iv(g')®s (9v

,

(g')`) - OSp (,D, g'))

S1CV1 or V2

Lemma (6.1.2) (gl)`) - esp(,D,gl)) + tv(9')9v4'(0) -,D(0) SoCV1 or V2

Gsp(gv1gv(D,9')+lcv(9')

=1cv2(9')

SpCV2

SpCV1

esa(4,g')-iv2(91) E @Sp('Fv24'1(9')L) S/CV1

SpCV2

Part III Preliminaries for the quartic case

174

=kV (91) > 19Sp(-q17V4,(91)1)+kV,(9') 1: es'(9v29vp,g1) spcv1

S19cv2

-kvJ91) >2

as"(b,91)

s1cv1

spcv2

Proof. This lemma follows from the following relation (6.1.3)

kv(91) > -v4((91)`xl, 0) + kV(91) > -07v4'(0, (91)`x2) x1EVlk\{0}

E

x1EV1k\{0}

40, 91x2) + kV(91)9V4D(0) -'b(0)

'D (91x1, 0)

x1EV1k\{0}

x1EV2k\{0}

= kv2(91) > 9v1Fv'D(91x,,0)+kV(91)

-

9vD(0,(91)'x2) x1EVlk\{0}

x1EV1k\{0}

> ID(91x1, 0)

>2

- sV2(91)

x1EV1k\{0}

.v24D(0, (91)`x2)

x1EV2k\{0}

= kV (91) > -Fv4)((91)`x1, 0) + kv1 (91) > - kv1 (91)

>

91x2)

x2E V2k\{0}

x1 E V1k \{0}

'Fv1'((91)1x1, 0) - > t(0, 91x2)

x1EV1k\{0}

x2EV2k\{0} Q.E.D.

Consider a 3-sequence 0 such that 1(a) = 1 and So

V1, V2.

Lemma (6.1.4) Let g1 = a()1)g° be as before, and X as in §3.1. Then the integral

fA/Gk

w(91)X(91)es, (4,,

91)8(90,

w)dxgl

is well defined for Re(w) >> 0.

Proof. Since Sa St V1i V2, for any N1, N2 >> 0, there exists a slowly increasing function hN1 iN2 (g°) of g° such that es, (4, 91) << hNl,N2 (90) inf(Ai 1 A 1 N2 ).

W e choose N1, N 2 large enough so that the function inf(A tai N2)X(a(.X1)) is integrable on T+. We fix such N1i N2. If Re(w) >> 0, the function hN1,N2 is integrable on the Siegel domain. Therefore, the convergence of the above integral (9°)9(90, w)

follows. Q.E.D.

The following lemma follows from the same argument as the above lemma.

Lemma (6.1.5) Let X be as in §3.1. Then the integral v,8t(D,w,X,w) = JG/Gk w(91

)X(91)ev(,91)g(9°,w)dxl

6 The case G = GL(2) x GL(1)2, V = Sym2k2 ® k

175

is well defined for Re(w) >> 0.

Definition (6.1.6) Let 0 = (a) be a (3-sequence such that Sp C V1 or V2. Let X be as in §3.1. We define J GA/Gk

w, X, w) =

ai>1

w(91)X(91)es5 (D, 91)f

(9°,

w)dg'

foA/Gk w(91)X(91)esa (41, (91))8(9°, w)dg1

S,6 CV1i

SOCV2i

ai>1

<(D, w, X, w)=

fGA/Gk w(91)X(91)esp (4), (91)`)x(9°, w)dg1

SoCVI,

J GI/Gk w(g1)X(91)esp (-D, g')e(g°, w)dg1

SpCV2-

ai<1

X1<1

If Sp C V1, we define (6.1.7)

Va,1((D, w, X, w) = --- 0, >- (9V, 90b, U), r. V2X, W), 77

0,2 (D, LO, X, W) = Ea,> (b, w, X, w), 0,3(.D,

X, w) = =-a,
0,4 (-P, w, X, w) = -=D,< (gv, -P, w, Kv1 X, W).

Also if Sp C V2, we define (6.1.8)

Ea,1(b, W, X, w) = za,>

w) 'vX,w),

0,2 (4), w, X, w) = ca,> (6 V2 Y', w, K'V2 X, w),

=a,3(,,, w, X, w) =0,4 (p, w, X, w) _ ---D,< OD, w, X, w)-

We also define =S, at C W, X, w ,

a,at,i N', W, X, w

similarly by considering SQk, Sf,$tk.

Proposition (6.1.9) The distribution 'Ea,;,(4), w, X, w) is well defined for all i if Re(w) >> 0.

Proof. Since the argument is similar, we prove this proposition for =a,1(4p, w, X, W)

for Sa C V1. Let g1 = a(X1)g°. By (1.2.6), for any N >> 0, there exists a slowly increasing function hN(g°) such that Osa (4D, g1) << Al NhN(g°)

for g° E 60. We fix such N so that AINX(a(.ll)) is integrable on the set {Xl A, > 1}. Since hN(9°)&(g°,w) is integrable on 67° if Re(w) >> 0, this proves the I

proposition. Q.E.D.

An easy consideration shows that

-1X-1 ,w). 0,I(C w,X,w) _ °0,4(9VC w -1 kV

Part III Preliminaries for the quartic case

176

Therefore, (,D, w, X, w) for i = 1, , 4 can be constructed from any one of them. Let Sa C V1, and a = (,3). Suppose that YO = Za for all Sa C V1 or V2. Notice that this condition is satisfied by our two representations.

It is easy to see that (6.1.10)

X(9a)t(9a)-2pa Gza (RaID, 9a).ua (9a, w)dg

W, X, W)

.

cAJInMDA/Mok a1>_1

We define Js(4),gl) similarly as in (5.1.4) Let 4

W) =

(6.1.11)

totOD, W, X,

W, X, W), SacV1,V2 i=1

I(-I), W,X,w)= J A IGk

W(91)(91)Js(,91)e(90,w)dg1.

We have proved the following proposition.

Proposition (6.1.12) Suppose that YD = Za for all Sa C Vl or V2. Then X, W) =

E f i(a)=1,SagV1,V2 -

A/Gk

W(91)X(91)Kv(91)Gsa (, (91)(90

fA/Gk 1(a)=1,S5QV1,v2

W(91)X(91)Gsa(-D,91).(90

w)dxgl

w)dxgl

+ tot(,', W, X, w) + "V,st* W-1, /£V'X-1, w)

W, X, w).

§6.2 The spaces Sym2k2 ® k, Sym2k2 ® k2 Let V1 = W = Sym2k2, V2 = k or V2 = k2 and V = V1 ® V2. We consider these cases in this chapter and the next chapter. We write elements of V in the form x = (x1, x2) = (x10, x11, x12) x20), where xl E Vi, X2 E V2, or

x = (Xl,X2) = (x1o, x11, x12, x20, x21), where x1 E V1i X2 E V2.

Let G1 = GL(2), Gi = Ti = GL(1) for i = 2, 3, and G = G1 x G2 x G3. The group G acts on V by (91,t2,t3)x = (t291x1,t3x2) or (t291x1,t391x2) for g = (g1,t2it3) E G.

Let T C G be the kernel of the homomorphism G -+ GL(V). Consider a rational

character Xv(91,t2it3) = (detgl)°'t22t33 of G. This character is trivial on T if

6 The case G = GL(2) x GL(1)2, V = Sym2k2 ® k

177

and only if al = a2 or 2a1 - 2a2 - a3 = 0. We choose such a,,a2ia3 > 0 so that they are coprime integers. Then we can consider Xv as a character of G/T

and it is indivisible. We consider (GIT,V). We fix al = a2 = 5,a3 = 3 for V = Sym2k2 ® k, and al = 5, a2 = 3, a3 = 4 for V = Sym2k2 ® k2, because these are the normalizations which we will use in later chapters.

We have to choose T+ C T+ (see §3.1) so that T+ = T+/T+. We can choose T+ _ {(1, t2, t3) E T2+ X T3+ I t2 t3 = 1} or T+ = {(1, t2, t3) E T2+ X T3+ I t2 t3 = 1}.

In Chapters 6 and 7, t11, t12, t2, t3 are elements of Al. We define t1(A1) _ a2(A1 1, Al) and t2(.\2) = (a2, A2 5) or (A2, 2 3) for 1i 2 E IIB+. Let

d(A1,\2) = (t1(A,),t2(\2)) Then T+ = {(1, t2(A2)) I A2 E 1R.+}. Let t1 = a2(tll, t12). We define (6.2.1)

t ° = (tl, t2, t3), to _ (t, (A,), 1)t °, tl = d(\1, A2)t °.

We use this notation in Chapters 6 and 7. Let dxto - dxtlldxt12dxt2dxt3, dxt° = dxaldxt°, dxtl = dxAldxA2dxt°. Let GA, be as in Chapter 3. Any element of G' is of the form gl = t2(\2)9° where g° E G. We use dxt1 for the diagonal part to define a measure dgl on

G. Let dg = dxAdgl. We define an action of GA on VA by gx = Aglx, where we consider the ordinary multiplication by A. Let eij be the coordinate vector of xij. Let yij be the weight of ei3 i.e. teij _ yij (t)eij for t E T. We identify t with {(zl, z2) E R2 I zl + Z2 = 0} x R by d(A1, ).2)(-a'a;b) = A2a.>,2

Let

((a, -a; b) (a, -a ; b'))

-

r 2aa' + lbb' s

Sl

2aa' + 14 bb'

Sym2k2 ®k,

Sym2k2 ® k2.

Then this is a Weyl group invariant inner product on V. We use this particular inner product, because it is the inner product which we will consider in Part IV. We can identify yij's with elements of R2 as follows.

(/ 3/)

(v2,

1

(7

14

3

14)

Note that the origin is above the line segment joining the two points (f, 14)

and (- 1f , -

3 ) 14

Part III Preliminaries for the quartic case

178

Let be the parametrizing set of the Morse stratification. Then elements in S \ {0} for V = Sym2k2 S k are as follows. Cl

C2

C3

C4

I Elements in 93 \ {0} for V = Sym2k2 S k2 are as follows.

C3

C4

6 The case G = GL(2) x GL(1)2, V = Sym2k2 ® k

179

C7

We choose SL(2) x {(t3,t-5)} or SL(2) x {(t4,t-3)} as G" in §3.2. Then Zp etc. are as follows. Table (6.2.2) (for V = Sym2k2 (D k) ZO R N1. = 2 (-1, 1; -2)

x12, x20

,32 = (0, 0; 3)

x10, x11, x12

03 = (-1, 1; 3)

x12

,34 = (0, 0; 5)

x20

WO

Moll

-

{(a2(t-l, t), t3, t-5)} SL(2) x {1} (t-3, t3),

{(a2

t-6, t10)}

SL(2) x {1}

Table (6.2.3) (for V = Sym2k2 (D k2) (3

Nl = 16 (-1, 1; -6) RR

Q2 =

q

(-1, 1; 2)

Z13

W13

Moll

X12, x20

X21

{(a2(t-3, t3), t4, t-3)}

x11, x21

x12

{(a2(t-l, t), t-4, t3)}

Q3 = 10 -L(-7,7; -2)

x12, x21

X34 = (0, 0; 4)

x10, x11, x12

05 = (-1, 1; 4)

x12

,66 = (- 2,

-3)

2; X37 = (0, 0; -3)

X21

X20, x21

-

-

t2), t28, t-21)}

{(a2(t-2

SL(2) x {1} {(a2(t2, t-2), t4, t-3)} {(a2(t3, t-3), t-4, t3)}

SL(2) x {1}

Consider V = Sym2k2 ® k. It is easy to see that x E V is k-stable if and only if (x10, x11, x12) is k-stable with respect to the action of GL(2) and x20 # 0. In particular, Vk 0 0. Let Di = ((31) for i = 1, , 4. By Table (6.2.2), the weights of x12i x20 with respect to M'1 are t5, t-6 respectively. Since there are both positive and negative weights, Z1 1k = {x E Za,k I x12,x20 0 0}. We identify Z.O. with W, and consider Zazk, Z02 O etc. Let Eat (IF, w, s), Ea2i (W, w, s) be the zeta function, the adjusted zeta function respectively. We also use the notation Tae (R02iOT, w, s) etc. for the adjusting term etc. Let Zero (resp. Z'1,,0) be the subspace spanned

0}. Let by {ell, e12, e20} (resp. {ell, e20}). Let Z'i. Ok = {x E Zc,Ok x11, x20 Hv = H.O. be the subgroup generated by rG and Tk. Then Vgtk = Gk X H,,, Z'i, ok, I

and Sa2,stk 2-'- Gk XHa2,, Z'aZ,ok Since Ma acts trivially on Z?, for D = D3, 04i Zak _

180

Part III Preliminaries for the quartic case

Zak \ {0} for 0 = 03, N. We do not consider /3-sequences of length > 2. Let Rv,o4>

be the restriction to Zv,o Consider V = Sym2k2®k2. Note that if a convex hull of a subset of ryij's contains

the origin, it contains a neighborhood of the origin. Therefore, Vk8 = Vk. Let Q be the binary quadratic form which corresponds to (x10, x11, x12), and l the linear function which corresponds to (X20, X21). Then it is easy to see that x is semistable if and only 1 does not divide Q. In particular, Vk8 # 0. Let Di = (/3i) for i = 1, , 6. By the above table, the weights of x12, x20 with respect to M0"1 are t10, t-6, the weights of x11i X21 with respect to Mat are t'4, t4, and the weights of x12, x21 with respect to MQ3 are t32, t-19. For these cases, there are both positive and negative weights. Therefore, Talk = {x E Za1k I x12, x20 ss

752k = x E Za2k I x11, x21 54

0}, ,

ss

Za3k = x E Zask I x12, x21 54 0}.

We identify Z54 with W, and use similar notations to D2 of the previous case. Let Ha4 be the subgroup generated by TG and Tk. Then Sa4,stk - Gk XHs4k Z'a,,ok Since Ma acts trivially on Za for 0 = 05, D6, Zak = Zak \ {0} for 0 = c05, c06. As in Chapter 5, ZR1k = 0. We do not consider /3-sequences of length > 2. In both cases,

Vk\{0}=Vk8llflSe,k. We define ez,.o u0) etc. for cl = D2 in the first case and for 0 = Z4 in the second case as in (5.2.4). /kx)3. We define w(g') as in §3.1. It is Let w = (wl, w2, w3) be a character of (Ax easy to see that rcv(gl) = XZ 4 rcvl (g') = X2 9 and 1v2 (gl) = X2, for V = Sym2k2®k. Also ICv(gl) = X2 6, r,Vi (gl) = A 2 12' and K y2 (gl) = a2 for V = Sym2k2 ®k2. For V2 = k2 or k, let [ , ]' be the standard inner product, and [x, y]v2 = [x, rGy]',2. We defined a bilinear form [ , ]v for V = Sym2k2 in §4.1. We use this bilinear form as [, ]v1 in §6.1. We define J8(4),g1) similarly as in (5.1.4). Let (6.2.4)

w(gl)(gl)J8(,gl)dg',

Il((D, W, X) = J AIL,.

I(4), w, X, w) = f

A/Gk

w(gl)(gl)J8(,gl)e(g°,w)dgl.

(J$ (,D, g1) = J(41, g1) for Sym2k2 (9 k2.) We define the zeta function using (3.1.8) for L = Vk for the first case, and L = Vk8

for the second case, and use the notation Zv(,D, w, X, s) etc. Let X(t2(1\2)) = \2 for some c E C. By the Poisson summation formula, Z01, w, s) = Zv+(-D, w, X, s) + Zv+($, w-1, rcv1X-1, N - s)

+ JR+ ABIl(`I'a,w,X)d'A,

6 The case G = GL(2) x GL(1)2, V = Sym2k2 ® k

181

where N = 4 for the first case, and N = 5 for the second case. By (3.4.34), - CGA(w; p)I1(4, w, X)

1(4),

As in Chapter 5, we assume that 4P = Mv,4,-P in this chapter and the next chapter.

§6.3 The principal part formula In this section, we prove a principal part formula for the zeta function for the space Sym2k2 ® k. Since the zeta function is almost a product of the Riemann zeta function and the zeta function for the space of binary quadratic forms, we only outline the proof.

We assume that c # -5, -2,0,1,3,4,9. Let T E 9(Zv,oA). Suppose that q = (ql, q2) E (Al )2, uo E A, and p = (91,µ2) E 1R+. Let a(uo) be as in §2.2. Let dxq = dxgldxg2i and dxµ = dx pjdx p2. Let w1, W2 be characters of Ax/kx, and w = (w1, w2). We define w(q) = wl(gl)w2(g2). Let fV (`L) p, qi uo, S1, S2) = µ23 a(uo)$2

(1µ2g1, Alµ2gluo, p1/t2 5q2)

Definition (6.3.1) For complex variables s, sl, 82 and tI1, w as above, we define (1)

TV(1,w,s,S1,S2)

=f

w(q) '1fv(IF,p,q,uo,Sl,s2)dxpdxgduo, 2 x (A1) 2 x A

(2) Tv+(`F, w, s, S1, S2) = JR2 X(A1)2 XA w(q)µifv(IF, µ, q, no, S1, S2)dxpdxgduo, 1>1 TV' (W, w, S1, s2)

(3)

= J +x(Al)2xA w(q)fv('I,1, P2, q, no, sl, s2)dxl.12dxgduo.

As in (5.2.2), Tv(1Y, w, s, s1, s2) has a meromorphic continuation. We define TV ('F,

si)

Tv+('W,

Si)

d ds2 s2=0

TV('F, W, S, S1, S2),

d ds2 s2=0

d

T' (W, w, Si)

Tv+(W, LO, S, S1, S2), 1

ds2 92=0

In our case, Ya = Za for all 0. Therefore, all the distributions in (6.1.12) are well defined. Let 1)6(w3)TV (IZVOl), (wl, 1), c, 2 (1 - s)) EV,st(4), w, X, S) = 6(wlw2

s-1

Then by considering r = rG, s = ST, (6.3.2)

1

°,v,st(-P, w, x, w) -

fRe(s)=r>i

w,

,

s)(s)A(w; s)ds.

Part III Preliminaries for the quartic case

182

Let b(wl)b(w2)

Ear 0') w, X, s) =

1

E2(Ra,11 (W2, W3),

10

2

1

(S+1), 10 (3s + 3 - 2c)).

Then similarly as in Chapter 5, by considering r = rG, s = sr, w(g')X(g1)OSal ('p, 91)8(9°, w)dxgl

(6.3.3)

FGA'/Gk 1

2Ir

fRe(s)=r>1

Ear (-P, w, X, s)O(s)A(w; s)ds.

Let

Jz(,D,w,X)=

w2 a2 W (( 21,

3

11),

3)

By (3.4.31), (6.3.4)

02,2 (P, w, X, w) + "02,4(4P1w, X, w) - CGA(w; p)

w, X).

Let b(w3)b(w1w2 1) Ta2+(Ra2,oRa2I,w1, 3, L01 X, s) =

S-1

3

2(1 - s))

+ b(w3)b(wlwz 1) Ta2+(Ra2,oga2Ra2'P,wi 1,3- 3, 2(1 - s)) 3 s-1 Then similarly as in Chapter 5, by considering T = TG, s = sT, (6.3.5)

, w, X, w) +a2,st,4 (`P, w, X) W)

02,st,2 ( 1

27rV -1 Re(s)-r>1

Ea2,st (D, U)' X, s)¢(s)A(w; s)ds.

Let ED, (4D, w, X, s) be the following function

b(wl)b(w 2b(w3) (

E1(Ra,,P,w2, 2(s+1)) + E1(Ra,.

3s-2c+3

v%

41,w21, 2(s+ 1))

3s+2c-15

Then similarly as in Chapter 5, by considering r =TG, s = sT, (6.3.6)

w, X, w) + 'y a3,4

a3,2

N

1

-1 Refs)-r>1

w, X, W)

Ea3 ('b, w, X, s)O(s)A(w; s)ds.

Let

J4(D,w,X) =

Q32b(w1)b(w2)

5

+

C

E1+(Ra4Cw3,-5)

J2b(w1)b(w2) 1 E1+(9o,Ra,4),w3 5

5(c+5)).

1

6 The case G = GL(2) x GL(1)2, V = Sym2k2 G k

183

Then by using the Mellin inversion formula, (6.3.7)

-=D,,2 (-D, W, X, W) + Ea4,4(1b, w, x, w) ^' CGA(w; P)J4(,D, w, X),

where 26(w1)6(w2) 5

c

E1(Ra4D,w3,-5) (--D(O)

+ !I 12b(wl)b(w2)6(w3)

JIFc 50)

+

All these computations are valid replacing (1) by $, X by'cj1X-1 etc. by the assumption on c. We define Ea1i(j) (1D, w, x, so) etc. similarly as before.

Easy considerations show that (6 3 8 ) .

.

E vsc, (o) ( , w, X, 1) _ E al,(O) ( 41 , w, X, 1 ) =

- b(wl w 2 )5(w z

3 ) T% ( R vo , P , (w1 , 1 ),

c),

,P , (w2i w3),

1,

(w2, w3),

1,

z

3b(W

50 (w2) E 2 (1,-1) ( Ra1 a(wl)b(w2)

10

E 2,(o,o) (R al

,

5 ( 3 - c)) 1

5

(3

- c)).

Also

(6.3.9)

J2($, w-1, I£V'X-1) + J4($, w-1, kV1X-1)

w-1, v1X-1, 1) + Ea3,(o) (I, w-1, kV1X-1, 1) - J2(4', W, X) - J4(I, W, X) + Ea2,$t,(o)

- Ea2,st,(o) (b, W, X, 1) - Ea3,(o) (P, W, X, 1) 6(W3)

3

+

(ED2,ad (Ra2

212b w1)b(w2

$) (w2 1 , wl 1) ,

) (El(RD4&w1,

4

3

c 5

c) - Ea2,ad (Ra2 -P, (w27 wl), 3))

)-

E1

c (Ra4 1b w3 - 5)

Definition (6.3.10) ZV,ad(', W, X, S) = Zv(4, w, X, s) - b(w1w22 )s(w3)Tv(Rv,o4), (w1,1), s, c).

We use the terminology `adjusted zeta function', `adjusting term' for this case also. Let F(4i, w, X, s) =

b(wl)b(w2)E2,(o,o)(Ral , (w2, W3),1, 1(3 - c)) 2(5s + c - 8)

36(wl)b(w2)E2,(l,-1)(Rol' , (w2, w3),1,1(3 - c))

+

+

2(25s + 5c - 40)

4b(wl)b(w2)E2,(o,-1)(Ra1 , (w2, w3), 1, 1(3 (5s + c - 8)2

- c))

184

Part III Preliminaries for the quartic case + b(W3)

Ea2,ad (Ra2 , (W2, W1), 3 )

3s-c

+ 32b(W1)6(W2)

E1(Ra,',W3i-5) 5s + C

Then by considering relations as in (5.6.2), we get the following proposition.

Proposition (6.3.11) Suppose b = Mv,,4). We also assume that

c0 -5, -2,0,1,3,4,9. Then Zv,ad(4',w, X, s) satisfies the following principal part formula W, X, s) = Zv+('D,W, X, S) +

W-1, /£V1X-1, 4 - s)

6(W1W2 1)b(W3)Tv+(RVo4), 2

6(W1W21)b(W3) 2

(wi, 1), s, c)

Tv+(Rvo(D, (W1

1

1),4-s,4-c)

- F(,D, w-1, '4 1X-1, 4 - s) - F(-D, w, X, s).

When we apply this result in Chapter 13, we consider the situation s = -3, c =

-1.

Chapter 7 The case G=GL(2) x GL (1)2, V=Sym2k2® k2 In this chapter, we consider the case G = GL(2) x GL(1)2, V = Sym2k2 ® k2 using the formulation in Chapter 6. We use the notation in §6.2.

§7.1 Unstable distributions In this section, we define distributions which arise from unstable strata.

Definition (7.1.1) Let w be as before. We define (1) bal (w) = 6(w1w2 2)b(wlw3 1), wal = (w2, w3), (2) 6a2 (w) = 6(w1w2 1)b(w3), wag = (w2, 1), (3) 6,3(w) = b(wl)b(w2w3), w03 = (w2,w3), (4) 604(w) = b(w3), bay,st(w) = 6(w3)6(wlw21), way = (w2,w1), wa,5t =w2, (5) 6as(w) = 6(w1)b(w2)6(w3), was = w2,

(6) 6#(w) = bas(w) = 6(w1)6(w2)6(w3)

Definition (7.1.2) Let 4), w, X be as in §6.1. For a complex variable s E C, we define X, s) =

(1)

2 bat (w)

E2(Ral , wal ,

3s-3-c , 2s - 3 - c), 2

3s - 7 + c

(2)

Eat (D, W, X, s )

(3)

Ea3(C w,X,s) =

(4)

Eay (1), w, X, s) = 6 4 (w) Ea4,3,d (Ray D, way, + s 4 4

(5)

Eas,l (D, W, X, s) = bas (w)

>

=

bal(w)

4

63(w) 10

E2(Ra2$1 wag,

E2(Ra3 C , w a3,

4

3s + 3 + c 2s + 2 - c),

E1(Ra54i7 w55,

4(s + 1)

-

s - 1),

10

5

1)

2c2

E1(Ra5YVl-D,wa51, 8+1 )

X, s) = bas (w)

(6)

4(s - 5) + 2c E1(Ras 4), s + 1)

Eas,l (4), X> w> s) =bas (w) Ea6,2 (4), X, w, s) = bas (w)

3s + 3 + c E1(Rae9v24), s + 1)

3s - 3 - c

We define Eay,ad (4), w, X, s), E04+(", w, X, s) similarly. The distributions Ea; ((D, w, X, s) for i = 1, , 3 and Eay (4), w, X, s) have at most

a double pole at s = 1. The distribution Eas,1((k, w, X, s) has at most a double pole at s = 1 for l = 1, 2. The distribution Eae i, (4),w, X, s) has at most a simple pole at

s=1for1=1,2.

We define E*,,(j) (,P, w, X, 1) etc. similarly as in (5.3.4).

Part III Preliminaries for the quartic case

186

§7.2 Contributions from unstable strata We consider I(,D, w, X, w) in this section. Let pi = (ai, si) be a path such that si(1) = 0, and pz = (0j,si) a path such that si(1) = 1 for i = 1,2,3. Let 0 = 01, 02 or a3. It is easy to see that w(g1)X(g1)kv(g1)eS, (&

fA/Gk

fA/Gk

(g1)`)-,(go, w)dxgl

w-1(g1)X-1(g1)rv1(g1)esa($,gl)S(g° w)dxgl

because g((g°)°, w) = g(g°, w). Therefore, we only consider fG/G,.

w(gl)X(g1)©s, (-P, gl)S"(g°, w)dxgl.

Lemma (7.2.1) For i = 01, 02, 03, JA/Gk

w(g1)X(g1)OSa (4', g1).I(g° w)dxgl

JAnTA/Tk

w(t1)X(t1)(t°)-2"ic i(tl)eZa (R,4), tl)&N(t°, w)dxtl.

Proof. Clearly,

fG/G,.

w(g1)X(g1)esa (,D, gl)8(g° w)dxgl w(t1)X(t1)(t°)-2Pe ,(Ra(D,t1n(u))8(t°n(u),w)dxtldu.

fAnBA/Bk

For a E k, we define

fl,a(tl) =

JNA/Nk

f2,a(tl) = f

9y, (-D, tl(n2u), 1)) < au > du, Oy,2 (,(D, tl(n2u), 1)) < au > du,

NA//N,

f3,a(w, t1) =

JNA/Nk

8'(t°n2(u), W) < au > du.

Then, by the Parseval formula, we only have to show that

fe/Tk

w(t1)Ac E fi,a(tl)f3,-a(W,tl))2 dxtl ,.. 0 aEkx

for i = 1, 2. We fix a constant M > w°. By (3.4.3), there exists 6 > 0 such that if l >> 0,

lf3,-a(w,tl)l « Vi

7 The case G = GL(2) x GL(1)2, V = Sym2k2 ® k2

187

for w° - S < Re(w) < M. Let 41 be the partial Fourier transform of Rat 4 with respect to x21 and the character <>, and 1)2 the partial Fourier transform of R,2 4) with respect to x12 and the character <>. Then

f1,-W)

f

/k Z12,x2oEkx,x21Ek

u

Jru 'D(712(tl)xi2, 72o(tl)x2o, 721(tl)(x21 + ux20)) < au > du

xiz,xzoEkx

_Ef X12,xzoEkx

_

'D(712(tl)x12, 72o(tl)x2o, y21(tl)(x21 + ux20)) < au > du

/k x21Ek

(712(tl)xi2,72o(tl)x2o,721(t1)x2ou) < au > du

A

i(712(tl)xi2, 72o(tl)x2o, 721(11)-1x20 a)

)11 1A2

x12,x2oEkx

Similarly,

f2,,(tl) = Al2A24

'P2(711(tl)x11,721(11)x21,712(tl)-1x12 a) x11,x21Ekx

Therefore, by (1.2.6), for any N1i N2, N3 > 1, (7.2.2)

«AiiA201A2)-N1(A11A23)-N2(A-lA3)-N3,

Ifi,a(tl)j aEkx

If2,a(tl)I «A12A24A24N1(AiA23)

N2(Al2)24)-N3

aEkx

Since the convex hull of each of 1(2,4),(-l,-3),(-1,3)1,1(0,4),(l,-3),(-2,-4)1 contains a neighborhood of the origin of R2, for any N >> 0,

E Ifi,a(t1)I, E l f2,a(tl)I << rd2,N(A1, A2)aEkx

aEkx

We fix 1. Then JA/Tk

w(tl).2

fi,a(t1)f3,-a(w,11)A1dx11

aEkx

is bounded by a constant multiple of

f Rrd2,N()1, A2))2+lAcdxAldx)2 2 for i = 1, 2. This proves (7.2.1). Q.E.D.

Let r be any Weyl group element. For i = 1, 2, 3, we define (7.2.3)

Epi,T(,Ib,w,X,w)= J p/Tk w(tl)opi(tl)Oz.(RD ,t1)eN,T(t°,w)dxtl.

Part III Preliminaries for the quartic case

188

By a similar proof as in (6.1.4), Bpi (4D, w, X, w) is well defined for Re(w) >> 0. By (7.2.1), the first two terms of (6.1.12) 3

w) - =pi,T

VX i=1

w, X, +u

T

Easy computations show that apl (t1) = XiX2+3 Qp2 (t1) =X24, and Op3 (t1) _ A2Xc 1 2

Proposition (7.2.4) Let r, s = sT be as before. Then pi,T (I), w, X, w) -

27f fR)=

Eoi (4), w, X, s)AT(w; s)ds.

Proof. Let t' = (ti, t2) = (µ1q1, µ2q2) E (AX )z, where Al, µ2 E ]R+, q1, q2 E A1. Let

dxt' = dxtidxt2.

(/J(A1

Suppose that f is a function on (A1/kx)2. It is easy to see that

f

Al/kx)4 w(t °)f (ti2t2, tllt3)(tx t ° = 6" (w) J Al

///

/kx)4 w(t °)f (t11t12t2, t12t3)dxt ° = 62 (w)

f

Al

J

\A1

w11(q)f (q)dx 4,

/kX)2 w02 (q)f (q)dxq,

(w(t °)f (t2tz, t12t3)dx t ° = 613 (w) fAl /kx)2 w13

/kx )4

after the change of variable (ql,q2) = (ti2t2, t11t3), (t11t12t2,t12t3), (t12t2, t12t3) respectively. We make the change of variable µ1 = A2A4, 92 = X11Xz 3 for p1, µ1 = X2, µz = X1123 for P2, and µ1 =XiX2, µ2 = X1X23 for P3. Then dx µldx µ2 = 2dx X1dx X2, X1 = µi N'2, A2 = µ1 2µ2 1 for pi,

dxµldxµ2 = 4dxA1dxA2, X1 =µiµ2, X2 = µi for p2, and dxµldxp2 = 10dxA1dxA2, X1 = µ'1 N'2 , X2 = Al µ2 10 for p3.

Clearly, £pi,T(t°, w) = &N, (t', w). It is easy to see that 3a-3-c

Al op,

2

(tl)(to)TZ+n =

2s-3-c

µ2

z = 1,

A2-1

2 = 2,

° µ1

4

3a+3+c U

10

4e+4-2c N'2

10

i=3.

7 The case G = GL(2) x GL(1)2, V = Sym2k2 $ k2

189

Therefore, w(tl)(t1)Ozo(Ra,t1)N,T(to,w)(to)-2pdxtl

J

= E(,w) X,A

/Tk

for i = 1, 2, 3. This proves the proposition. Q.E.D.

Proposition (7.2.5) Ep;,T (1b, w, x, w) - 0 if T 54 TG for i = 1, 2, 3.

Proof. If T # TG, we can choose r >> 0. Then Z(r) < wo. Q.E.D.

Next, we consider So, for i = 4,5,6. Since

-1 -1

-1 Ea.,1w-

w, X, W) =

a,,4

)

r--V1 X- 1

1

,

,

w),

we only consider Ea w, X, w) for j = 2, 4. The following proposition is a direct consequence of (3.4.31)(1).

Proposition (7.2.6)

w w-

(1)

a4,2(

(2)

14,4 (D , w, X, W)

,

, X,

c).

G

)

(

4

,

CGA(wi P)

w 4 as+(R as, a4,

Eaa+ (ga4 Rio, 4), aal, 3 - 4

Let c

Eaa,st,2(D) w, X, s) = 604,st(w) Ea4,st,4(4), w, X, s) = ba4,st(w)

1-s

7a4+(Raa,oRa4wa4,st, 2 )

4(s-1)

,

Taa+(Ra4) o`aaRaa'b ,wa41st,3- 41 123) 4(s- 1)

The proof of the following proposition is similar to (5.4.3)-(5.4.7), and is left to the reader.

Proposition (7.2.7) Let r = TG, and s = sT0. Then (1)

Ea4,st,2

1Xs)(s)A(w;

w, X, W)

e,->1

s)ds,

>1

f

(2)

aa,st,4 (4), w, X, w)

27f

-- J

R ,->)1

Eaa,st,4 ('D, w, X, s)0(s)A(w; s)ds.

Since YO, = Za; and M-6; T for i = 5, 6, E0;,2(4), w, X, W) =

J T/Tk

w(t1)X(t1)()za; (.b, t1)c"'N(t°, w)(to)-2pdxtl,

A1>1

a.,4(4b,w,X,w)

_f

T A/Tk A1<1

w(t1)X(tl)eza;(,D,tl)f°'N(t°,w)(t°)-2pdxtl,

Part III Preliminaries for the quartic case

190

for i = 5, 6. Let r be a Weyl group element. We define

i

w, Xf w) =

(1 .2.18)

f

T.l/Z.k A

w(tl)X(tl)e

w)(to)-2Pdx

,5 (D,

t'.

al>1

i = 1,

We define E S,i,T (4i, w, X, w) for i # 2, and 8a5 larly.

, 4 simi-

Proposition (7.2.9) The distribution ^aSiT(-P,w,X,w) is well defined for i =

1, ,4 if Re(w) > 0, and (1)

.a5,2,TR, w, X, W)

w, X, w) ti

(2)

2r

Refs)=r

vas (w)

27rv/---l

E a5,1Rf w, x, s)AT(w; s)ds, E05,2 (C w, X, s)AT(w; s)ds.

/ yo

Proof. Let t' = µq, where p E R+, q E A'. Suppose that f is a function on Al /k x . It is easy to see that

f

Al

/k5)4

w(t o).f (tzt2)dx t ° =b(w) Al f /kX

wp

after the change of variable q = t212t2- We make the change of variable p = X1A2, A2 = A2. Then dxudxA2 = 2dxaldxA2, Al = µA22 It is easy to see that X(tl)(to)T -P = Y" -A2 2(s+l)+C Therefore,

fl

w(tl)X(tl)ez,s (Ra54,f

tl)(to)TZ-Pdxt1

=

w, Xf s)

T /Tk a2>1

This proves the first statement. By the remark after (6.1.9), the second statement can be obtained by replacing (D by 9v I), wa by wa 1, and c by 6 - c. Q.E.D.

Proposition (7.2.10) .^aeijiT (w, 4), X, w) is well defined for i = 1,

, 4 if Re(w) >>

0, and as,4,T(, (1) , X, w) -29r

(1 )

(2 )

1z

06,2,,r (w, 4) ,

X, w) ^

2 7r

Aer>1 r 1

->1

ae,1( 4, f Xf s )A T ( w; s )ds,

fR)=a5,2

(

, X) s )A T ( w; s )ds.

Proof. Let t' = pq, where p E R+, q E Al. Suppose that f is a function on Al/kx. It is easy to see that

JM /k-)4 w(t e).f (t12t22)dx t e = b#(w)

Jl

/kx

.f (q)dxq

7 The case G = GL(2) x GL(1)2, V = Sym2k2 ®k2

191

after the change of variable q = t12t22. We make the change of variable µ = A1.\23, A2 = .\2. Then dxpdxA2 = dxAldx\2, Al = µA2 It is easy to see that X(tl)(to)rz-' = us+1X23+3+c

Therefore,

fTp /Tk

w(tl)X(tl)Oz 6 (Ra6,D,

t1)(t0)'

-Pdtl = Eb6,1(oD, X, s).

a2 <1

This proves the first statement. The second statement can be obtained similarly. Q.E.D.

Proposition (7.2.11) If r O rG, `-'a6,i,r (D, LO, X, w),

(-P, w, X, w) ". 0

Proof. We only have to choose r >> 0. Q.E.D.

§7.3 The principal part formula We define 3

Jl

w, X, S) _ E 1"a, 0, w, X, S) i=1

+

c (w) Ta4+(Ra4,oRa4'P, w04,$t, 4 4(s-1)

1-S 2

+

c

4(s-1) + +

J2w,X) J3

(

U), X) =

4

1 - s) 2

W, X, S) + Ea6,2 (4, w, X, S)

w, X, S) +

w, X, S),

Sa44' (Ea4+(Ra44),wp4,4)+Ea4+(ga4R044"LON,3

4))'

c

- ba4,st8 (w)

T04+(Ra4,oR04D1 wa4,-t,

4)

1 44,-t (U)) Ta4+(Ra4io'a,Ra4Cwa4 8

st,3- 4c

Let T = TG, and s = sr. Then by (7.2.1), (7.2.4)-(7.2.11),

I(4,w,X,w)

2 seta>-r (J1(

,

w-1, Kv1X-1, s) - J1(D, W, X, s))O(s)A(w; s)ds

.moo

+ CGA(w; P)(J2($, w-1, r.V1X-1) - J2(4), w, x)).

Part III Preliminaries for the quartic case

192

Nv1X 1, s)-Jl(,D,w, X, s) must be holomorphic

By Wright's principle,

at s = 1. Therefore, 27r 1

-1

Recs>=.

(J1(

,

w-1, 1V1X-1, s) - Ji(lb, w, x, s))O(s)A(w s) ds

.moo

,., CGA(w, p)

(Ea.,(o) ($, w-1, rcv1X-1, 1) -

w, X> 1))

i=1,2,3

+ CGA(w; p)

(4', w-1, v1X-1> 1) -

w, X, 1))

i=5.6 1=1,2

+CGA(w,p)(J3($,w-1,, 1X-1) - J3(D,w,X)) This implies that

Il (4', w, X) _

w-1

(Ea ,(o)

v1X-1,1) - Ea,,(o)

w, X, 1))

i=1,,2,3

+ L,

Ea,,d,(o)

w, X, 1))

=1,2

i=2,3

Since the reason for studying this case is to apply it to the quartic case, we restrict ourselves to X satisfying the condition 4, 64c # 0, 2, 3. In Part IV, we will consider the situation c = 2, so this condition is satisfied. By assumption, Ea6 w, X, s), Ea6 w-1, v1X-1, s) for i = 1,2 are holomorphic at s = 1, and their values are E1(Ra641, 2) 606 (w) sa6 (w)

6+c E1(Ra6,D, 2)

12-c

-Sa6 (w) -bah (w)

E1(Ra69y2ID, 2) c

E1(Ra62) 6-c

respectively. The following lemma is an easy consequence of (4.4.7).

Lemma (7.3.1) If -D is K-invariant, Z

E,(Ra64D,2) = Z k)g04$(O)l

).D (0).

The above relations hold by replacing 4) by . Also by assumption, bas (w) El,(-1)(Ra5..v14), wa5 1,1)

E05,2,(O) (D, W, X, 1)

8

+

(4

- 2)2

1 sD5(w) E1,(o)(Ra5. y1',was ,1)

8

(4 - 2)

The above relation holds by replacing 4b, w, X by $, w-1, rcV1X-1 respectively.

7 The case G = GL(2) x GL(1)2, V = Sym2k2 ® k2

193

If c#4, bas (W)

E1,(-1)(R,5 )w05,1) + E1,(o)(Ras4),waa,1)

8

(4 -1)

(4 - 1)2

/

If c = 4, x, 1) =

6as8W)

E1,(1)(Ra54i, Wa5,1).

If c ¢ 4, we define b54 (W)

J4('P,W, X)

Wa4, C). 4).

4

If c = 4, we define bag8w)

6a44W)

J4('p, W, X)

Ea4,ad,(o) (Ra44', W041 1) +

E1,(1)(Rab 4), WD" 1).

By the principal part formula for Ea4,ad (Ra4 -P, w1,,, s) and the above considerations, we get following proposition.

Proposition (7.3.2) X) _

(Ea;,(o)(4', W-1, v1X-1, 1) i=1,2,3

-

W,

X,1))

+ J4* W-1, icv1X-1) - J4(P,W, X) If c 4, as in §1.7.

J4W,X) =

If c = 4, J4(4)x,W,X) = A-1J4(4',W,X)

Definition (7.3.3) Zc, P2(X) = 4, P3(X) = 1 10 c P4(X) =

P1(X)

(1)

7

4c ,

7

q1= 2, q2=4, q4=1.

(2)

Easy computations show the following relations and the proof is left to the reader.

Lemma (7.3.4) For i = 1, 2, 3, (1)

E

(,pa,

W, X, s) =

-q;(s-1)-p;(x)Ea; (4p, W, X, s),

E1 (Ex, W-1, kV/X-1, s) = \q;(s-1)-(5-p;(e 'x-'))ED ($, W-1,

(2)

KV1X-1

The following proposition is an easy consequence of (7.3.4).

Proposition (7.3.5) For i = 1, 2, 3, (1)

Ea;,(o)(%, w, X, 1) 0

1

(_qi logA)`A-p:(x)Ea:,(i)(4), W, X, 1), (2)

ED,, (0) (4'x, W-1, Kv1X-1, 1) 0

1

(qi

log X)-iXv'x

v1X-1, 1).

s).

Part III Preliminaries for the quartic case

194

Definition (7.3.6)

F,(-P,w,X,s)-

(q,)-jba;(S)Eai,(j)(q),U,X,1) pi)1-j

(

i=1 j=-2 X)

F2(F,w,X,s)=

3-4

These considerations imply the following principal part formula.

Theorem (7.3.7) (F. Sato) Suppose 4D = M,4P. Also assume that 4, 64` 0 0, 2, 3. Then Zv(1b, w, X, s) satisfies the following principal part formula Zv (4', w, X, s) = Zv+(,D, w, X, s) + Zv+(&, w-1, kv X-1, 5 - s) i=1,2

Because of the above formula, the following functional equation follows.

Corollary (7.3.8) Zv(4D, w, x, s) = Zv(&, w-1, r.v1X-1, 5 - s). Since we want to use (7.3.7) for the quartic case, we describe F, ((b, w, X, s) explicitly here. We assume that c = 2. So 6 - c = 4. We define Fa1 (4b w s) = w, s) _

Fat

1 E2(Ra1 4(D, wa1, -1, -3)

s+3

2

1 E2,(o,o)(Ra21', wag, -2, 0)

s-2 1

4

3 E2,(1,-1)(Ra2 , wa2, -2,0)

+

s-2 1

16

7 E2,(o,-1)(Ra24',wa2,-2, 0)

+16 F03(4',w,s) _

(s-2)2

1 E2(Ra3'D,wa3, 45 > 25)

s-56

10

1

Fa1

( w s) =

F02(

w s) =

1 E2(Ra14, wa1 , -2, -5)

11-s

2

1 E2,(o,o)(Ra2 , wa21, 0, 0) + 3 E2,(1,-1) (Ra2 4 , wa21, 0, 0) 16 4 wa2', 1 E2,(-1 1)(Ra2$, 7 E2,(o,-1)(Ra2$1 wa21, 0, 0) 0, 0)

4-s

4-s

+3

4-s

7 E2,(-1,0) (Ra2 $, wa21, 0, 0)

+ 12 P03 (

w s) -

(4- s)2

1 E2,(o,o) (Ra3 $, wa31, 1, 0)

4-s

10

+ 3 E2,(1,-1) (Ra3 $, U)0311110)

4-s

7 E2,(-1,0) (Ra3 $, wa31,1, 0)

+ 30

(4-S)2

4-s

40

2 Ez,(-1,1) (Ra3 , wa31 1, 0)

+15

(4-s)2 49 E2(Ra2 $, wa21, 0, 0) + 48 (4-S)3

+16

7 E2,(o,-1) (R03 , u'-111, 0)

+40

(4-s)2 1111

49 E2,(-1,-1) (Ra3 $, W+ 120 (4-S)3

0)

7 The case G = GL(2) X GL(1)2, V = Sym2k2 ® k2

195

Then (7.3.9) i=1,2,3

Therefore, the poles of ZV(4, w, X, s) are s = -3, 2, 54, In Chapter 12,13, we show that the Laurent expansion of Zv w, X, s) at s = -3 will contribute to the rightmost pole of the zeta function for the quartic case. The function ZV (C W, X, s) has a simple pole at s = -3, and the coefficient of the order two term of the principal part of the quartic case turns out to be a constant multiple of E2(Ral lb, wal, -1, -3).

Part IV The quartic case In the next six chapters, we consider the quartic case G = GL(3) x GL(2), V = Sym2k3 ® k2. In Chapter 8, we study the stability and the Morse stratification of our case. In particular, we explicitly describe all the ,Q-sequences we need. In Chapter 3, we proved that the distributions associated with certain paths are well defined. However, we have some paths which are not covered there. The representations Zpe, Zp8 of MQ6, Mp8 (in §8.2) are reducible and require a handling similar to Chapters 6 and 7. We prove that certain distributions associated with N6, ,as are well defined in §9.1. In §9.2, we prove some special estimates of the smoothed Eisenstein series which are required in Chapter 10. In Chapter 10, we prove that we can ignore the non-constant terms associated with unstable strata. We will prove that the distribution Ep (oD, w, w) is well defined for all p and Ep'='p (OD, W, W) " 0. p

For this purpose, some cancellations between different paths have to established as in §10.4, 10.7. In Chapters 11-13, we compute the constant terms EEp(-D, W, W) associated with paths explicitly, and prove a principal part formula for the zeta function of our case. In our case, we can fortunately use Wright's principle. This case is of complete type. Let Zv(4D,w,s) be the zeta function defined by (3.1.8) for L = Vks = Vk. The principal part formula for the zeta function for this case is (13.2.2). The location of the poles is s = 0, 2, 3, 9, 10, 12. The orders of the poles at s = 2, 10 do not coincide with the multiplicities of the corresponding roots of the b-function for this case.

Chapter 8 Invariant theory of pairs of ternary quadratic forms §8.1 The space of pairs of ternary quadratic forms Let V_be the space of quadratic forms in three variables v = (vl, V2, v3). We identify V with k6 as follows: 2

2

Qx(U) = x11v1 + x12V1V2 + X13V1V3 + x22,2 + X23V2V3 + x33,3

-* x = (x11, x12, x13, x22, x23, x33)

The group Gl = GL(3) acts on V in the following way: Q91..(v) = Qx(vgl), for g1 E G1. _ Consider V = Sym2k3 ® k2 = V ® V. Any element of V is of the form

Q = (QS1, Q,), where x1 = (x1,111 x1,12, x1,13, x1,22, X1,23, x1,33),

X2 = (x2,11) x2,12, x2,13, x2,22, X2,23, x2,33)

8 Invariant theory of pairs of ternary quadratic forms

197

We choose x = (x1i x2) as the coordinate system of V.

Let G2 = GL(2), and G = Gl x G2. The group G acts on V as follows. Let g = (91, 92) where g1 E G1i and 92 = I a d

)

E G2. We define

g ' Q = (aQ91'x1 + bQ91'X2, cQ91.x, + dQ91'x2 ).

Let T be the kernel of the homomorphism G -+ GL(V). We define

Xv(9) = (detgl)4(detg2)3. Then Xv can be considered as a character of GIT, and it is indivisible. In the next six chapters, we consider (G/T, V). Throughout the next six chapters, we use the notation (8.1.1)

a(t11, t12, t13; t21, t22) = (a3(tll, 42, t13), a2(t2l, t22))

Let T C G, G', t*, t' etc. be as in Chapter 3. In our case, G°A = GA, and therefore, t* = t°*. We identify t* with {z = (z11, Z12, Z13; Z21, Z22) E W I z11 + z12 + Z13 = 0, Z21 + Z22 = 0}.

We use the notation z1 = (z11) z12, z13)/, z2 = (z21, z22), and write z = (z1i z2). For z = (z11, Z12, z13i z21, z22), z' = (z11, z12, z13; z21, z22), we define (Z, Z) = z11zi1 + z12z12 + Z13Z13 + z21z21 + z22z22'

This inner product is Weyl group invariant, and we use this inner product to determine the Morse stratification in the next section. Let 1111 be the metric defined by this bilinear form. We recall that G' = Ker(Xv). We choose G" = SL(3) x SL(2) for G" in §3.1. The weights of xl,ij, x2,ij for 1 < i < j < 3 are as follows: 4 1

(13 3 1

(3'

2

2 -2.1 -1)

X2,11

3' 2 -2)

X2,12

1

2

1. 1

3' 3' 2'

2, 1

4

1 1

1. 1

2

(-3,3'3'2'2 4, 1 2

-3'2

1

2

3' 3' 2'

X2,13

2

(-3' 3' 3' 2,

1

X2,22

x2,23 1

2

X2,33

() (3+-3'-3+-2,2 2._1 1

1

3' 3' (1

2

1

3'

1,

3' 3'

3'

(2 4 (2

-2.

2' 2 1 1 2' 2 ) -1 1)

1 1._1 1) 2' 2)

31 3 ' 3 '

(2 -3,-3'2 4-11) ' '

Therefore, with our metric, the weights of coordinates look as in the picture on the next page. Let 7l,ij,'Y2,ij be the weights of xl,ij, X2,ij respectively. If t E T, we denote the value of the rational character determined by ryl,ij (resp. 'Y2,ij) by yl,ij (t) (resp. 72,ij(t)). This should not be confused with ty-j etc. which is the adelic absolute value of ry1,ij(t) etc. We use this notation throughout the next six chapters.

Past IV The quartic case

198

§8.2 The Morse stratification We study the stability over k. Consider a point x = (x1i x2) E Vk as before. Let Q1 = Qxl, Q2 = Qx2. The quadratic forms Q1, Q2 define subschemes V(Q1), V(Q2)

of Pk respectively. We define Zero(x) = V(Q1) fl V(Q2) and call it the zero set of x. The stability with respect to G', G" are the same. We proved the following proposition in [84] ((1) was proved in [59]). Here, we give a proof based on geometric invariant theory.

Proposition (8.2.1) (1) P(V)k = P(V)80)k and is a single Gk orbit. (2) If X E Vk \ {0}, ir(x) E P(V)7 if and only if V(Q1) n V(Q2) consists of four points.

Proof. Let fB be the set of minimal combination of weights. Table (8.2.2)-(8.2.4) are the list We first show that V, 0. It is easy to verify that if x E Vk and x E Sp, for some 1 < i < 10, then Zero(x) has at least a double point or contains a line or a conic. Therefore, if we can show that Sp; = 0 for i = 11, 12, any x E Vk such that Zero(x) consists of four distinct points should be semi-stable. Such x clearly exists, so Vss 0 0. So we consider SO,l, Sp12.

Consider Spl,. Let ap,, (a) = a(a-1, a-1, a2; a, a-1) for a E GL(1). Then by Table (8.2.3),

MPI, =

{((9112 1) I2)

ap11(a) 91,12 E SL(2), a E GL(1) } .

Elements of the form aQ11 (a) act trivially on Zp11, and the action of 91,12 on Z01,

can be identified with the standard representation of SL(2). Therefore, ZQ11k = 0. Consider S012 By Table (8.2.3),

Mp12=I

(

\91,12

1),92)

91,12, 92 E SL(2) I .

Elements of the form (( 91,12 1) ,1 ) act on ZQ12 trivially, and the action of 92 on Z1912 can be identified with the standard representation of SL(2). Therefore, Zp12k = 0. This proves that V88 0. Note that we did not use a relative invariant polynomial.

8 Invariant theory of pairs of ternary quadratic forms

Table (8.2.2)

Strata

Convex hull

Conics

Sol

2 identical non-singular conics

i i

SRZ

X 2 identical reducible conics

Sa3

040

K

2 identical double lines

SR4

1 common component

199

Part IV The quartic case

200

Strata

Convex hull

Conics

SP5

1 common double line

Sas

2 multiplicity 2 points

Sol

C-) The local ring = k[e]/(E4)

Sag

1 multiplicity 2 point

8 Invariant theory of pairs of ternary quadratic forms

Strata

Convex hull

Conics

Sp9

1 multiplicity 3 point

Sp1o

The local ring = k[Ei, E2]/(Ei, E2)

Sp11

Sp12

201

Part IV The quartic case

202

Table (8.2.3) ((71,11 ' ) (71,12

13 1

Q1 = (0, 0, 0; - 12, 12)

,132 = (-3, 3, 3;-2, 21)

('12,33 ' N))

-1, -1,1,1,1,1,1,1)

2(-1,

s(-11,-5,-5,1,1,1,-5,1,1,7,7,7) !(-11, -11 ,1 -11, 1 11, - 5-5,77 - 5 7, 17)

1 1) /j3=(-37-231 4. 3,-2, 2)

g

/34 = (- s, - s, 3. 0, 0)

2 1 7. 1) a5-(-3,12,12,-4,4

6

06 =

1(-5 -5 -1, -57, -17, 1, ,1,, 1,, 5, 1, 5) 24 , , ,

2

1

(-1 _1 1.-18,8 1)

/37 = (

24,

4,

241 121

O,

4,

4, 4)

1) 14,14 211 - 2 1211-L.--L 1 (39 = (-1 5, 0, 1' 3, - 1 10 , 10)

NR8

_

,Qlo =

2, 3, 0, 0) 1 _ 13; 1 _ 1 1) g, 61 2, 2

/iii = (_q, _2, 4 _j 1) 1312 = 333, 2, 2)

1(-2, -2,1, -2,1, 4, -2, -2,1, -2,1, 4)

1(-19, -10 -4, -1, 5 11,-13, -4,2,5,11,17 ) 12 4

(-3,-2,-l,-1,0,1, -1,0,1,1,2,3)

1(-11 -5 -5,1, 1 ,1 ,-5,1,1,7,7,7) 10(-5, -3,, -1,, -1,1,3, -3,-1,1,113, 1)

42

,

,

1(-4, -1, -1, 2, 2, 2, -4, -1, -1, 2, 2, 2)

1(-5 -5, -2,-5,-27 1 1 1 4 1 4 17) g 1(_4 -4 2 -4 2 8, -4, -4 2 -4 2 8) 3 >

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

Table (8.2.4) Zp

Wp

a

I1,31I2

,Ql

2

x2,j,j2 for 31, j2 = 1, 2, 3

02

g

x2,22, x2,23, x2,33

,Q3

6

x2,33

04

6

,Q5

12

X1,33, x2,23

-L

x 1,33, x 2,ji,j2

06

24

-

x1,13, x1,23, x2,13, x2,23

X1,33, x2,33 X2,33

for ilj 2 ,

=

1, 2,, ,

3

x2,13, x 2,23, x 2,33

x1,33, x2,13, x2,22

X2,23, x2,33

42

x1,22, x1,23, x1,33, x2,12, x2,13

X2,22, x2,23, x2,33

10

x1,23, x2,13, x2,22

x1,33, x2,23, x2,33

2

3

xi,jij2 for i = 1, 2, il, j2 = 2,3

Nil

3

X2,13, x2,23

012

3

x1,33, x2,33

Q7

4

,Q8

09 a1o

-

X2,33

8 Invariant theory of pairs of ternary quadratic forms

203

Let w = (Q1, Q2) where Q1 = V2V3 - V03, Q2 = V1v2 - v2v3.

Easy computations show that Zero(w) = {(1, 0, 0), (0, 1, 0), (o, 0, 1), (1,1,1)}.

If a conic in p2 contains three points on the same line, it contains the line. Therefore, if Zero(x) consists of four points, they are in general position. There exists an element of GL(3)k which sends Zero(x) to Zero(w). Therefore, we may assume that x = (Q1i Q2) where Q1 = x1,12v1v2 + x1,23V2V3 + x1,1003, Q2 = X2,12V1V2 + X2,23V2V3 + x2,1003, and

x1,12 + x1,23 + x1,13 = X2,12 + X2,23 + X2,13 = 0

Without loss of generality, we can assume that x1,12 0. By applying a lower triangular matrix in GL(2)k,, we can assume that x2,12 = 0. After a coordinate change, we can assume that Q2 is a constant multiple of v2v3 - V1V3. Then by applying an upper triangular matrix, we can assume that Q1 = V1V2 - v2v3. Thus, the set of x such that Zero(x) consists of four points is a single Gk-orbit. Since the set of such points is an open set, this proves (8.2.1). Q.E.D.

A straightforward argument shows that x E Sp, for some 1 < i < 10 if and only if Zero(x) has the corresponding geometric property. However, we do not logically depend on this statement. All we have to know is the fact that So, # 0 for i = 1, , 10 and the inductive structure of Sp; for i = 1, , 10. So the verification of the geometric interpretation of Table (8.2.2) is left to the reader. We now consider the strata Sp;. In our situation, there are only a few possibilities for parabolic subgroups of G. Let

Pl = {

91 , 12

0

P2 = { (t01

0

t13) 9 023)

91,12 E GL(2), t13 E GL(1)

l

t11 E GL(1), 91,23 E GL(2) } .

Standard parabolic subgroups of G1 are G1, P1, P2, and B1. Standard parabolic subgroups of G2 are G2 and B2. We consider MR in §3.2. (a) Sp, We can identify the vector space Zp, with the space of quadratic forms in three variables, and Mpl SL(3) = {(g1i I2) 91 E SL(3)}. We discussed this case in §4.1, and Zplk consists of non-degenerate quadratic

forms.

Part IV The quartic case

204 (b) SR2

We can identify the vector space Zp2 with the space of binary quadratic forms. Let ap2 (a) = a(a2, a-1, a-1; a-2, a2) for a E GL(1). Then

MQ2

1

0

0

91,23

'

I2

91,23 E SL(2), a E GL(1) } .

a192 (a)

Since elements of the form ape (a) act trivially on Zp, a point in Zp is semi-stable if and only if it is semi-stable under the action of 0

SL(2) '= {((;

det 81,23 E SL(2) } .

91,23

I2/

Therefore, ZQzk consists of non-degenerate forms again. (c) SR3

In this case, we can identify Zp, with the one dimensional affine space. Let aR3 (a) = a(a, a, a-2; a-2, a2 )

for a E GL(1). Then

Ol

MQ3=!((91,12

12

a,3,

Since MO3 acts trivially on Z 3, ZQ3k = Zp,k \ 101(d) Sp4 We can identify the vector space Z$4 with M(2, 2), i.e. the vector space of 2 x 2

matrices. It is easy to see that Mp4 = SL(2) x SL(2)

{((912

01)

, 92 f

(

91,12, 92 E SL(2) } .

Therefore, ZZk is the set of rank two matrices. /

J

(e) SRS

The vector space Zp5 is spanned by coordinate vectors of x1,33, x2,23. Let

ap5 (al, a2) = a(al, a2, al 2a2 l; al Sat 1, al a2) for al, a2 E GL(1). Then MR5 = {ap5 (al, a2) al, a2 E GL(1)}.

The weights of the coordinates x1,33, x2,23 determine the characters al 9a2 3a3 of ap5 (al, a2) respectively. These characters depend only on a3 = a3ja2i and there

are both positive and negative weights of a3. Therefore, Z7 k = {(x1,33, x2,23) E (kx )2}. (f) SRe

Let ape (a) = a(a, a, a-2; a-1, a) for a E GL(1). Then Mrs Re

91,12

_

0

0 1

92 )

a p, Re (

)

91,12, 92 E SL(2), a E GL(1)

.

8 Invariant theory of pairs of ternary quadratic forms

205

Let V1 (resp. V2) be the subspaces spanned by coordinate vectors of {x2,11, x2,12, x2,22} (resp. {x1,33}).

Clearly Zp6 = V1 V2. Elements of the form ap6 (a) act on V1, V2 by multiplication

by a3, a-5 respectively. Therefore, we have already discussed this case in Chapter 6. So Za6k 0. Let ZZ6k be the set of k-stable points, and ZR0 Stk = Zpek \ Zpek. As we defined in Chapter 3, SQ6k = GkZp6k, Sp6,stk = GkZQ6,stk' (g) So,

Let ap, (al, a2) = a(al, al 2a2 2, aia2; a2i a21) for al, a2 E GL(1). Then Mp,7 = lap, (al, a2) I al, a2 E GL(1)}. The coordinates x1,33 x2,13, x2,22 determine characters aia2, aia3, ai 4a2 3 respectively. The convex hull of {(2, 5), (2, 1), (-4, -3)} contains a neighborhood of the origin of R2. Therefore, Zpk = Zp,k = {(x1,33 x2,13 x2,22) E (V )3}. (h) Spe

Let ape (a) = a(a-2, a, a; a2, a-2) for a E GL(1). Then Mpe

=

{(@ 91,23)I2) aps(es)

91,23 E SL(2), a E

GL(1)}.

Let V1 (resp. V2) be the subspace spanned by coordinate vectors of {x1,22, x1,23 x1,33} (resp. {x2,12, x2,13})

Clearly, Zp8 = V1 ® V2. Elements of the form aRs (a) act on V1, V2 by multipli-

cation by a4, a-3 respectively. Therefore, we have already discussed this case in Chapter 6. In this case, ZQsk # 0, and Zaek = Zpek. (i) Sp9

Let a,% (a1, a2) = a(al, a2, al a2; a1 2a21, aia2) for al, a2 E GL(1). Then MR0 =laps (a1, a2) I al, a2 E GL(1)}. The coordinates x1,23, x2,13, x2,22 determine characters a1 3a2i ai, aia2 respectively. The convex hull of {(-3, -1), (2, 0), (2, 2)} contains a neighborhood of the origin of 2 . Therefore, Z7.k = Zp9k = {(x1,23,x2,13x2,22) E (kx)3}. G) Sp1U

We can identify Zp10 with Sym2k2 0 k2. It is easy to see that

M' = SL(2) x SL(2) - { I

I

0

9 023)

,

92) 91,23,92 E SL(2) }

Therefore, we have already discussed this case in Chapter 5. So ZQ10k Let Zp10,k be the set of k-stable points, and ZQ10,stk = ZR10k \ defined in Chapter 3, Sp1o,k = GkZp10,k, Splo,stk = GkZalu,stk'

.

0 Zp10,k.

As we

Part IV The quartic case

206

§8.3 /3-sequences of lengths > 2 In this section, we describe /3-sequences of lengths > 2 for our representation explicitly. Let 81, , )31o be as in §8.2. Let (31,1 = 02 - /31, /1,2 = /33 - /31. Then (/31i 01, 1), (01, /31,2) are the /3-sequences

of length 2 which start with /31. Let 31,1,1 = (33 - 02. Then only /3-sequence of length 3 which starts with /31.

is the

Let /32,1 = ,33 -(32. Then (/32i /31,2) is the only /-sequence of length 2 which starts with ,32.

There is no /3-sequence of length > 2 which starts with /33. Let /34,1 = /32 -/34. Then (34i /34,2) is the only /3-sequence of length 2 which starts with 04. We do not consider /3-sequences of lengths > 2 which start with /35. We can write any element of T+ in the form vp6 (A1)ape (A2)(a3(-

1,

A3,1),1),

where vpe is a 1PS proportional to )36 and A1, A2, A3 E R+. If we identify Pp. (A1), ap.(A2), (a3(\ 1, A3, 1), 1) with 1PS's, these are orthogonal to each other. Let tp8 = {z E t 1 (/36i x) = 0}. We identify t% with {(-a, a; b) I a, b E R} = 30 so that 1))(-a,a,b)

(vo,(A1)aRe(A2)(as(A 1,A3, 1),

= AbA2a 2 3

Since 11(1,1, -2; -1,1)112 = 8 and 11(-1,1, 0; 0, 0)112 = 2, 11(-a, a; b)112 = 2a2 + 8b2.

If (/t6i /3) is a /3-sequence, /3 is an element of tts. Since the metric on t;. is the same metric as we considered in §6.2, /3 corresponds to /3-sequences which we considered in §6.2. Let /36,1, , (36,4 be the elements which correspond to /31, . . , /34 in §6.2. Then /3-sequences of length 2 which start with ,Q6 are ((36, /6,1), , 036036,4) We do not have to know /36,1 for our purpose. We can describe /36,2, , /36,4 explicitly as follows: 1

(36,2=( 3,

1

3,24 33 2

3

2

1

3;- 2)

1

2)-/3s,

2

1

3

22

241 332

1

1

2

We do not consider /3-sequences of length 3 which start with /36. We do not consider /3-sequences of length > 2 which start with /37.

We define t% similarly as in the case S6. Since II(-2,1,1; 2, -2)112 = 14, the metric on tQ8 coincides with the metric we considered in §6.2. , /36 in §6.2. Then 08,6 be the elements which correspond to /31i Let /-sequences of length 2 which start with /38 are (/38, /38,1), , ((3s, /38,6). We do not have to know /38,1, /38,2, /38,3 for our purpose. We can describe /38,4i /38,6 explicitly as follows:

21

1

1

1

,38,4 = (-3, 3, 3; 2, - 2) - Qa, (38,5

2

=

3

'_2

4 1

3 3;

1

2

3

33

1

-1

2, 1

2) - /38, 1

22

q

8 Invariant theory of pairs of ternary quadratic forms

207

We do not consider ,3-sequences of length 3 which start with X38. We do not consider ,3-sequences of lengths > 2 which start with ,39. We define 1

1

1310,1 = (0,4, 4,

1

1

4, 4),

1310,2 = 132 - 1310,

,310,3 = 133 -,610-

Then (13io,131o,1), (010, 010,2), 0310, 010,3) are the ,3-sequences of length 2 which start with ,310. Let 310,2,1 = 1310,3. Then (310, 010,20310,2j) is the only 3-sequence of

length 3 which starts with ,310.

Chapter 9 Preliminary estimates §9.1 Distributions associated with paths Let (P, w, s, GA, g etc. be as in §3.1. In our case, the condition dim G/T = dim V is satisfied. Therefore, (G/T/V) is of complete type. We consider the integrals in (3.1.8) and use the notation Zv(4), w, s) etc. for L = V. Note that since G1 = GA, we have a canonical measure on G. _ Let [ , ]i, be the bilinear form for V = Sym2k3 which we defined in §4.1. We define a bilinear form on V by [(X1, X2), (yl, y2)]V = [x1, y2]V + [x2, y1]v

Then this bilinear form satisfies the property [gx, g`y]V = [x, y]v for all x, y E V where g` = TGtg-1,rG. We use this bilinear form as [, ]v in §3.1. Let J(oD, g°) be as in (3.5.6). We define (9.1.1)

1°(`li, w)

= J s/Gk w(g°)J(, g°)dg°;

1(4D, w, w) = f

w(g°)J(, g°)(g°, w)dg°. /Gk

The Poisson summation formula implies that _ 1 (9.1.2) Zv(OD, w, s) = Zv+(4), w, s) + Zv+(&, w-1,12 - s) + f A sI°(,

w)dx.

Since the first two terms are entire functions, the last term is the issue. As in previous parts, we assume that P = Mv,,,D. Also we assume that V)(z) = and V)(p) 0 0, so (g°, w) = 9((g°)`, w) By (3.4.34), (9.1.3)

1(-b,w,w)

CGA(w;p)I°(-D,w)

(CG=T2-1T31).

So we study I(4, w, w) as a function of w. By (3.5.9) and (3.5.20), (9.1.4)

I(4), w, w) =S#(w)A(w;

4D(o))

p,l(p)=1

We know that 8p (4), w, w) +-":"-, ((b, w, w) is well defined for Re(w) >> 0. If p = (r), s)

and 0 = (i31), (/32), (03) or ((310), Ep (-D, w, w) = 0. If p belongs to class (1) or (3) in §3.5, cp (,D, w, w) is well defined for Re(w) >> 0. This applies to p = (0, s) where

D = (04), (as), (07), (08) or (f39). Therefore, all the distributions in (9.1.4) are well defined for Re(w) >> 0 except for p = (D, s) such that i) = (,35). We prove that (,D, w, w) is well defined for Re(w) >> 0 and 8p (,D, w, w) - 0 for this case in §10.3.

Since the rank of our group is 3, we have to consider paths of lengths up to 3. It turns out that if a is a /3-sequence of length > 2 which starts with /3k, (32, /34, ,3lo,

9 Preliminary estimates

209

we can apply our results in Chapter 3. However, if a is a 3-sequence of length > 2 which starts with 36 or /38i we have to use a method similar to Chapters 6 and 7. For the rest of this section, we define certain distributions associated with /36i /38, and prove that they are well defined. Let p = (a, s) be a path such that a = (/36) or (/38). In both cases, Za = Va1® Va2 where Valk = Sym2k2, Va2k = k for /36, and -01k = Sym2k2, Va2k = k2 for /38. We define a bilinear form [, ]v,; on Vai for i = 1, 2 similarly as in §6.1, using the longest element of the Weyl group of M.D. Let ° v,; be the partial Fourier transform with respect to [ , ] v,, for i = 1, 2. In both cases, Ap2 = Aa (see (3.3.11)). By Table (8.2.3), AD = {da(.\1) I Al E R+}, where 6

3,

3

2

1

R) ,

2

a = (as

-1)

da(A1) _

1-1) -1 1-1

Let kv,; be the rational character of Ma defined by the determinant of GL(Vai) for i = 1, 2. For ga E G°O fl MsA, we define rv,1 (go) = Ikv,, (9a)I-1 for i = 1, 2. By (6.1.1), (6.1.2),

Oz, (Rasp, 9a) = ral(9a)Oz, (A-sRa4)p, ea(9a)) +AaRsDp(6) - Rs4p(o)

+

(ica1(9a)Os,, (AaRa-Dp,ea(9a)) - Os,, (RalDp,9a)) Z,, QVa1,Va2

+

(Ial(9a)es,,("aRDDp,Ba(9a))-Os,,(Ra-Dp,9a)) Z,, CV01

+

(ral(9a)Os,,

es,, (Ra4Dp,9a)) ,

Z,, C Va 2

and (ic

Os,,

(9a)Os,,

Z,,CV,1,Va2

+ .`'aRa,Dp (6) - Ra,Dp (6)

= Kv,2 (9a) E Os,, (9v,1 9aRo Dp, 9a) Z,, C Va 1

Os,,(9sRa4'p,Ba(9a)) Z,, C Va 2

T,

OS,, (Ra,Dp,9a) - 'cv,2(9a)

OS,, (` V,2Ra4'p,ea(9a))

Z,, C V02

Z,, C V01

_k01(90) E Os,,("aRoDp,ea(9a)) Z,, CV01

+acv,,(9a)

p,9a) Z,,CV,2

-' v 1 (ga)

Os,Z,,CV,1

Rasp+9,(9a))

-

es,, (Rasp, 9a), Z,,CV,2

where a' runs through /3-sequences of length 2 and a -< D.

Part IV The quartic case

210

Let X be a principal quasi-character of go E G°A f1 MSA/Mak, and W E .©(ZDA). We define JAaoM,'A/Mat X(9a)esa,(IF,9D).p(go,w)dga

sD' C Vol,

J AaoMOS/Mat X(9a)©sa, ('y,9a(9a))ep(go,w)dgD

So' C Vat,

("vat (sa) -

"Va1 00-1?1

So, C Vol,

X(9D)©s,, ('y, 0a(9a)).p(go,w)dga f AaoM' /Mat 1 W, X, W) _

"Va1 (go)- <1

J AaoMOA/Mat

Sa, C Vat.

X(9a)Gsa, ('y,9a).p(9a,w)dgo

"V,1 00-1 <1

Lemma (9.1.6) The distributions

are well

defined if Re(w) >> 0.

Proof. Let go = Raga, where A5 = do(A1) is as in (9.1.5), and ga E MMA. Let ao(X2) = ape(A2) or ap8(22) for A2 E R+. Then we can write ga = aa(A2)ga, where

9a E M. The element Aaaa(A2) acts on VD1A, V02A by multiplication by ep1(As)Aa, ep1(As)A ',

respectively, where a, b > 0 are constants. Then'v01 (go)-1 = By (1.2.6), for any N >> 0, there exists a slowly increasing function hN (go) of go such that (epl(As)Aa)dim V02

go) < (epl(Ab)Aa)-NhN(9a) 0sa, ('1'e 9a(9a)) << (ep1(Aa)-lA1)-NhN(9o) Os,, ('F,95(95)) << (epl(A,)Aa)NhN(9o) es,, (IF, go) << (ep1(A5)-lA2)NhN(9a)

esa,

for Zo, C Vol, for Z5, C V52, for Za' C Vat, for Za, C Va2.

Therefore if

Let p1 = ep1(Xa), µ2 = IV,, (9D)-l. Then A2 = Ill N >> 0, there exists a slowly increasing function h'N(pl,ga) such that go) <
for Zo, C Vol,

esa, ('1`, 05(g5)) << /12 NhN(Al, ga)

for Zo, C VD2,

e5,, ('I',95(go)) «p2

for Zo, C Val,

esa,('1',9a) <
for Zo, C Va2.

Let 9

o _ (( 91,12

-

43

) Ia2(t21,t22)) or

((t11

91,23)

,

a2(t21, t22))

for d = ()36), ()38) respectively, where tll, t13, t21, t22 E A', 91,12, 91,23 E GL(2)A. By

(3.4.31), there exist constants rl, r2 > 0 such that if Ml, M2 > w0i µ11(Re(w)-wo)Itll(91,12)tl2(91,12)_lI-r2(Re(w)-wo)

e ( 9a, w ) « 1j 1,11(Re(w)-wo) l It12(91,23)t13(91,23) p

1

I-*2 (Re(w)-wo)

a = (Q6), a =

('38),

9 Preliminary estimates

211

for M1 < Re(w) _< M2 and ga in the Siegel domain. We choose Re(w) >> N >> 0. Then the function Itll(91,12)t12(91,12)-1I-r2(Re(w)-wo)

g,)lt'1(Re(w)-wo)

X(9a)µ2 NhN(fti,

is integrable for p, < 1, µ2 > 1, and ga in the Siegel domain. Other cases are

similar. This proves the lemma. Q.E.D.

Let' = Ra4bp. If Si,' C V 1, we define (9.1.7)

w, w)

w, op, w),

p,a',3 (b, w, w)

(go IF, w, Ic010'p, w),

w, w) _ p,a',<

w, /CVal Qp, w).

Also if Si,' C Va2, we define Ep,a',1(C w, w) = =p,a',> (90 IF, w, Ka1op, w),

(9.1.8)

0 ,a',2

(', w, w)

(Va2 'I, W1 KV02 Qp, w),

=p,a',3 (), w, w)

w,'Vai op, w),

(..

p ,a',4 (-t, w, w) _ Ep,a',< (IF, w, op, w)

We define 4

(9.1.9)

p,tot('D,w,w) _

(-1) iFlC

W, W).

Za'CV1,Va2 a=1

For a' such that Za' ¢ Vat, Va2, we define rp' (Ra' gyp', w, w), gyp' (Ra gyp', w, w) in

exactly the same manner as in (3.5.8). Then by (3.5.5),

8p'(Re'(P,',w,w)+,=p'w,w) is well defined for Re(w) >> 0.

These considerations show that (9.1.10)

"p+(-D, w, w)

-,p(op,w,w) = E

+

(2p'(R,'4p',w,w) +'p'(Ra',Pp',w,w)) p-
'(p')2

Zp' a Val Va2

+ =p,tot (4D, w, w)

We will prove in Chapter 10 that the distributions =p' (Ra' gyp' , w, w), 'p' (Ra' gyp' , w, w)

are well defined for Re(w) >> 0.

Part IV The quartic case

212

§9.2 The smoothed Eisenstein series In this section, we prove some estimates of the smoothed Eisenstein series which are not covered in Chapter 3. We will use these estimates in Chapter 10. We first fix some notation. For u = (ul, u2) E A2, let (9.2.1)

((I2

np, (u) = nP2(u) =

U

\ \tu

For u = (u1, u2i u3) E A3, let 1

nB, (u) =

(9.2.2)

u1

((U2

J2)

1

U3

1

For u E A, let 1 (9.2.3)

n'(u) =

u

)I2)n"U)=

1

1

1

I2

1

U

1

j1

nB2 (u) _ (I3, (U

For) ER,t11, ,t22EA1,let 1

(9.2.4)

dpi (A,) = a(A1 1 , .\1 , A2 1, 1),

dp2(A1)=a(A 1 t0

For u = (u1i

,

2

Al,1i1,1), A

= Cl(tll,t12,t13it21,t22)

u4), let nB(u) = (nB, (u1i u2, u3), nB2(u4)). Also let dxt o =

ndxti;.

In our situation, the Eisenstein series EB (g°, z) is a function of g° E GA, z = (z1, z2), where z1 = (z11, z12, z13) E C3, z2 = (z21, z22) E C2 and Z11 + Z12 + Z13 =

0, z21 + z22 = 0. Let e(g°, w) be the smoothed Eisenstein series. We choose the constants (Cl, C2) in (3.4.3) so that C1 = 1, C2 = C > 100. Let I = (I1i I2), T = (Ti, T2), v = (v1i v2), sI(l) = (s 1,i, (1), 52,12 (l)) be as in (3.4.24)-(3.4.28). Note that v2 is always 1. Consider an element of the form A = dpi (A1)a(p1, µ1 1, 1iµ'2,µ2 1),

where A1, µ1,µ2 E R+- We can write any g° E G°A in the form g° = kAt °nB(u), where k E K and u = (u1i , u4) E A4. We write at ° = (t1i t2), where t1 E (g°, w) for various T°A, t2 E T20A. For the rest of this section, we estimate cases. We recall that by (3.4.30), if q E t*, for any 6 > 0, M > w°i

9 Preliminary estimates

213

for L(q) + 5 < Re(w) < M, where h(A) is a slowly increasing function independent of 5, M. Moreover, if v = 1, h(A) = 1. Let q = (ql, q2) _ (qil, q12, q13; q21, q22) E Dj,r. We define rl = qll - q12, r2 = q12 - q13, and r3 = q21 - q22 Then 2r1 + r2 ( 9.2.5 )

qll =

-r1 + r2 3

qlz =

3

,

q13 = -

rl + 2r2 3

The following table shows the condition for q to belong to the domain Dj,,. Table (9.2.6) Il = 0

1

no condition

(1,2)

rl > 1

rl > 0

(1,3)

rl>1

rl>1

(2,3)

r2 > 1

(1,2,3)

rl>1 rl+r2>1

(1,3,2)

rl+r2>1

-

--

rl>0

r2 > 0 rl>1 rl+r2>0

-

r2>1

r2>1

rl>O r2>1

-

r2>0

r2>1

-

I1 = {1,2}

I1 = {2}

I1 = {1}

Tl

rl+r2>0

-

-

--

r2>0

It is easy to see that Al rl-2T2 idrl

t11g1+P1 =

(9.2.7)

Ai 3µ X

Al rl-2r2µ1 rl 2r1+r2 -r2 A 1

Al

Pi rl+r2 rl+r2

µl µl

2r1+r2 r2

Al

A T1+r2µl -Tl-r2 1

Also 1 µi 2111 (9.2.8)

-$1,1111 -

-

tl

t2S2,12

2

(1)

-

3

t

µl

1 -111 )

3111

_ 1

111

µ'1 µ12111(A -3p1 -')-112 - A1112µ112-2111

-212. /2 2

Suppose vi = (1, 2). Then (9.2.9)

t1

1o

3µl)-111 = A1111µ1111

2111 (Al 3/11)-112 = A1112µ1111-112

Ii = {2}, I1 = {1, 2}.

Part IV The quartic case

214

Proposition (9.2.10) Suppose I # (0, 0). (1) Suppose that r1 (1, 3), (1, 3, 2), or T = (1, 3) and Il = {1, 2}, or r = (1, 3, 2) and I1 = {1}. Then for any N > 0, e > 0, there exists a constant b > 0 such that if

M>w0, ej,T,1(9°, w) <<) I,42µ23

for w° - S < Re(w) < M, where cl > N, C2 < -N, C3 < f(2) Suppose that 12 = {1}, and ri = (1, 3) or (1, 3, 2). Then for any e > 0 there exists a constant S > 0 such that if M > w°i &r,T, l (9°, w) << )i' Itµc3

for w° - S < Re(w) < M, where c1 > -2 +8 C, C2 < -(-2 + sC), and c3 < f-

(3) Suppose that (I1i 7-1) = ({2}, (1, 3)) and 12 = 0. Then for any e > 0 there exists a constant S > 0 such that if M > w°i 2+c1 p1: c2

0

1"'i,T,l(9,w) K Al

c3

µ2

for w° - S < Re(w) < M, where Ic1I, IC2I, IC3I < e.

Proof. If 12 = 0, tT2 Z2+P2 2

-1

1+r3

ll

µ'2-r3

T2=1, T2 = (1,2).

If r2 = 1, we choose r3 = -1, and if r2 = (1, 2), we choose r3 > 1 close to 1. If 12 = {1}, tT2s2+P2-82,12(t) 2

1-r3-212.

=2

We choose r3 = a for this case. In all the cases, Lj(q2) < 1 + e1i where e1 > 0 is a small number. Also t22z2+P2 = µ2'3 for some c3i where C3 < e in all the cases and

IC3I > 0. If T1 = (2, 3), we fix r2, and choose r1 << 0. If

T1=(1,2,3), wechoose rl= 2-r2>>0. Then L(q)=3+(1+e1)C<w°=4+C if el is sufficiently small. Then ti1g1+P' satisfies the condition of (1). Since v = 1, we can fix 1. This proves (1) for r1 # (1, 3), (1, 3, 2). If I1 = {1, 2}, we choose r1 = r2 = 2, and 11, >> l12 >> 0 in (9.2.8). Note that we can do this because v = 1. Consider the case Ti = (1, 3, 2), Il = {1}. We choose r2 >> 0 and ri = 1 - r2. Then L(q) < w°, and

ti

- -A A

1

2r2-1 -1-211' 1

P'1

tC

Therefore, if we choose 11, >> 0, we have an estimate of the form (1).

Next, we consider (2). If Ti = (1, 3), we choose r1 = r2 = 1 + 16C and r3 = 2. Then L(q) = 4 + 2C < w°i and T191+P1

-

Al16

µl-16

9 Preliminary estimates

215

We have already considered the case I1 = {1, 2} in (1). If I1 = {1} or {2}, we choose 11, = 2 in (9.2.8). Note that we can choose any 11, > 1 in this case. This consideration shows that we have an estimate of the form (2). If rl = (1, 3, 2), we choose r1 = 0, r2 = 1 + 8C, and r3 = 2. Then L(q) < w°i and

-2+8C -IC

= Al

t1

Al

We have already considered the case I1 = {1} in (1). Therefore, we have an estimate of the form (2). Finally, we consider (3). We choose r1 = E2, r2 = 2 - e3, 111 = 1 + E4, where E1,' , E4 > 0. Then L(q) = w° + 2(E2 - E3) + CE1, and "I,r,I(9°, w)

<< A

2+2E2-E3+3E4µ13+E4pC31

where Ic3I < E.

, E4 so that L(q) < w°i we have an estimate of the

Since we can choose El, form (3).

Q.E.D.

Proposition (9.2.11) (1) Suppose I2 = {1}. Then there exist a constant 6 > 0, a slowly increasing function

h(a1i µ2), and a constant c > 1 such that if M > w°i 'I,r,v(9°, w) << P'1 `h(A1, P2)

for w° - 6 < Re(w) < M. (2) Suppose that I1 E) 2 and v1 = (1, 2). Then there exist a constant 6 > 0 and a slowly increasing function h(A) such that if M > w°i for any 1 >> 0, ei,r,v(9°, w) << h(A)A11µ-1

for w° - 6 < Re(w) < M. Proof. Since we assumed that C > 100, we covered the case when v = 1 in (9.2.9). Therefore, we assume that v1 = (1, 2) (v2 = 1) and prove (2). If I1 = {1, 2}, by (3.4.30), 4'I,r,v(9°,

w)

11-112 << A3`12µ21 1

A(rq+P)h(A),

where h(.\) is a slowly increasing function. So we choose any q E Dj,r so that L(q) < wo, fix 111, and choose 112 >> 0.

If I1 = {2}, by (3.4.30), I,r,v (9°s w) <<

1111 p-111 Av(rq+P)h(A),

where h(A) is a slowly increasing function. So we choose any q E DI,, so that L(q) < w°, and choose 11, >> 0. Q.E.D.

Chapter 10 The non-constant terms associated with unstable strata In this chapter, we prove the following two theorems.

Theorem (10.0.1) Let p = (D, s) be a path. Then 8p (4D, w, w) - 0 for (1) (2) (3)

= ()35),

(4)

_ (08),

(134), (137),

(5) r) = (138,138,1), and s(2) = 1, (6) a = ()38,138,2), and s(2) = 1, (7) -0 = ()3io, /31o,1)

Theorem (10.0.2) Let p1 = (131, 51), p2 = (132, s2) be paths. Then Epi

W, W) + Epz 'p2

=p1

(1) 01 = (136), 02 = (138,138,1), (2) 01 = (N8, 13s,2), 02 = (,39),

for

W, W) - 0

and sl(1) = 52(1), 52(2) = 0, and 5i(1) = 52(1), s1(2) = 0.

We will prove (10.0.1) (1) in §10.1, (2) and (7) in §10.2, (3) in §10.4, (4) in §10.5, (5) in §10.3, and (6) in §10.6. Also we will prove (10.0.2) (1) in §10.3, and (2) in §10.6. Corollary (10.0.3) Ep'= p (4), W, W) - 0. p

In this chapter, g8 always denotes an element of G°A f1 M0A. We use the measure dgo on G°A fl MDA which we defined in §3.3 (just after (3.3.11)). Let dxtO be the measure on TA° such that dg° = dkdx t°du, where dk, du are the measures on K, NA which we defined before.

§10.1 The case 0 = (,34) In this section, we prove the following proposition. Proposition (10.1.1) Let p = (0,s) be a path such that D = 134. Then yp (4', w, w) 0.

Proof. Let n(ut,) = npl (ua) for ua = (u2i u3) E A2. Consider E )y, (`I', 9an(ua)),9(9on(uo), w)du8

(10.1.2)

J(A/k)2

for IF = Ra4,p.

Let da(A1) = dpl(A1) for Al E I1 (see (9.2.4)). Let 913 = (91, 92), 91 = ( 91,12

t13)

'

10 The non-constant terms associated with unstable strata

217

where I det 91,121 = It13I = I det 921 = 1. Let go = d(A1)g°00.

For a = (al, a2) E k2, let < a ua >=< alu2 + a2u3 >. We define (10.1.3)

J1,«(9a) = I(A/k)2 OYa (IF,9an(ua)) < a ua > duo,

,fz,a(90,W) = fA/k)2 e(9an(ua), w) < a ua > duo. (

By the Parseval formula, (10.1.2) is equal to

8,«(91).2,-«(9a,w) aEk2

We consider 91,12, 92 in the Siegel domain as follows: (10.1.4)

9112 = k1

F(tltll

1

1)'

µ11t12) (u1 (P2t21

92 = k2

1

u4

µ21t22

1

,

where tll etc. are chosen from a fixed compact set in A', and P1,µ2 are bounded below.

We can write any element y E Yjk' in the form y = (yl, y2) where y1 EZak, Y2 E Wak and

yl

X1,13

X1,23

x2,13

X2,23

)

X

'

y2 = ( x2,33 )

Then 9an(ua)y = (A,t13g2y1tg1,12,)it1392(y2 + yltua)) Therefore,

fl,a(9a) = f A2

W(A1t1392y1t91,12,

A4't12

g2y1tua) < a ua > dua

ylEZak

392yltua) < a ua > duo.

_ yl EZnk

JA2

For y2 = t(x1,33, x2,33), y2 = t(x1,33+x2,33), we define [y2, y2] = x1,33x1,33 +

x2,33x2,33 Let IF, be the partial Fourier transform of IF with respect to y2 and the above bilinear form. Then t

fl,«(9a) = 1 8

4 2t -1t -1t 1 t13 92 ty1 a)

yl EZak

Let (10.1.5)

f3,a(9a) = A1

8

W1(A1t1392y1t91,12, A1 4t 13 t92

lty1 lta),

v1 EZak

-1,13#0

14 ,«(90)

_

Alb

A14t13t92 lty1 lta ) v1 EZak -1,13=0

Part IV The quartic case

218

Then fl,a(9a) = f3,a(9a) + f4,a(9a) Lemma (10.1.6) Suppose that µl, µ2 are bounded below. Then for any N > 1,

E If3,a(9a)I << rd3,N(Ai, pl, p2) aEk2\{0}

Proof. Let a' = (a', a2) = ayi 1. Then T1(A1t1392y1t91,12, A1 4t13 t92 lta )

f3,a(9a) _ A1 $ aEk2\{0}

v1 E ZDk

«'Ek2\{0}

x1,13#0

An easy consideration shows that Al 4t21 t9 2

ltai

= tk2 l \ 4t 12

il-163x2) )

/62 lt2i

A2t22 a2

J

.

Let

f5(90) = E

If3,a'(9a)I, f6(91) = E

If3,a'(91)I

a2=O,aiEkx

aiEk,a2Ekx

We choose 0 < \Ir2 E .9(ZaA), 0 < `P3 E 9(WDA) so that IWl(yl, y2)I << W2(yl)'F3(y2)

By (1.2.8), for any N > 1,

E

XF 3(-1

4t 13

t92

ltx)

<< sup(1,)lp2)(Al 4P2)-N,

«4Ek «2Ekx

3(A1 4t13 t92

lta) « (Aiµ2)N,

«i Ekx 1

«2'4 =0

E W2(\1tl392yit91,12) <
.1,13#0

where hl(A1, /61, P2) = sup(1, Al'Ptl 1tt2) sup(1, A1 lftlft2

1)

sup(1, )1 l/t1ft2).

So for any N1, N2 > 1, f5(9a) << Al shl(\1, /tl, /t2) sup(1, A4 1p2)(\l/1i/A2)-N1(Al 4U2)-N2,

f6(9a) << Al shl(A1, Al, /t2)(A1p1Et2)-Nl (Al 4µ2l)-N2.

Therefore, for any N1, N2, N3 > 1, f3,a(9a) << h2(Al, Ml, /62)('\1µlµ2)-Nl ((Al 4p2)-N2 + (Al 4/62 aEk2\{0}

l)-N3),

10 The non-constant terms associated with unstable strata

219

where

h2(A1, µl, µ2) = Al 8 sup(1, \1µ2)hl (X1, µl, µ2).

If A1 > 1, we choose Nl >> 0, and if Al < 1 , we choose Nl = 2N2 = 2N3 >> 0. Since Al, 92 are bounded below, we get the proposition (10.1.6). Q.E.D. Next, we consider f4,a(ga). Consider yi E Zak such that x1,13 = 0. This implies that x1,23 0 0 and x2,13 0 0. An easy computation shows that t

-ltyi-lt a = det x-

92

1

L 1t21 ((x2,23 + x2,13u3)al - x1,23«2) _1 _E2 22 x2,13a1

Let

f7(9a) =

Y,

1.f4,a(9a)I, .f8(9a) = E

If4,a(90)I

a1=0,o2Ekx

aiEkx,a2Ek

Lemma (10.1.7) Suppose that µl, µ2 are bounded below. Then we have the following estimates. (1) For any N > 1, f7(ga) << rd3,N(A1, Al, A2)-

(2) For any N > 0, f8(9a) «

12 Al-2N

sup(l, Al lµl/t2)µi µ2 Proof. By (1.2.8), for any N1, N2, N3 > 1, f7(9a)
sup(1, A4A2) (1µ1µ21)-N1(A1µ1µ2)-N2(A14µ2)_N3

X

We choose N1 = N4 + N3i and N2 = N4, where N3, N4 > 1. Then 2

f7(9a) « A-' sup(l, A-1µ4µ2) sup(l, AW2)A12N4 (A1 2 µi

/J-N3

Since 91, µ2 are bounded below, we get (1). Consider f8(ga). By (1.2.8), for any Ni, N2, N3 > 1, f8(9a) « Al 8 sup(1, Al 1µ1µ2)(A1µ1µ2 1)-N1(A1µ1

1µ2)-N2 (a14µ21)-N3.

We choose N1 = 1 + N, N2 = 2 + N, N3 = 1, where N > 0. Then f8(9a) << Ar

2

sup(l, ai 1µ1µ2)µi µ2 .

This proves (2). Q.E.D.

We continue the proof of (10.1.1). By the consideration as in §2.1,

E 1f2,a(91,w)I <EiI,7,v(9a,w), aEk2`\{0}

L, a1=0,a2Ekx

jf2,a(9a,w)j «

Part IV The quartic case

220

where I = (11,12) and either Il or 12 is non-empty. Therefore, there exists 6 > 0 such that if M > wo, EEk2\{o} If2,a(90, W)I is bounded by a slowly increasing function for wo - 6 < Re(w) < M. This implies that If4,a(9o)f2,a(9a)It(9o)-2p,

If3,«(9o)f2,a(9o)It(9a)-2pa,

«Ek2\{0}

«iEkx,a2Ek

are integrable for go in the Siegel domain. By (9.2.10), there exists 6 > 0 such that if M > wo, (go,w) «sup(Ail µ i2µz3,A

2+C

1µ12µ23),

where cl > 8, c2 < -4 and Ic3I, Icil, Ic2I, Ic3I are small. Note that we are restricting ourselves to the Siegel domain, so we can assume that Ic3I, Ic3I are small. t(ga)-2p = It is easy to see that A1912µ22. Therefore, for any N > 0, t(go)-2pfs(9o)

E

If2,-a(9o, W) I

a1=°,a2Ekx

is bounded by a constant multiple of 1

sup(1, A1 µ1µ2) 3llp(A 1

2-2N+c1 -2+c2 -2+c3 µl µ2 , A1 ^1

2-1

-2+cs -2+cs µ1

µ2

)

If A, > 1, we choose N >> 0, and if Al < 1, we choose N = 0. Hence,

f

a M,A/Mak

f1,a(9o)f2,-a(9a)t(9o)-2pdga _ 0.

wp(go)

aEk2\{0}

This proves (10.1.1).

Q.E.D.

§10.2 The cases a = (,Qs), and 03 = (01o) We consider paths pi = Let 01 = ()35), 02 = (01,51)4P2 = (02,52), and p3 = 03,53) such that 51(1) = 52(1) = 53(1) (i.e. P3 P2)- For 0 = 01,D2, Ma = T. Therefore, we use the notation go = to = (tl, t2) where tl E T°A, t2 E T20A.

We first consider pl. Let p = p1, 0 = 01.

Proposition (10.2.1) The distribution 8p(ID,w, w) is well defined for Re(w) >> 0, and , p (4), w, w) - 0 .

Proof. Let W = RaPp, and n(uo) = nB(uo) for uo = (u1,u2,u3,u4) E A4. It is easy to see that Oy, (W, t°n(uo)) does not depend on u1i u2. For a = (al, a2) E k2, let

< a, uo >=< a1u3 + a2u4 >. We define (10.2.2)

fl,«(t°) = J

OYa (IF,t°n(ua)) < a uo > duo, A/k)4

f2,a(to, w) = J A/k)4

ea(t°n(uo), w) <

uo > duo.

10 The non-constant terms associated with unstable strata

221

By the Parseval formula,

I

fin (t°)f2,-a(t°, w).

y.('W,t°n(uo))8(t°n(uo),w)duo = aEk2

We can write any element y E Yak as y = (yl, y2) where yi = (x1,33, x2,23) E Zak, y2 = x2,33 E Wok. Then

`1(t°n(uo)y) < a uo > duo.

fl,a(t°) = (A/k)4 Y2Ek

YiEZak

An easy computation shows that t°n(ua)y = (71,33(t°)x1,33, 72,23(t°)x2,23, 72,33(t°)(x2,33 + X1,33U4 + x2,23u3)) Let u4 = x1,33u4 +x2,23u3, ui = ui for i = 1, 2, 3. Then u4 = x1,33(u4 - x2>23u3) We define

(10.2.3)'(t°, u4, y) = i'(ry1,33(t°)x1,33, 72,23(t°)x2,23, 72,33(t°)(x2,33 + u4)).

Since t°n(uo)y does not depend on ui,u2,

fl,a(t°) _ YiEZak

f(A/k)2

T V/(t°, u4, y) < alu3 + a2x1,33(u4 - x2,23u3) > du3du4. Y2Ek

Since < alu3 + a2x1,33(u4 - X2,23, u3) > _ < (al - a2x1,33x2,23)u3 + a2x1,33U4 >,

the above integral is 0 unless x1,33 = al 1a2x2,23. Therefore, /

fl,a(t°) =

Ek -1

ZA V (t

0

-1

/

/

, u4, yl) < a2x1 33u4 > du4.

-1,33-°1 -2-2,23

Let W1 be the partial Fourier transform of W with respect to the coordinate x2,33

and < >. Then fl,a(t°) is equal to (72,33(1°)

(-1

W1(71,33(t°)x1,33, 72,23(t°)x2,23, 72,33(t°)-1a2x1,33) a1 E ZOk

-1 -1,33=°1 -2-2,23

Since X1,33, X2,23 54 0 for yl E Zak, the above integral is 0 unless al, a2 # 0 or al = a2 = 0. We choose 0 < q'2 E .9(YSA) so that 011 <''2. Then

E If1,a(t°)I C aEk2\{0}

I72,33(t°)I-1e3('P2,71,33(t°),72,23(t°),72,33(t°)-1),

Part IV The quartic case

222

Let Pl = 171,33(0°)I, P2 = I'Y2,23(t°)I,µ3 = I'Y2,33(t°)I-1, and (10.2.4)

d(µ1, P2, µ3) =a(µ1 21

_1 _12

a

µi1

,

,E

µ2

1 _ 41

µ121

11

i

µl

1 21

2 ).

Then to can be written in the form to = d(pi, P2, µ3)t °. By (1.2.6), for any N1i N2, N3 > 1, Ifl,a(t°)I << IL-NjIL-N2 A1-N3

(10.2.5) aEk2\{0}

By the consideration as in §2.1, 1f2,a(t°, w)I E iI,r,1(t°, w)

r

aE(kx )2

where Il = {2}, 12 = {1}.

Lemma (10.2.6) Suppose that Il = {2} and I2 = {1}. (1) If Ti = (2, 3) or (1, 2, 3), there exists a constant S > 0 such that for any N > 0, M > w° there exist cl, c2i c3 > N satisfying ei,r,l(to,w)

<< Pi1P'22P33

for w° - 6 < Re(w) < M. (2) If Tl = (1, 3), there exists a constant S > 0 such that if M > wo, «P11µ2P3

forty°-S 0, M > WO,lll,121 > 1,

<<

(P2P3)-111(P1P3)2-2r3-ia1 x

1(911,913,912)+P1

Ti = (2,3),

1(912,913,911)+P1

Ti = (1, 2, 3),

1(913,912,911)+P1

T1 = (1, 3)

for L(q) + S < Re(w) < M. An easy computation shows that ti' = Pl ° p2 lp3 ' . We fix r3 = 2, 111 = 121 = 2. Then Pl I,r,1(tO,w)
X

2*1

4r2 P2 r1-r2

P3

2r1-4r2

Ir1-qT2 -T2 -qrl-q r2

P1

P2

P3

Irl+Zr2 r2 -4r1+lr2

Pl

P2 P3

In the first case, we fix r2 and choose rl << 0. In the second case, we choose r1 = -r2 + S >> 0 where b > 0 is a small number. Then it is easy to see that the statement of (1) is satisfied. In the third case, By choosing q so that r2 = 10, rl = 1,

10 The non-constant terms associated with unstable strata

we get the estimate of (2). Since L(q) = 22 + L(q) < w°. This proves (2).

223

2C, w° = 4 + C, and C > 100, Q.E.D.

By (10.2.5), (10.2.6), for any N, Nl, N2, N3 > 1, 0

0 -2p «rd3,N(/tl,/t2,/-13)+µla -Nl 9-N2 4-N3 P2 µ3

0

Ifl,a (t )f2,-«(t ,w)I(t) aEk2\{0} Therefore,

0 JA/Tk

wp(t°) E fl,a(t°)f2,-a(t°,w)(t°)-2pdxt° - 0. aEk2\{0}

This proves (10.2.1) (We have also showed the absolute convergence for Re(w) >> 0). Q.E.D.

Next, we consider P2 such that 52(2) = 0.

Proposition (10.2.7) ;42 ((DI WI W) - 0' Proof. Let tY = Rae 4)p2. Let nag (u3i u4) = (n" (U3), nB2 (u4)) for u3i u4 E A. Then 43 (t°na2 (u3, u4), w) < 0lu3 + a4u4 > du3du4 = f2,a(t°, w). J(A/k)2

Since ep21(t°) = (t°)010 = Al /.t2µ3 and 52(1) = sl(1), we are considering the integral

J

wp2(t°)

fl,a(t°)f2,-a(to,,W)(t°)-2pdxt°

El/ aEk2\{0}

TA/Tk

µ1D2 µ3 <1

But we have already estimated this integral in a larger domain. Therefore, p2 (4i, w,

W) - 0. Q.E.D.

Next we consider P2 such that 52(2) = 1.

Proposition (10.2.8)

p2 (4i, w, w) - 0.

Proof. Let eYa2 (`11, t°Ba3 (na2 (u3, u4))) < l3 + 2U4 > du3du4,

(10.2.9) fl,a(t°) = f A/k)2

f2,a(t°, w) = J

r'p3 (9D3 (t°)na2 (u3, u4), w) < a1u3 + a4U4 > du3du4.

A/k)2

Then, by the Parseval formula, Ep2 ($, w, w) is equal to

f

wp2 (t°) E TA Tk

;,,;.2 3>1

aEk2\{01

fl,a(t°)f2,-a(t0

W)((t°)-2Pa3 IGa31(t°))-1(t°)-2pa2 dxt°.

Part IV The quartic case

224

We can estimate EaEk2\{0} I fi,a(t°)I in exactly the same way as in (10.2.5). Let µ4 = µ1µ2µ3. Then µ1

Ni

µ2 N2µ3

NS

= µl2

-N1 -f'2 µ32 fl -N3 P4

Therefore, for any N > 0, ,«(t°)I << 4 Nrd2,N(µl, µ2) aEkx

Since EaEkx I f2,_a(t°, w)I is a slowly increasing function, well defined for Re(w) > wo - 6 for some 6 > 0, °p2 (4>, w, w) - 0. Q.E.D.

§10.3 The cases D = (/36), In this section, we consider the cases al = (/36), D2 = (38,,38,j). Let (l3 = (08)-

such that si(1) _

Consider paths pl = (D1,51)42 = (22,52), and p3 = 52(1) = 53(1).

We first consider D = t)1, s = si. In this case, Pa = P1 x B2. The proof of the following lemma is easy, and is left to the reader.

Lemma (10.3.1) The convex hull of each of the sets (1) {71,33, 72,11, 72,22, -72,33}, (2) {71,33, 72,11, 72,22, -72,13},

(3) 171,33, f 12,111 72,22, -72,23},

(4) 01,33, 72,121 -72,13, -72,23}

contains a neighborhood of the origin. For ua = (u2i u3, u4) E A3, let n(ua) = (npl (u2i u3), nB2 (u4)). Let 91

091,12

0

0 t13

,

t2

t21

=1\ 0

0 t22

where I det gi,12I = It13I = It21I = It22I = 1. Let ga = dpl (,i)a36 (µ2)(g1, t2), where a36(µ2) is as in §1.2. Let 91,12, ui etc. be as in (10.1.4), and go = kt°n the Iwasawa decomposition of ga. We can write any element y E Yakl in the form y = (yi,Y2) where y1 = (x1,33, x2,11, x2,12, x2,22) E Zak and y2 = (x2,13, x2,23, x2,33) E Wak. Let

IF = Ra4)p. Then DYa('P,9an(ua)) =CY; (1I',9an(uo))+ey,

(11,9an(ua))

We recall that these are theta series associated with subsets of {(y1, y2) E Zak, Y2 E Wak}, where y1 E Zak, Za Stk respectively. By assumption,

ey, ('1', 9an(ua)) = w}1(k)CYa (W, t°nn(ua))

So when we estimate various functions, we assume that k = 1.

10 The non-constant terms associated with unstable strata

Let

. We define (10.3.2)

fl,«(90) = J

e; (, 9n(uo)) < a u> duo, A/k)3

e(9n(u), w) < a u> duo.

f2,. (90, w) = f A/k)3

Proposition (10.3.3)

J

f1,«(9o)f2,-«(9a,w)t(9a)-2PDd9a ^ 0.

wp(9o) n M,A/Mok

«Ek

0}

Proof. We define 2

2

Q,, (u2, u3) = x2,11u2 + x2,12u2u3 + x2,22u3, lye (u2, u3) = x2,13u2 + x2,23u3

Then n(uo)y = (yl, x2,13 + au2Qy1, x2,23 + au3Qy,, x2,33 + lye + Qy, + x1,33u4),

where 8u2 etc. are the partial derivatives. Let u'2 = 8u2Q,1, u3 = 8u3Qy1. Then

-

C

2X2,11 X2,12

X2,12

U2

2x2,22) ( u3

So if A(yl) is the determinant of the above matrix, U2

( u3) = 0(91)-1 Note that if yl E ZaSk, A(VI)

2X2,22

-X2,12

2x2,11)

u3

)

0.

As before,

fl,«(90) = ylEZok

I I'(9an(uo)yl) < a uo > duo. A3

Let A2,13 = A2,13(9a, y1, u1, u5) = 72,13(t0)au2Qy1,

A2,23 = A2,13(9a, Y1, u1, uo) = 72,23(t0)(au3Qy1 + ul(x2,13 + au2Qy1)),

A2,33 = A2,13(90, yl, u1, ua) = 72,33(t0)(42 + Qyl + xl,33u4).

Then

fl,«(go) = yl EZak

f

IF (9o y1, A2,13, A2,23, A2,33) 3

< a uo > duo.

225

Part IV The quartic case

226

Lemma (10.3.4) There exists a constant 6 > 0 such that for any N > 0, M > wo,

E Ifl,a(9s)f2,a(9s, w)I < rd3,N(Al, 111, P2) aEk3\{0}

forwo-S
dinate x2,33 and < >. Then 1F(gon(uo)yl) < a3u4 > duet JA y2,33(to)-1«3x1,33)

IF1(goy1, A2,13, A2,23,

< -a3(1 ,2 + Qy1) >

We choose 0 < "2 E 9(Z-OA x A) so that

fA2

IW

1(yl,x2,13,x1,33)Idx2,13dx2,23 «W2(y1,x1,33)

Then `I'2(9ay1,y2,33(t0)-1«3x1,33)

Ifl,a(9a)I << X 6N'2 T, a3Ekx

yi EZok

Note that this bound does not depend on al, a2. By (10.3.1), for any N > 0,

E Ifl,a(9a)I <
By the consideration as in §2.1,

E If2,a(91,w)I << E

(go, W),

n1,a2Ek

a3Ekx

where I=(I1i12)and12={1}. Therefore, there exists 6 > 0 such that if M > wo, 1,a2Ek,a3Ekx If2,a(gs, w)I is bounded by a slowly increasing function for wo - 6 < Re(w) < M. So for any

N>0,

E If1,a(gs)f2,a(9a,w)I <
(10.3.5)

a1,a2Ek a3Ekx

for wo - 6 < Re(w) < M. We consider a such that a3 = 0. Let

al = A(yl)-1(2«1x2,22 - «2x2,12), a2 = A(yl)-1(-alx2,12 +

2a2x2,11)

Then alu2+a2u3 = aiu'2+a2u3. The condition a 54 0 is equivalent to the condition a'=(a',a')0.

10 The non-constant terms associated with unstable strata

227

We define 'I'3(yl, x2,13, x2,23) fA2 `1'(yl, x2,13, x2,23, x2,33) < x2,13x2,13 + x2,23x2,23 > dx2,13dx2,23dx2,33

Then

fl,a(90)=I72,33(t°)I-1 E 'F3(9ay1,72,13(t°)-la' -a2u1,72,23(t°)-lag yi E Zo k

We choose 0 < 'P4 E So(ZDA),0 <'I's E So(A2) so that I

3(y1, x2,13, x2,23)I <'I'4(y1)W5(x2,13, x2,23)

By (1.2.6), for any N1, N2, N3 > 1,

E T4(9ay1) <<

I71,33(t°)I-N1I72,11(t°)I-N2I72,22(t°)I-N3,

V1 EZnk

'k5(72,13(t°)-la' - a2u1, 72,23(t°)-la2) << sup(1, I72,13(t°)I)I72,23(t°)INI, -1 Ek .Z Ekx

E 'I's(72,13(t°)-1x1,0) <<

I72,13(t°)IN1.

aiEkx

Therefore, by (10.3.1), for any N > 0,

E

(10.3.6)

Ifl,a(9a)f2,-a(9a, w)I << rd3,N(.Al, µ1, P2)-

(-1--2 )Ek2\fo)

°3=0

Hence, (10.3.4) follows from (10.3.5) and (10.3.6). Q.E.D.

Now Proposition (10.3.3) is an easy consequence of (10.3.4). Q.E.D.

Next, we consider

(A/k)3

©y.,., (`F, 9an(ua))e(9an(ua), w)dua.

Let (10.3.7)

d(Al, µl,µ2) = a(Aj 1µ1,'l

lµi 1

, Air L 2, µ21),

and t° = d(A1i pi, µ2)E °. Let go = n'(u°)t° for u° E A. Let Ha be the subgroup generated by T and the element ((1, 2), 1). We identify ZOk with the space Sym2k2 ® k in Chapter 6. We define Za °k to be the subspace

Part IV The quartic case

228

which corresponds to Z,,ok in Chapter 6. We define Z'a3ok similarly. Then Zl,.tk Mak XHak Z'asok. We define

Xb = {kn (uo)to I k E K fl MbA, Al >

(10.3.8)

a(uo)}.

Then as in §4.2, if f (go) is a function on GO fl Maw/Hak,

f °1fMae/Hak f(9)d9=

f(9)(to)-2dxodo.

f

ar1Be/Tk

We define (10.3.9)

1,

6z;,0 (W, 9an(ua)) =

T(9an(ua)(yl, y2)),

51=(-1,33,0,-2,12,0), Y2 E Wak -1,33,-2,12 Ek x

f3,«(96) = f(A/k)3 6z6,a (W, 9an(ua)) < a ub > dub.

Then fA,MA/M,k (A/k)3

eya,., (`f', 9an(ua))e(9an(ua), w)dgbdua 0

f3,«(9a)f2,-a(9a,w)(t) Xa f1BA/Tk ,,Ek3\{0}

2p 2 x t0duo. Al

By a similar consideration as before,

E 51(13

f3,a(90) =

'(9an(ua)yl)dub.

fA

-1,33,-2,12Ekx

As in the proof of (10.3.4), If3,a(9a)I << A 6µ2

'1'2(901,

72,33(t0)-1x1,33a3)

51=(-1,33,0,-2,12,0)

-1,33,-2,12 Ekx

This bound does not depend on a1, a2. We choose 0 < 9'6i T7 E 9'(A2) so that '1'2 (9ay1, 72,33(t0)-1x1,33a3) 12µ21x2,12, A12µ21x2,12u0)q'7(A1i

<< '6(A

For it El

,uoEA, let hi(µ,uo) =

W6(µx2,12,ux2,12uo) x2,i2Ek5

We also define 'F7(A4iE2X1,33, AI 4A-2X-,I

h2 (Al, P2) X1,33,a3Ekx

10 The non-constant terms associated with unstable strata

229

Then If3,a(ga)l «A16/12h1(A12121,uo)h2(.A1,µ2)

(10.3.10) a3Ekx

Lemma (10.3.11) For any 81i S2,83 E C, the integral As' A12 2 a(uo)s3hl(t12u21,uo)h2(A1,IU2)d-Ajdxµ2duo

LXA converges absolutely.

Proof. Let µ3 = A 1 2µ2 1, µ4 = A1/12 Then A1 = pi µ4 , /12 = 113 2µ4 1 By (1.2.6), for any N1, N2 > 1, h2()1iµ2) << Therefore, our integral is bounded by a constant multiple of µ4N'(µ34µ43)-N2.

µ4N2+siµ33N2-N1+s2 18+

a(u°)

Re(83 )

X

x

hl(P3,uo)d µ3d µ4du°,

xA

where si = Re(2s1 - 282), s2 = Re(2s1 - 82). Let r1i r2 E R, and c > 0. Then the integral µs1 a(uo)r2 h1(µ3, uo)dX p3duo

converges absolutely and locally uniformly on the domain of the form {(rl, r2) E R2 I r1 > 2+e,r2 > -e} by (4.2.9). So if µ4> 1, we choose N1>>N2>>0, and if µ4 < 1, we choose N2 >> N1 >> 0. This proves (10.3.11). Q.E.D.

By the consideration as in §2.1,

If2,a(g1,w)I «I: 4,Ty(ga,'w), a3Ekx '11°2Ek

where 12 = {1}. By (3.4.30), and (9.2.11)(1), (10.3.10), (10.3.11), we get the following proposition.

Proposition (10.3.12) Xa nBA/Tk

wp(t°) E

a1,a2Ek -3Ekx

Note that

E f3,a(n(uo)t°)f2,-a(n(uo)t°,w) '1,°2Ek -3Ekx

is invariant under the action of Tk. We consider a such that a3 = 0.

f3,a(n(uo)t°)f2,-a(n(uo)t°,w)(t°)-2pµidXt°duo

0.

Part IV The quartic case

230

Proposition (10.3.13) X, nBk/Tk

Wp(9a)

E

f3,a(9a)f2,a(9a,w)(t°)-2P1A2ld"t°duo

0.

a1 EkX

a2Ek a3=0

Proof. It is easy to see that (t°)-2Pµ1 = A61/12 2, r'02 (to) = A l sµ2. Let c4 = -a2x2 i2, as = -a1x2 i2. Then f3,a(9a) is equal to

-6 3 1 µ'2

T3(9ay1,'Y2,13(t°)-la1 -'Y2,23(t°)-1a2u0, 72,23(t°)-la2) -1,33,-2,12 Ek X

We choose 0 < T8 E .'(A2 ), 0 < IF9 E .'(A3) so that 1W3(90Y1, x2,13, x2,23) <<

2µ2 1x2,12, A 1 2µ2 1x2,12u0)

9(A1 u2x1,33, x2,13, x2,23)

Let h3 (p, uo), h4(t°, u°) be the following functions

E T8 (µx2,12, AX2,12u0) x2,12Ekx

E q'9(A411!2x1,33, 72,13(t°)-1a1 - 72,23(t°)-la2uo, 72,23(t°)-la2) -1,33 Ek x ai Ek

a2Ek%

The condition al # 0 is equivalent to the condition a'2 54 0. So If3,a(9a)I << A1 6A2h3(.1 212 1, u0)h4(t°, 460) a1Ekx a2Ek

By (1.2.8), for any N1i N2 > 1, h4(t°,uo) «sup(1,A1/.i1µ21)(A4N2)-Ni(Al

lµ1112)_N2

By the consideration as in §2.1,

E If2,a(91,w)I < E 1FI,T,V(9a,'w), I'T'V

EkX a2Ek *1 a3=0

where I1 3) 2, vl = (1, 2). Therefore, by (3.4.30), there exists a slowly increasing function h5(t°) and a constant b > 0 such that if M > w0i

> If2,a(9a,w)I << h5(t°) a1Ekx a2Ek a3=0

10 The non-constant terms associated with unstable strata

231

forwo-b
1)h5(t°).

Then, for any N1, N2 > 1,

E If3,a(90)f2,-a(90,w)I '1 EkX a2Ek

3=0

<< h3(A1 2/.12 1, uo)h6(t°)A3N2-2N1(A1 2/12 1)N1+N2µ1 -N2

If Al > 1, we choose N1 >> N2 >> 0, and if al < 1, we choose N2 >> N1 >> 0. Then the right hand side is integrable on Xa f1 BAITk. This proves the proposition. Q.E.D.

We consider a such that al = a3 = 0. This implies that a2 = 0. Let h7(t°) _

19(A1/_12x1,33, 72,13(1°)-1a1A31,33,aiEkx

Then for any N1, N2 > 1, h7(t°) << (A

p2)-Nl (Al lµl lµ2)-N2,

and

If3,«(9a)I << A16µ2h3(A12µ21,uo)h7(t°). '1='3=0 a2Ekx

By the consideration as in §2.1,

E If2,a(91)I C £I,,,1(9a, w), °1=a3=0 .2Ekx

where Il D 2. Therefore, by (3.4.30), there exists a slowly increasing function h8 (t°),

and a constant b > 0 such that if M > wo and 1 >> 0, If2,a(9a)I << h8(t°)(Aiµ1)1 a1=°3v0 a2Ekx

for wo - 6 < Re(w) < M. Let h9(t°) = A1 -6µ4h8(t°). Then for any N1i N2 > 1, l >> 0, (10.3.14)

E If3,«(90) f2,-«(9a) I '1='3-0 .2Ekx « h9(t°)h3(A12µ21, uo)(Ai/12)-Nl (A1 lµ1 lµ2)-N2 (AlA1)i = h9(t°)h3(A12u21 u0)Ail-2N1+3N2 1 N2(\j2µ21)N1+N2.

Part IV The quartic case

232

Lemma (10.3.15) The integral

f

u'p(t°) A/TkxA

(uo)t°,w)(to)-2Ppidxtoduo

f3,a(n (uo)t°)f2,-,(n °1=-3=0 -2Ekx

is well defined if Re(w) >> 0.

Proof. By (3.4.30), there exists a slowly increasing function h10(t°) such that

An+r2 1

T1+r2

N1

Airl+r2µ2

it,-r,I(9a, w) << hlo(t°) { Il

r, +r2

Tl = (2, 3), T1 = (1, 2, 3),

r2

This is possible because v in (3.4.30) is 1 for our case. If Ti = (2, 3), we fix r2 and consider rI << 0. If Ti = (1, 2, 3), we fix r1 + r2 and consider r1 >> 0. If Ti = (1, 3), we fix ri and consider r2 >> 0. It is easy to see that the convex hull of the set

j(-2,0,-l),(4, 0, 1), (-1, -1, 1), (-1,1, 0)} contains a neighborhood of the origin.

Let to = (t1, t2) where tl E T°A, t2 E TiA. In all the cases, there are a finite number of points ql,

-

, qj E D11,1.1 such that

If3,«(n(uo)t°)I inf((tl)Tlq,)(t°)-2puihlo(t°) °1=°3=0 -2Ekx

is integrable on TA°lTk x A. Then our integral is well defined if Re(w) > L(qi) for

i=1...,j.

Q.E.D.

Lemma (10.3.16) -Pi

w)

L2IT.xA

-2Ekx

Proof. Consider the integral

TA/TkxA Cl1

fdxt°du0. f3,u(9a )f2,-a(90)(t°)-2Pµ2

Wp1 (t°)

( 0)

f3,«(9a)f2,-.(9a)(t°)-2Pµidxt°du0.

Wp1 (t°)

.1 =.3 =O

-2Ekx

We use the estimate (10.3.14). If Al > 1, we choose Ni >> N2, 1, and if Al < 1, we choose 1 >> Nl >> N2. Then the above integral converges absolutely. Therefore, f3'.(gD)f2'_.(gD)(t1)-2p[12

WP' (to) TA/TkxA µl< a(u0)

a1=°3 e0 -2Ekx

Q.E.D.

10 The non-constant terms associated with unstable strata

233

We define

f4,a2,a4(t°) = J f3,(°,a2,°)(t°n'(ul)) < a4u1 > du1, A

f5,

2,4(t°,w) =

fT2/TkxA

ITVTk

IA/k

f2,(°,2,°)(t°1Z (ul),W) < 4u1 >

f3,a(go)f2,-a(ga)(t°)-2pµ2ldxt°duo

Wa1 (t°)

'2Ekx

Wp1 (t°) T f4,a2,a4(t°)f5,a2,a4(t°, w)(t°)-2pµdxt° .2Ek x a4Ek

By (10.3.1)(4), for any N > 0,

If4,a2,a4(t°)I «rd3,N(A1,91,µ2)

T,

a2,a4Ekx

Since we have an estimate of Eat a4Ekx If5,1112,a4 (t°, w) I by a slowly increasing function as before, we get the following lemma.

Lemma (10.3.17) JTA/Tk

W pl

(t°)

f4,a2,a4(to)f5,a2,a4(t°,w)(t°)-2pdx to - 0.

2Ex '4Ekx

Let f6,a2 (t°) = f4,a2,0(t°),, f7,a2 (t°, w) = f5,a2,o(t°, w). Then

&-2 (to) =

JA/k

f7,a2 (t°, w) =

/k

OY02 (Ra2(Dp2,ton" (u3)) < a2u3 > du3,

J3

(t°n"(u3), w) < au3 > du3.

Consider P2 such that 52(2) = 0.

It is easy to see that ep21(t°) = (t°)"8 =

Al µ'1 µ2 ' . So Ap2°Ma2A = {t° E TXX I ,1µi lµ21 < 1}. Therefore, p2 (1 , W,

W) = f

Wp2 (t°) E fs,a2 (t°)f7,-a(t°,

al

TAO

/Tk

µll

w)(t°)-2pdx

to.

a2Ekx

µ2k1

Hence, (10.3.18)

Ep1 E 1("'' , W, w) + EpJ42 ((D, W, w) Ep1

J

Tk/Tk a1µ11µ21>1

f6,a2(t°)f7,-a2(tO,w)(t°)-2pdxt°.

Wp1(t°)

a2Ekx

Part IV The quartic case

234

Note that E2 = -flLemma (10.3.19) There exists a constant S > 0 such that for any N > 0, M > wo, I f6,a2(t°)f7,-a2(t°,w)I << (Alpl lµ21)-Nrd2,N(/-1l,P2) a2Ekx

forw°-6
>1, l>1,

E

Ifs,a2(t°)f7,-a2(t°,w)I

°1=°3=O a2Ekx «hll(t°)(A1/1'2)-N1(A11/111/12)

N2(A12µ21)-N3(A1/11)1

Let µ3 = \lpl lµ21 Then the right hand side of the above inequality is (10.3.20)

hll(t°)111

(4N1-2N2-2N3-41) µZ (5N1-3N3-31)µ3 (4N1-N2-2N3-31)

We choose N1, N2, N3,1 in the following manner:

N2i N3,1 fixed, and Nl >> 0 N2i l fixed, and N1 = 1 N3 >> 0

Al > 1, 92 > 1, Al > 1, 92 < 1,

N2iN3fixed, and l=4N1>>0

µk1,µ2>1,

N2 fixed, and N3 = 41, Nl = 3l >> 0 8

Al < 1, µ2 < 1-

This gives us the bound of the lemma. Q.E.D.

The following proposition follows from (10.3.18), (10.3.19).

Proposition (10.3.21) If 51(1) = 52(1) and 52(2) = 0, Ep1"p1(4),w,w)+EP2yP2(4)'w,w)

0.

Consider P2 such that 52 (2) = 1. Then -P2 (4), w, w) is equal to

f

TA Tk

a1M1 1µZ 1>1

Wp2(t°) T,

fs,a2(t°)f7,-a(t°,W)((t°)-2hD3ra31(t°))-1(t°)-2P°2d"t°.

a2Ekx

By (10.3.19), we get the following proposition.

Proposition (10.3.22) If 52 (2) = 1, `^'p2 (D, ca, W) - 0

§10.4 The case 0 = (/37) In this section, we consider a path p = (a, s) such that 0 = (07) An easy consideration shows the following lemma and we omit the proof.

10 The non-constant terms associated with unstable strata

235

Lemma (10.4.1) The convex hull of each of the set (1) 171,33, 72,22, 72,13, -72,23)1, (2) 171,33, 72,22,'Y2,13, -72,33} contains a neighborhood of the origin.

Let = Ra4p. In this case, Pa = B, so we write go = to _ (tl, t2) where t1 E T 10A, t2 E T 20A. We choose an isomorphism d : R+ -* T. Therefore, we can write to = d(A1, A2, A3)t °, where Al, A2, .\3 E R+. Let n(uo) = (nP, (u2, u3), nB2 (u4))

for uo = (U2, u3, u4) E A3. For fixed to and ul E A, Oy, (IF, t°n'(ul)n(uo )), e(t°n'(u1)n(uo),w) are functions on (A/k)3.

For a = (a1i a2, a3) E k3, let < a uo >=< alu2 + a2u3 + a3u4 >. By the Parseval formula,

f

OYY (%F, t°n (ul)n(uo))e (t°n'(ul)n(uo), w)duo

A/k)3

_

fl,a(tOn'(ul))f2,-a(t°n'(ul),w), aEk3

where

©y, (P, t°n (ul)n(uo)) < a uD > duo,

.f1,a(t°n'(ul )) = f A/k)3

f2,a(t°n'(ul), w) = J

e(t°n'(ul)n(uo), w) <

u> duo.

A/k)3

Proposition (10.4.2) There exists a constant 6 > 0 such that for any N > 0, M > wo,

E I fl,a(ton

(ul))f2,-a(ton (ul), w)I <
aEk3\(0)

for w0-6
By definition, n (ul)n(uo)y = (y1, x2,23 + x2,13U1 + 2x2,22u3, x2,33 + x2,22u3 + X2,13U2 + x1,33u4)

Let

2

u2 = x2,13u2 + x2,22u3 + x1,33u4, ui3 = x2,13u1 + 2x2,22u3,

ui¢ = u¢.

Then

u2 = x 2,132 u 1

u3 = 2 i

U4 = u¢.

4-1x-1 x-1 u' 2+ 2-1x-1 ul u 4-1x-1 x-1 xX 33u4, 2,22x2 13 u21 - X2,13 32,22 2,13X2,22 3

-1 / 2 22x2,101, x2,22u3 - 2-1X-1

Part IV The quartic case

236

Clearly, n'(ul)n(uao)y does not depend// on u/4. Therefore,

fl,a(t°n (ul)) _

J W(t°n'(ul)n(no)yl)du2du3. yiEZ'k

A2

Let V), (a, ul, u') = < aix2,13uI + 2-1a2x2,22u3 > x < ai(-4-1x2,13x2,22u32 + 2-1x2,22uiu'3) >

X < (a3 - alx 2,l3X1,33)u4 >, 02(a, u1) = < -4-1aix2,-122x2,13u1 - a2x2,22x2,13u1 -1 >, 1x-1 x-1 u' 2+ 2 2,22 3 u+ al( -4 03( a ,u 1, u'3) = < 2,13 2,22 3 1

2-lax-1

2-1x-1

2,22

u u3) 1

>

.

Then fi,a(t°n'(ui)) is equal to T(61, 72,23(t°)u3, 72,33(t°)u2) 1(a, u1, u)du2du' ,b2(a, ul) y1EZak

AZ

The above integral is zero unless a3 - a1x2,13x1,33 = 0, because n'(ui)n(u,)y does not depend on u4. Let W1 be the partial Fourier transform of T with respect to the coordinate x2,33

and < >. Then u, u)du' du3

fA2 W (t°y1,72,23(t°)u,

=

We choose 0 < T2 E .'(Z,A x A) so that IA Since I02(a,u1)I =

I03(a,u1,u3)I = 1,

Ifl,a(t°n (ui)) «i02(t°)

`W2(t°y1,0'1x2,1372,33(t°) Y1 EZOk

s2,13=-1 a3 T1,33

If the condition x2,13 = alai 1x1,33 is satisfied, ai $ 0 implies a3 # 0. So

1f1,a(t°)I <
Y'EZDk

o,Ekx

This bound does not depend on a2. By (10.4.1), for any N > 0,

E Q1 a3Ek'

Ifl,a(t°)I << rd3,N(Al, A2, A3)-

10 The non-constant terms associated with unstable strata

237

There exist a slowly increasing function h(t°) and a constant 6 > 0 such that if

M>w0,

I f2,a(t°, w)I << h(t°) Ekx .2,3Ek

for w0 - 6 < Re(w) < M. Therefore, for any N > 0,

E 1f1,a(t°)f2,-a(t°, w)I << rd3,N(Al,'2, A3)ulEkx

°2,3Ek

Next, we consider terms such that al = a3 = 0, a2 # 0. Let 'F3(yl, X2,23) _

T (yl, x223, x2,33) < x2,23x223 > dx2,23dx2,33

Then fl,a(t°n'(ul)) is equal to K02(0) E 'F3(t°y1, 2-1a2x2,2272,23(t°)-1) < -2-1a2x2,22xz,13ul > yiEZnk

By (10.4.1), for any N > 0,

E I fl,a(t°n (ul))I <
2Ekx

al=°3=O

The rest of the argument is similar. This proves (10.4.2). Q.E.D.

The following proposition follows from (10.4.2).

Proposition (10.4.3) Ep (1), w, w) - 0. §10.5 The case 0 = (/3s)

In this section, we consider a path p = (D, s) such that a = (/38).

Proposition (10.5.1) cp(4P,w,w) - 0. We devote the rest of this section to the proof of (10.5.1). For A1, 91, µ2 E Rte, we define (10.5.2)

d(A1, µl, µ2) = a(A1 2, 1u1, lµl

µ2, µ2

We write ga = kd(Ai, µi, µ2)t °n"(u3) where k E K fl MOA. As in §3.4, when we estimate various functions, we assume that k = 1. Let to = d(A1, µl, µ2)t °. Let n(ua) = (np2 (u1, u2), nB2 (u4)) for ua = (ul, u2, u4) E A3. For N = (Nl, N2i N3), we define (10.5.3)

h1,N(9a) =

71,22(t°)I-N"I'i2,13(t°)I-N2I72,23(t°)IN3,

h2,N(9a) = 71,22(t°)I-Nl I72,13(t°)I-N2I72,33(to)IN3,

h3,N(9a) = h4,N(9a) =

71,23(t°)I-Nl 172,12(t°)I-N2I72,23(to)IN3,

I71,23(t°)I-"I72,12(t°)I-N2I72,33(t°)IN3

Part IV The quartic case

238

Then

h1,N(ga) = A (2N,-N2-2N3)A1 -(2N1-N2)P2-(Ni-N2+N3) 1

h2,N(ga) = Al (2Ni-N2-2N3)µ1 -(2N1-N2+2N3)µ -(Nl-N2+N3) 2 (2N1-N2-2N3)µl-N2 -(Nl-N2+N3) h3,N(9a) _ Al N2 (2N1-N2-2N3)µ= (N2+2N3)µ2 (Nl-N2+N3)

h4,N(9a) = Al

Consider the condition (10.5.4)= For any N = (Nl, N27 N3) satisfying N1i N2, N3 > 1, If (9a)I << hi,N(ga). Lemma (10.5.5) Suppose that f (go) is a junction on G°Af1MMA/Mek which satisfies

Condition (10.5.4)= for some 1 < i < 4. Then for any N > 0, If (9a)I << pi Nrd62, N(Al, p2)

for go in the Siegel domain. Proof. Since Iy2,23(t°)I = P I72,33(t°)I >> 1%,33(0°)I, we only have to consider i = 1, 3.

Suppose that (10.5.4)1 is satisfied. Then we choose N1, N2, N3 in the following manner:

N2,N3fixedand Nl>>0,>>0

ifal>1,µ2>1,

N3 fixed and N1 = 2N2 >> 0 N3 >> Nl » N2

if Al > 1, µ2 < 1,

N1=N2,N3=4N2>>0 3

ifAl<1,µ2 _ 1-

if Al < 1, µ2 1 1,

Suppose that (10.5.4)3 is satisfied. Then we choose N1, N2, N3 in the following manner:

N3 fixed and Nl »N2>>0 N3 fixed and Nl=2N2>>0 Nl fixed and N3 » N2 Nl fixed and N3 = 33N2

ifal> 1,µ2> 1, if Al>1,µ2<1, if Al < 1, µ2 > 1, if )1 < 1, µ2 < 1-

Then we have the estimate of the lemma. Q.E.D.

Let T = Reap. As usual, we can write any element y E Yak' in the form y = (y1, y2) where yl = (x1,22, x1,23) x1,331 X2,12, x2,13) E Zak and Y2 = (x2,22, x2,23, X2,33) E Wok

For a = (al, a2i a3) E k3, we define < a ua >=< alul + a2u2 + a3u4 >. Let (10.5.6)

fl,.(ga) = f

DY, (W,9an(ua)) < a ua > due,

A/k)3

f2,«(9a, w) = fA/k)3

(gt n(ue ), w) < a u-0 > dua.

(

The proof of the following lemma is easy, and is left to the reader.

10 The non-constant terms associated with unstable strata

239

Lemma (10.5.7) Let P(y1) = x2,12x1,33 - x2,13x2,12x1,23 +x2,13x1,22 Then P(y1) is a relative invariant polynomial under the action of M.D. Let

A(y1) =

2x2,12

0

x2,22

x2,13 0

x2,12 2x2,13

x1,23

u'1 ,

x1,33

u2

u2

= A(y1)

( u2 )

.

u4

u4

Then n(ua)y = (yl, x2,22 + ui, x2,23 + u2, x2,33 + ui)

Therefore,

fl,.(9a) = E f yiEZak

iP(9an(ua)y1) < a ue > due.

3

A

Let (ai, az, as) = (a1, a2, a3)A(y1)-1 (det A(y1) = P(y1)

0). Then

< a1'u'1 + a2 u2 + a3u4 >=< alu1 + a2u2 + a3u4 > .

It is easy to see that 9an(ua )y1 = (9ay1, ui, u2 + 2u1u1', u4 + uiui + u1u2).

For u' = (u'1, u2, u4), we define

'I"(9a, y1, u) _ IF(9ay1, ui, u2 + 2uiu'1, u4 + uiui + u1u2). Also we define 11 U3 = u2 + 2u1u1)

l

2

'u4 = u4 + u1u1 + u1u2.

Then

u2 = u3 - 2u3ui, ) = u"4+ u32u'1 u 4= u"4 - u2u 32u3u13 1 - u3(u"

-

- u3u'' 3

Therefore,

< a' u >=< (a' - 2a2u3 + au3)ui + (a2 - a3u3)u2 + a3u'4 > . Let

V)a' (ul, u2) =< (a' - 2a2u3 +a3, u3)ui +(a2 - a3u3)u2 > .

Let If1 be the partial Fourier transform of W with respect to the coordinate x2,33

and < >. Then

J(9n(u)y1) < a ue > due yiEZ3

= 112,33(t°)1'

yiEZa k JA2

1(9ay1, u1u3, a3-Y2,33(t0)(ui, u2)duidu2.

Part IV The quartic case

240

We choose 0 < 4f2 E 9(ZZA x A) so that

f

I'F1(Y1, ul, u3, x2,23)Idu' du" < qf2(y1, x2,23)-

2

A

T hen If1,o (9a)I << K

(go) E 'P2(9ay1, a3

Y2,33(t0)-1),

yi EZak

By (1.2.8), (10.5.5), for any N > 0, Ifi,a(9a)I << Pi Nrd2,N(Al, µ2) a3Ekx

for g, in the Siegel domain. This bound does not depend on a', a2. Therefore, by a similar argument as before, there exists a constant 6 > 0 such that if M > wo, (10.5.8)

If1,a(9a)f2,-a(9a,w)I << P1 Nrd2,N(Al, P2) «i.n2 Ek a3 EkX

for go in the Siegel domain and wo - 6 < Re(w) < M. Next, we consider a such that a3 = 0. Let P3(y1) x2,22, x2,23) _ JA2 'Y(yl, x2,22, x2,23, x2,33) < x2,23x2,23 > dx2,23dx2 33

We choose 0 < q14 E 9'(ZOA x A) so that 1.

IW3(y1,x2,22,x2,23)Idx2,22 < T4(y1,X2,23)

Then If1,a(ge)I «!a2(9a2) -o °3n3EkX

W4(9ay1,a272,23(t0)-1),

-3=o y1EZok n2 EkX

By (1.2.8), (10.5.5), for any N > 0, Ifi,a(go)I < P1 Nrd2,N(Al, P2)

3- 0 o3EkX

for go in the Siegel domain.

Therefore, there exists a constant 6 > 0 such that if

M>w0, (10.5.9)

E

a3=0 a2Ekx ai Ek

I fl,a(9a)f2,-a(ga, w)I «N'1 Nrd2,N(A,, /22)

10 The non-constant terms associated with unstable strata

241

for go in the Siegel domain and w° - 6 < Re(w) < M. Finally, we consider a' such that a'2 = a'3 = 0. Let 'y5(y1, x2,22) =

Jf3

"' (yi, x2,22, x2,23x2,33) < x2,22x2,22 > dx2 22dx2 23dx2 33

We choose 0 < is E Y(ZZA x A) so that I'y5(yi,x2,22)I <_ 'P6(y1,x2,22) Then If1,0(9a)I <<'

'I'6(g0y1,'Y2,22(t°)-1a11)

(90) yiEZn%

ai Ekx

Clearly, a = (c4, 0, 0)A(yl) = (2x2,12ai, 0, x1,22ai). If Y1 E Za8k, x2,12 # 0 or x1,22 0 0.

Suppose a1 i4 0. Then If2,0(90, w)I

Z3

1(9,,W) I,T

.i Ekx

°1#0

where I = (11i 12) and I1 3 2. So by (3.4.30), there exists a constant 6 > 0 and a slowly increasing function h(t°) such that if 1 >> 0, M > wo, If2,n(9a,w)I << h(t°)(Xiµl)l.

(10.5.10) 2alEkx °1*0

Consider the following conditions (10.5.11)1 For any N1, N2 > 1,1 >> 0,

f(9a)

«I71,22(t°)I

N1I72,13(t°)I-V2(A13ul)`;

(10.5.11)2 For any N1, N2 > 1,1 >> 0, f(90) <<

Iy1,23(t°)I-N1I72,12(t°)I-N2('A1p1)t.

Lemma (10.5.12) Suppose that f (go) is a function on G°, fl MJA/Mak which satisfies either (10.5.11)1 or (10.5.11)2. Then for any N > 0, If(9d)I<
for g, in the Siegel domain. Proof. If f (go) satisfies (10.5.11)1i we choose

N1>> N2>>l>>0

Al>1,1L2?1,

N1 = 2N2 »0, lfixed

Al

1,µ2

1,

N1 = 2 N2, 1 = N2, N2 >> 0

A,

1, µ2

1,

N1=3N2, 1= 1N2, N2>>0 Al<1,µ2<_1.

Part IV The quartic case

242

If f (go) satisfie s (10.5.11)2, we choose

>1,µ2?1,

Nl»N2>>l>>0

Al

Nl=4N2>>0, (fixed

Al < 1, µ2 > 1,

Z N1=N2,l=2N2,N2»0 Al >1,µ2<1,

N1=3N2,l=4N2,N2>>0 Al <1,µ2<1. Then we have the estimate of the lemma. Q.E.D.

By (10.5.10), (10.5.12), for any N > 0, (10.5.13)

If1,.(ga)f2,-.(ga, w)I << µ1 Nrd2,N(A,, µ2) #2 - 3 ai Ekx

n1#0

for go in the Siegel domain.

Suppose that al = 0, a3 0 0. If a2 = a3 = 0, a2 = 0. So If2,.(go,w)I S 1: I°I,,,1(go,W) .1_°2_°3=0

I,T

a1 Ekx '300

where I = (11,12) and 12 = {1}, I1

1. Consider 8j,1.,1(go, w) for such I. Let 113 = I72,13(t°)I,µ4 = 172,22(0°)I. Then 03 = Al lµ1 1µ2 1, µ4 = A2 A2'U 1 and

3 3 A1 = µl lµ3 3 µ4 , A2 = µ3 µ4 . Let

(10.5.14)

d(µ1,µ3,µ4)=(µ13µ43,p3A3,p 2A33A43;

aA3gµ3

Then to = d(µ1,µ3, µ4)t °. By (1.2.6), for any Ni, N2, N3 > 1, (10.5.15)

I'W6(gay1,72,22(t°)-lai)I «µ3N1-N2µ4N` v1 E ZDk

-1,22.x2 13Ek%

Ekx

Lemma (10.5.16) Suppose that I1 ; 1, 12 = {1}. Then there exist a constant b > 0 and a slowly increasing function h(µ3, µ4) such that if M > wo, l > 1, ,fi,r,l(ga, w) « µl 5h(µ3, µ4)(µ33 µ43 )'

for go in the Siegel domain and w° - 6 < Re(w) < M. Proof. Let q E DI,,, rl, r2i r3 be as in (9.2.5). There exists a slowly increasing function hr,r(P3,µ4) for each T such that Al(rl+r2)hr,, (µ3)

4

T , 1(ga , w)

P4)

µ1r2hT,r(µ3, µ4) 2(rl+r2)hr,r(µ3,

2

<< µ i (µ3 µ4 ) x

µl

1

µlr1 hr,r(µ3, µ4)

µl

I µu

2,1

h,.,, (p3, A4) 1r

r (µ3, µ4)

T1 = 1,

rl = (1, 2), µ4)

Ti = (1, 3), Ti = (2, 3), Ti = (1, 2, Ti = (1, 3, 2).

10 The non-constant terms associated with unstable strata

243

Note that if II = {2}, we have an extra 11 21' factor for some 1' << 0, but since µl is bounded below, for ga in the Siegel domain, we can ignore this factor. We choose r3 = If Ti = 1, we choose rl, r2 << 0. If -r1 = (1, 2), we fix r1 and choose r2 << 0. If Ti = (2, 3), we fix r2 and choose r1 << 0. If T1 = (1, 2, 3), 1.

we choose r1 = 1 - r2 >> 0. If TI = (1, 3, 2), we choose r2 = 1 - r1 >> 0. Then L(q) < w°i and the estimate of the lemma is satisfied. If Tl = (1, 3), we can choose

r1=r2=2. Then L(q)=8+2C<4+C since C> 100. Q.E.D.

It is easy to see that (t°)-2P = µi sµ3 31.t4 3 and K02(90) = µ6µ43. Therefore, by (10.5.15), (10.5.16), for any N > 0,

E

(10.5.17) (t°)-2'lc02(90)

W)

v1 EZab

s1,22,x2,13Ek5

-i Ekx

<< µl 8rd2,N(µ3, µ4)

Proposition (10.5.1) now follows from (10.5.8), (10.5.9), (10.5.13), (10.5.17).

§10.6 The cases Z = (38, 08,2), (09) Let 01 = (08,88,2), a2 = (09), and 03 = (08). Consider paths pi = (0i,5i)42 = (Z)2, Z2), and p3 = (03,63) such that sl(1) = 52(1) = 53(1). This means that p3 -< pl. Let (10.6.1)

d(.\i, Al, µ2) = a(A14p1, iµ2 Aiµl 1 µ2

3µl 1 ,

A3

and to = d(A1, µl, µ2)t °. For u2, u3, u4 E A, a = (al, a2, a3) E k3, Let n(u , ), <

a- u8>beasin§10.4.

We first consider pl such that sl(2) = 1. It is easy to see that ep11(t°) _ 2 (IC132(t°)(t°)-2Pa3)-1(to)-2p°1 (to)-Rs = A-1 and Clearly, 43 (t° w) = As µ12µ2 is the constant term with respect to np2(u2i u2).

Let %F = R,1 Pp1. We consider

IA/k

Oy,1 (W, ton "(u3))3 (t°n'(u3), w)du3.

For a E k, we define (10.6.2)

f1,a(to) =

f

Oya1 (W,t°n'(u3)) < au3 > du3, /kf

f2,a(t°, w) = J /k

-ep3

(toni(u3), w)

< au3 > du3.

Lemma (10.6.3) Let f (t°) be a function on TA/Tk. Suppose that for any N1, N2, N3 > 1, f(to) << Iy1,23(to)I-N1I72,13(t°)I-N2I-Y1,33(to)IN3

Part IV The quartic case

244

Then for any N > 0, f (to) «.Al Nrd2,N(µ1, p2). Proof. Easy computations show that I71,23(to) I-N, I72,13(to) I-N2 I71,33(to) I Na (N1+N2-N3)ll-l1 (-2N1+N2+3N3) lt2-(-N2+2N3)

- Al

We just have to choose N1, N2, N3 in the following manner:

N1=N2=N3»0

Al _ 1,/-t2

N1iN3fixed and N2>>0

hl>1,tt2<1,

1,

N2 fixed and Nl>>N3>>0 /L1<1,lt2>1,

N3 fixed and Nl»N2>>0 pl<1,µ2<1. Q.E.D.

The following proposition is an easy consequence of the above lemma.

Proposition (10.6.4) Suppose sl(2) = 1. Then pl (-D, W, w) =

fa

2/Tk

wp1(to) E A8111 2112 2f1,a(to)f2,-a(t°, w)dxto - 0. aEk x

,>1

Next, we consider P2 and pl such that sl(2) = 0. The proof of the following lemma is easy, and is left to the reader.

Lemma (10.6.5) The convex hull of each of the sets (1) (111,23, 72,22, 72,13, -72,33}l, (2) 171,23, 12,22, 72,13, -72,23}

contains a neighborhood of the origin.

Proposition (10.6.6) Epl

pl("),W,w)+Ep2p2((1),W,w) - 0.

We devote the rest of this section to the proof of the above proposition. Let W = Ra2Dp2. Any element y E Ya.k can be written in the form y = (yi, y2) where yl = (x1,23, x2,13, x2,22) E Za2k and y2 = (x1,33, x2,23) x2,33) E Walk Then n (ul)n(ua)y = (y1) x1,33 + u3, x2,13 + u4 + x2,13u1, x2,33 + u2),

where 7lr3 = x1,23u3,

u4 = 2x2,22u3 + x1,23u4, /

2

/

u2 = x2,13u2 +x2,22 U3 + u3u4.

Let

(10.6.7) f3,a(ton (u1)) =

f IF (ton'(ul)n(ua)(yl, 0)) < a ua > duo; 3

Y1EZa2k

A

I(ton (ul)n(u), w) <

f4,a(t°n (ul), w) = A/k)3

u> due.

10 The non-constant terms associated with unstable strata

245

An easy computation shows that

< a ua >= < alx1 u 2,13 - a l(x2,13x1,23x2,22u32 - x2,13u3(u4 - 2x1 23x2,22u3)) > 2 X < a2x1,23u3 + a3(x1,23u4 - 2x-1,23x2,22u3) >

Let Y 01(yl, u3, u4) = < -a, (x2,13x1,23x2,222132 - x2,13u3(1b4 - 2x1 23x2,22u3)) > /1

X < a2x1,23u3 + a3(x1,23u4 - 2x1 23x2,22"3) > .

We also define (to) yl, ul, u2, u3, u4)

= W (t°n'(ul )yl, -Y1,33(t°)u3, -Y2,23(t°)(u4 + x2,13"1),'Y2,33(t°)u2)

Then f3,a(t°n'(ul)) is equal to I WY'(t°, yl, u1, u'2) u'3, u4) < a1x2-,1 13u2 > W ,,/'a (y1l u'3+ u4 )du'. 3

yi E Za2 k

Let WW1 be the partial Fourier transform of IF with respect to the coordinate X2,33 and < >. We choose 0 < W2 E 9(Z-02A x A) so that

fA2

I ''1(yl, x1,33, x2,23, x2,33)Idxl,33dx2,23 < 'P2(y1, x2,33)

Then

If1,a(t°n (ul))I <<' 22(t°)

'1'2(t°y1,72,33(t°)-161x2,13) Y iEZazk

This bound does not depend on a2i a3. By (10.6.6), for any N > 0, E I f3,a(t°n'(u1))I <
Therefore, there exists a constant 8 > 0 such that if M > wo, (10.6.8)

I f3,a(t°1t (ul))f4,-«(t°n'(ul), w)I <
-2,n3Ek

for wo - 6 < Re(w) < M. Next, we consider a such that a1 = 0, (a2i a3) # 0. Let W3(yl, x1,33, x2,23) = 1A2

(yl, x1,33, x,23, x,33) < x2,23x,23 > dx23dX,33.

We choose 0 < qt4 E So(ZOA x A) so that IA I W3(yl,x1,33,x2,23)Idx1,33

'4(Y1,x2,23)

Part IV The quartic case

246

Then If3,a(t°n (ul))I << 6;022(t°) E 'W4(t°y1,"Y2,23(t°)-1a3x1,23). YlEZa2k

By (10.6.5), for any N > 0, I I f3,a(t°n (ul))I <
Therefore, there exists a constant 6 > 0 such that if M > wo, (10.6.9)

E I f3,a(t°n (ul))f4,-a(t°n'(ul), w)I << rd3,N(.\1, µl, /22) a3Ekx 1=0 2Ek

for wo - 6 < Re(w) < M. Finally, we consider a such that al = a3 = 0, a2 E k". Let `W5 be the partial Fourier transform of T3 with respect to the coordinate x1,33 and < >. Then W5 (t°y1,y1,33(t°)-1a2x1,23).

f3,a(t°n (u1)) = r-022(t°) y 1 EZ12 k

Thus f3,a(t°n'(ul)) does not depend on ul any more. Therefore,

E(t°n'(ul))f4,-(t°n (u1),J

/k '1=°3=0 a2EkX

f3,a(t°) LIk f4,-(t°n'(ul), w)dul, 2Ekx

and

(10.6.10)

J /k f4,-a(t°n (ul ), w)du1 = f2,-a2 (t°, w).

Let W6 be the partial Fourier transform of q/5 with respect to the coordinate x2,22 and < >. Let W7i `W8 be restrictions of %F5, `1'6 to the first three coordinates. Let f1(t0, x1,23, x2,13, x2,22) a2)

('Y1,23(t°)x1,23, Y2,13(t°)x2,13,-Y2,22(t°)-1x2,22, Y1,33(t°)-1a2X1,23)

We define 'f'5(A(t°,x1,23, x2,13, X2,22, a2)),

f5,-2(to) = lc,22(t°)I-Y2,22(t°)I-1 Y1EZ12k

f6,a2(t°) =

r-122(t°)I72,22(t°)I-l

'F8(A(t°, x1,23, x2,13, 0, a2)), X1 ,23,52,13 Ek x

f7,a2 (t°) = ic,22 (t°)

WY7(A(t°, x1,23) X2,13, 0, a2)) 51,23,x2,13Ekx

10 The non-constant terms associated with unstable strata

247

Then f3,(o,a2,o)(t°) = f5,a2(t°) + f6,a2(t°) - f7,a2(t°). We choose Schwartz-Bruhat functions 0 < W9i Wio on A4, A3 so that IT5I <-'I9, IT6I, I''7I < W10 Then

E If5,a2(t°)I, T, If6,a2(t°)k E If7,a2(t°)I a2Ekx

012Ekx

a2Ekx

are bounded by /ca22(t°)Iy2,22(t°)I-1'J4('y9, -Y1,23(t°),'Y2,13(t°), Y2,22(t°)-', Y1,33(t°)-1), '522 (t°) I-Y2,22(t°) I-1''3('y10,'Y1,23(t°),'Y2,13(t°), 'Y1,33(t°)-1)' ICD22

(t°)' 3('y10, 'Y1,23(t°),'Y2,13(t°),

-Y1,33(t°)-1)

respectively.

Proposition (10.6.11) (1) Suppose that f (t°) is a function such that for any N1, N2, N3, N4 > 1, f(t°) «I"Y1,23(t°)I-N1I72,22(t°)I-N2I-Y2,13(t°)I-N3I-Y1,33(t°)IN4

Then for any N > 0, If (t°)I << Al Nrd2,N(µl, µ2) for A1 > 1. (2) Suppose that f (t°) is a function such that for any N1, N2, N3, N4 > 1,

f(t°) <<

I"Y1,23(t°)I-'

I'Y2,22(to)IN2I-Y2,13(t°)I-N3I.Y1,33(t°)IN4

Then for any N > 0, If (t°)I << ANrd2,N(µl, µ2) for Al < 1.

Proof. Easy computations show that I"Y1,23(t°) I -N, I-Y2,22 (t°) I-N2 I'Y2,13(t°) I-N3 I-Y1,33(t°) IN4 (N1+7N2+N3-N4) N'-1 (-2N1+N2+N3+3N4) -(2N1 -N3+2N4)

Al

N'2 I'Y1,23(t°)I-N,

1-Y2,22(t°)IN2

(-Y2,13(t°)I-N3I`Y1,33(t0)IN4

Al (N1-7N2+N3-N4) N-1 (-2N1-N2+N3+3N4) /22-(2N1-N3+2N4)

We choose N1i

, N4 in the following manner.

Nl, N3 fixed, and N2>>N4>>0 µ1>1,µ2>1, N1, N2, N4 fixed, and N3 >> 0

Al > 1, p2 < 1,

N3iN4fixed, and N1>>N2>>0 µk1,/2>1, N2,N4fixed,and N1>>N3>>0 µ1<1,µ2<1. This proves (1).

We choose N1,

, N4 in the following manner.

N1iN4fixed, and N3=2N2»0 µb1,µ2>1, Nl, N2i N4 fixed, and N3 >> 0

Al > 1, µ2 < 1,

N3,N4fixed,and N2>>N1>>0 µk1,µ2>1, N1iN4fixed, and N2>>N3>>0

µ1<1,/2<1.

Part IV The quartic case

248

This proves (2). Q.E.D.

By (10.6.11)(1), (10.6.12)

Wp2 (t°) E f3,(O,a2,0) (t°)f2,-a2 (t°,

w)(t°)-2Pdxto

a2Ekx

- 0,

and by (10.6.11)(2), f5,a2(t°)f2,-a2(t°,

wp2(t°)

(10.6.13) JT'O /

Tk

A1<1

w)(t°)-2pdxt°

- 0.

a2Ekx

The following lemma is easy to prove, and we omit the proof.

Lemma (10.6.14) The convex hull of each of the sets (1) {(1, -2, 0), (1, 1, -1), (-1, 3, 2), (-4,1, 0)}, (2) {(1, -2, 0), (1, 1, -1), (-1, 3, 2), (-2,1,1)} contains a neighborhood of the origin.

Now we want to cancel out °p1 (4i, w, w) with "p2 (-D, w, w). However, we are still obliged to prove that 8pl (,D, w, w) is well defined for Re(w) >> 0. Consider el,,,l(t°,w) such that I = (1,,12) and 12 = 0, 1, = {2}.

Lemma (10.6.15) (1) If Tl = (2,3) or (1,2,3), there exists a constant 6 > 0 and a slowly increasing function h(t°) such that for any N >> 0, M > wo, ,rj,r,l(t°, w) << h(t°)(Ai/-L1 1)N

for w° - 6 < Re(w) < M. (2) If r1 = (1, 3), there exists a slowly increasing function h(t°) and a constant c > 0 such that if M2 > Ml > wo,

ij,r,l (t°, w) << h(t°)

(A1,,1 l µ2 1)cRe(w)

for Ml < Re(w) < M2. Proof. Let q E DI,,. Let r = (r1, r2, r3) be as in (9.2.5). By (3.4.30), there exists a constant 6 > 0 and a slowly increasing function h(t°) such that if M > w°i

1 4r1 -2r2 Alrlµ'2-r2 u I,r,l(t°, w) << 1,h( t°) x

-(r1+r2) Al2r1-2r2-rl µ1 µ2

12rl+r2 Al-(rl+r2)µ2-r1 for L(q) + 6 < Re(w) < M where 6 > 0, M > w° are constants. If Ti = (2, 3), we fix r2 and choose rl << 0. If Tl = (1, 2, 3), we choose rl = 1 - r2 >> 0. Then we get an estimate of the form (1). If T1 = (2,3), we fix r2 and choose rl >> 0. Then L(q) depends linearly on rl. Therefore, we get an estimate of the form (2). Q.E.D.

10 The non-constant terms associated with unstable strata

249

Since ep11(t°) = (t°)a8 = al, by the Parseval formula,

jpl (4), w, w) _

wp1 (t°) E f6,a2 (t°)f2,-a2 (t°, w)(t°)-2Pdx t°.

fA/Tk

a2EkX

a1 <1

By the consideration as in §2.1,

E

(t°, w),

If2,a2(t°,w)I << T

a2EkX

where I1 = {2},12 = 0. Therefore, the following proposition follows from (10.6.14), (10.6.15).

Lemma (10.6.16) The distribution 22p, (,D, w, w) is well defined if Re(w) >> 0. Therefore, by (10.6.10), (10.6.16), Ep1

w, w)

Pi (ID, w, w) + 1P2P2

Wp1 (t°) E f7,a2(t°)f2,-a2(t°,w)(t°)-2PdxtO.

N EP,

1T0ITk A1<1

It is easy to see that

a2EkX (t°)-2p

= A18Al 21'22 and rD22(t°) = A1 15µiµ2. Therefore,

for any N1, N2, N3 > 1, 2)-N1

If7,a2 (to) l << A3

X131'1 2A2 2

011'21'2 1)-N2

a2EkX

We have already estimated E I f7,a2 (t°) I (°I,T,I (t°, w) a2Ekx

for T1 # (1, 3), so we assume that T1 = (1, 3). By (9.2.10), for any E > 0, there exists a constant 6 > 0 such that if M > wo, 4I,T,1(t°, w) << ai+011'121'23

for w° - 6 < Re(w) < M where IC11, Ic21, IC31 < E. Therefore, (t°)-2P

(10.6.17)

«

1f7,112(t°)1'k1,1,1(t°,w) a2EkX A5+c1tt2+C2A2+C3

(A,t-tl 2)-N1(A1/-tll-t2 1)-N2 (A1 1tt3tt2)-N3.

We fix N1, N2 and choose N3 >> 0. Then there exists a slowly increasing function

h2(t°) such that for any N > 0, (10.6.17) is bounded by a constant multiple of h2(t°)(A11'1 31'22)"'. (10.6.17) is also bounded by a constant multiple of 2+C1 1'1

$+C3

1'2

3+C1

,1

1+02 -1+C3

1'1

1'2

, 12 +C1 1'12 +C3 1'22 +C3 )

by choosing (Ni, N2, N3) = (1, 3,1), (3,1, 2), (3, 2,1). Therefore,

wp2(t°) E f7,a2(t°)f2,-a2(t°)(t°)-2Pdxt° " 0. f""-ITk X1<1

a2EkX

This completes the proof of (10.6.6). Hence, we finished the proof of Theorem (10.0.1), (10.0.2). This implies that we can ignore the non-constant terms associated

with unstable strata.

Chapter 11 Unstable distributions In the next three chapters, we consider distributions ^ p (4i, w, w) and compute the principal part of the zeta function for the quartic case.

§11.1 Unstable distributions In this section, we define certain distributions associated with unstable strata. We remind the reader that we use the notation in previous chapters like n'(u), n"(u) in (9.2.3) or F' in (9.2.4). Definition (11.1.1) For a character w = (wl, w2) of (Ax /kx )2, we define (1) S#(w) = 6(wl)6(w2), (2) 6-0 (W) = s(w2), wa = (1,w1) fort = (i1), (3) sa(w) = s(wl), wa = (1,w2) ford = (04), (4) Sa (w) = S/(wi), wa = (wl, w2, w2) for (l = (or,), (5) Sa(w) = Slwi)S(w2), wa = (W2,w1,w1 lW2)fort = (37 ), (6) Sa(w) = 6(w1), wa = (W1,W2,W2) ford = (08), (7) 58(w) = S(w1')6(w2), wa = (1,w1 1, w1) for cl = (Qs),

(8) Sa(w) = S(wl),S8,st(w) = b(wl)S(w2), wa = (1,w2),wa,.t = (W2,W2) for Z _ (N1o)

Definition (11.1.2) We define subsets 01i 02 of permutations of {1, 2, 3} as follows. (1) 111 = {1, (2, 3), (1, 2, 3)}.

(2) 02 = {1, (1, 2), (1, 3, 2)}.

Let 0 be a (3-sequence for which Pa# = Pl x B2 or P2 x B2. The double coset

Pa#k \ Gk/Bk is represented by 01 x {1, (1, 2)} if Pa# = Pl x B2, and by 02 X {1, (1, 2)} if Po# = P2 x B2. The reductive part Ma of Pa# is of the form Ma = Ma1 X Mat and Mal = Mall X M812 where Mall = GL(2), Ma12 = GL(1) if

Pa#= P1xB2andMoll 'GL(1), Ma12=GL(2)ifPa#= P2xB2. By a standard argument, PlkrlBlk/Blk, P2kTlBlk/Blk are represented by the sets {7172 I yl E Ma11k/Mallk n B1k, Y2 E N 8}, {''1''2 -'1 E Ma12k/Ma12k n Blk, 72 E N,k}, I

respectively. Let dp (al) be as in (9.2.4), and di(\1i A2) = (dpi (.Al), a2(.\2, A21)) for i = 1, 2.

Let Aa = dl(A1, A2) if Pa# = Pl x B2, and as = d2(.\1, X2) if Pa# = P2 x B2. Any element ga can be written in the form ga = gaga where go E MDA. The element go is of the form (11.1.3)

t13) (t21

t22)) or

((tii

91,23) (t21

t22))

I

where 81,12, 81,23 E GL(2)00 and 1431 = It21I = It22I = Itiil = 1. The following lemma can be obtained by a standard argument and the proof is left to the reader.

11 Unstable distributions

251

Lemma (11.1.4) In the following two statements, Tr = (r,, r2).

(1) IfPo#= P1xB2, EB(gsn(us),z)dus fUs#A/UD#k MT(z)A/DZ+//EMa11nB1(91,12)z1T1(1),z1T1(2)). 'i E$11

,2=1,(1,2)

(2) If Po# = P2 x B2,

IUD#A/U,#k

=

E

EB(9an(u,),z)dua TZ+P

MT(z)A/D

EMc12nB1 (91,23, z1T1(2)) Z1T1(3))

'1E202

,2=1,(1,2)

The following lemma is a consequence of the Mellin inversion formula.

Lemma (11.1.5) Suppose that f (s) is a holomorphic function which is rapidly decreasing on any vertical strip contained in the domain Is Re(s) > 0}. Let I

E(g°, s) be the Eisenstein series on GL(2)00, and w a character of Ax /kx. Then 1

w(det g°) f (s)E(g°, s)dg°ds = 6(w) f (1). 1

GL(2)A/ GL(2)k

Refs)=r>0

Let W = Sym2k2, i.e. the space of binary quadratic forms. For IF E so(WA), w = (w1i W2) a character of (lax /k>1)2, and s E C, let Ew(W, w, s), Ew+('1', w, S), E W,ad(T ,W, s)

be the functions which correspond to ZV2 (',W, S), ZV2 (T, W, S), ZV2,ad (9', W, S) in

C W be the subspaces which correspond to Zo, ZO, in §4.2 respectively. We define 4,0k etc. in the same manner as in §4.2. Let Rw,O, R/w,0 be the operators which correspond to RO, Ro in §4.2. Let Hyl. be the group which corresponds to H in §4.2. Let XW C Al x GL(2)00 be the subset which corresponds to Xv in §4.2. Let TW(Rw,OiW, w, s, sl) etc. be the adjusting term etc. We use the same function a(u) as in §2.2. Let g E A x GL(2)11 = Rh x Al x GL(2)0A. For IF E .Y(WA), we define §4.2. Let Zw,o,

(11.1.6)

xEZ'IW,Ok

Gw,st('I', 9) _ 5 '1'(9x) xEZW,stk

Then (11.1.7)

Ow,st('1', 9) =

GZw,o (`F, N. 7EGklHw,k

Part IV The quartic case

252

Next, we consider distributions associated with unstable strata. (a) 'a = (,31). We can identify ZD with Sym2k3, i.e. the space of ternary quadratic forms. For T E 5°(ZDA), w = (W1i W2) a character of (Ax /kx )2, and s E C, let ED (IF, w, s), E'+ (q" w, s)

be the functions which correspond to Z,,3 (W, w, s), Zv,+(WY, w, s) in Chapter 4.

(b)D=(,34)

We can identify ZD with M(2, 2), i.e. the space of 2x2 matrices. For IF E 9(ZDA), w = (wl, w2) a character of (Ax /k x )2, and s E C, let ED (W, w, s) be the integral (3.1.8) for L = ZaZ. We discussed this case in §3.8. (c) D = 06), (D8)-

We can identify ZD with Sym2k2 ® k or Sym2k2 We considered these representations in Chapters 6 and 7. For IF E Y(ZDA), w = (w1iw2,w3) a character of (Ax /kx )3, X a principal quasi-character of GA /Gk, and s E C, let V.

ED (W', W) X, s), ED+(W) W) X, s)

be the functions which correspond to Zv (qD, w, X)s), Zv+(4,, w, X, s) for the case V = Sym2k2 ® k or Sym2k2 ® k2 in Chapters 5 and 6. Let t2(A2) be as in §6.2. Let XD be the principal quasi-character of Gj/Gk such that XD(t2(A2)) = A2-' for (06), and XD(t2(X2))= r.Tv,2A2 for (/38). We use the notation 'v1,nv2,gv1,gv2 instead Of'v,1,kv,2,.9v,1, as in Chapter 9. We identify ape ()t2), ape (.\2) in §8.2 with t2(\2). Then no, (t2(A2)) = X24 for (/36),

and /c,l(t2(A2)) _ A26 for (/38). We consider r,01 as a principal quasi-character of the group in §6.2. For D = (/36), let ED,ad (`Y, w, X, s) be the adjusted zeta function. We denote by RD,O, R" O the operators which correspond to Rv,o, R',,0 in §6.2. We define To (RD,OIF, w, s, s1) etc. similarly. Let HD C MD be the subgroup generated by T and for the element ((1, 2), 1). Let to = a(X11, \12, \13; A21, A22)t ° E TA° where Xzj E

all i, j. We define (11.1.8)

XD = {kn'(uo)t° I k E K fl MDA, uO E A, IA11A12 I ? a(uo)}.

Then GA fl MDA/HDk = XD/Tk. (d) D = (i31o) We can identify ZD with Sym2k2 ®k2. For 'P E 9'(ZDA), w = (wl, w2) a character

of (Ax /kx )2, and s E C, let ED('I'+w, X, S), ED+(1', W, X"5)

be the functions which correspond to Zv (ob, w, s), Zv+(4), w, s) for the case V = Sym2k2 ® k2 in Chapter 5. Let YD,o, ZD,o, ZD',0 C ZD be the subspaces which correspond to Yv,o, Zv,o, Z(,,0 in Chapter 5 respectively. We define RD,O etc. similarly. Let ED,ad (W, w, s) be the adjusted zeta function and TO (RD,01W, w, s, Si) etc.

the adjusting term. Let HD C MD be the subgroup generated T and the element ((2, 3), (1, 2)). Let to = a(Alli A12, A13; A21, A22)t ° E T. We define

(11.1.9)

XD = {kn"(uo)t°nB2(ul) I k E K fl MDA, u° E A, IA12A13 I ? a(u0)}.

11 Unstable distributions

253

Then GAf1MsA/Hak = Xa /Tk. Let h = (µl, µ2, ql, q2, uo) and hl = (1, µ2)ql, q27 uo) where Al, 92 E R+, q1i q2 E Al, uo E A. We define dh in the obvious manner. For lp E .9(Zs,OA), we define (11.1.10)

OZa,o

h) lY(µlµ2-1g1x1,337 µ1µ242X2,22> 2i 1µ24zx2,z2uo)

X1,33,52,22 Ekx

(e) As before, we use the notation Ew,aa,(2) ('I', w, so), Ea (2) (`I', w, s°) etc. for the

i-th coefficient of the Laurent expansion of Ew,ad(T,w, s), Ea(IF,w, s) etc. In the next three chapters, ga denotes an element of G°A fl MDA, and g°° an element of MaA.

Let dga, dga be the measures which we defined in §3.3. Let d"t° be the measure on TA such that dg° = dkd" t°du where dk, du are the canonical measures on K, NA. (f) Next, we define some distributions associated with the space W of binary quadratic forms, and distributions which are variations of the standard L-functions. Let T = (T,,T2) be a Weyl group element. Let x = xl, x2 = x117 x12> x137 x217 x22 E Q .

Consider the following substitution (11.1.11)

ST11 = x111(3) - Zlri(2), S7-12 = x111(2) - x111(1)7 sT2 = x212(2) - x212(1)

Let sT = (ST1l, 5112, ST2), and sr, = (ST117 ST12) We define ds7-1 = dsTlldsr12, dsT = dsTlds.2

_

Then dz = dsT.

We define functions L,(sTl), M1 (STl) of sT1 E C2 by the following table. Table (11.1.12)

Ll(srl)

Ti

'i T1(ST1)

-2(5111 + ST12)

1

(1,2)

-2sT1l

O(Srl2)

(2,3)

-2S 12

O

(1,2,3)

2sT11

7 f(ST11)/(ST11+

(1,3,2)

2ST 12

1

ST12)

0(ST12)0(STr 1 + ST12) .

1 + ST12)

2(5111

(1,3)

(Srii)

f1

0(ST11)0(ST12)w(ST11 + ST12)

We also define (11.1.13) L2(sT2) _ { -sT2

T2

ST2

MT2(sT2) =

1

T2 = 1,

_ (1, 2),

1

T2=1,

O(ST2)

T2 = (172)7

L(ST) = Ll(STl) +CL2(ST2)7 MT(ST)

M1(ST1)MT2(3T2),

fi(x)

A(w,sT) _

W - L(ST) Al(w7STl) =

(

AT(w;sT) = MT(sT)A(w; sT),

x17 III

w-LiT1(ST1)-V

)

A111(w7S11) = Al(w7STl)MT1(ST1)7

Part IV The quartic case

254

where z and sr are related as in (11.1.11). It is easy to see that L(z) = L(sr). We remind the reader that w° = 4 + C. We also remind the reader that we are still assuming that b(-TGZ) = O(z), O(P) 0 0.

We consider the domain which corresponds to Dr for Re(sr) and use the same notation D. For example, if T = ((1, 2, 3), (1, 2)), a point r = (ri, r2i r3) E Il83 belongs to Dr if and only if r1 > 1, ri + r2 > 1, r3 > 1. Let pr E R3 be the element which corresponds to p by the substitution (11.1.11). Similarly, let pi, E R2, P2r2 E R be the elements which correspond to Pt, P2 by the

substitution (11.1.11). It is easy to see that P2r2 = -1 if T2 = 1, and p2r2 = 1 if r2 = (1, 2). Also Pir1 = (-1, -1), (-2,1),(l, -2), (2, -1),(-1,2), (1, 1) for ri = 1, (1, 2), (2, 3), (1, 2, 3), (1, 3, 2), (1, 3), respectively.

Definition (11.1.14) Let i be a positive integer and IF E Y(Ai). Let 11(s,), li(sr) be linear functions of sr, and li+1(sr) a polynomial of sr. We define

Ei,sub(W, 1, sr) = li+1(sn)-1Ei(1F, 11(°Sr), ... , li(sr)).

(`sub' stands for `substitution.') Let G = GL(2), and G°o = GL(2)°A. Let lt(sr) be a linear function, and 12(sn) a polynomial. We assume that li(-8,111 8,11 + 8,12, 5T2) = li(STi11 S7-121 S.2) for i = 1, 2. Let g° E GL(2)00, t E All. Definition (11.1.15) LetT = (Ti,-r2) be a Weyl group element such that r1 E 232. For ' E 9(WA), we define (1)

Itli1(sr)

o

Esub (l , t , g, ST ) =

12(8T)

EB (t , go ,

- 8,11

),

_

3

(2)

4 .b,, (1 , t , g ° , w ) = ( 2nV -l f2w,T('I'11,w)=

(3)

Re(sr)=r

/k X XG O /Gk iti>1

jtj>1 7, QW, st ,r(Y',

Gwk:<(y.,tg°)esub,T(1,t,g°,w)dxtdg°,

A

Qw,T('I'11, w) = fX /kx XGO/G k

(4) (5)

)f

E 5b (l , t , g° , sr )An ( w;

Ows OF, tg°)19sub,T(11

t)

9°,

w)dxtdgo,

A

1, w) = f X /kX XGOA/Gk Gw, st('y1 t9 0 )4sub , r(l, t, g0 , w)dx td9 0 , t1>1

(6)

3

Ew,SUb+('F,1, ST) = fAX /kx XGOA/Gk Ow. ('I', t9°)Esub,T(l, t, g°, sr)dx

tdg°,

It1>1

where in (2), we choose r = (ri, r2i r3) E R3 so that r E Dr, l2(r) > 0, and r1 < -1. For the rest of this book, when we consider contour integrals with respect to sr, r is of the form r = (r1, r2, r3) E R.

11 Unstable distributions

255

The distribution Ew,sub+(q', s,) is well defined as long as Re(s,ll) < -1, and

_ (2)f 1

S2n,'1,w) =

E.W,SUb+(q'11, ST)AT(wl ST)ds,. Re(ar)=

,Y <-1, l2(*)>0

Clearly,

w) _ cv ('k,1, w) + 1lw,sc,,(W,1, w).

Since we have to deal with integrals in (11.1.15) many times in Chapter 12, we consider them simultaneously in this section and the next section. Consider (11.1.15)(5). Let ESUb(l, t, g°, s,) (resp. Esu6(l, t, g°, s,)) be the constant (resp. non-constant) term of Esub(l, t, g°, s,). It is easy to see that the nonconstant term EEdb(1, t, g°, s,) is holomorphic as long as Re(s,ll) < 0.

Definition (11.1.16) ST) =

(1)

f

OW,st(`

,

tg°)Esub(ll t, g°, s,)dxtdg°.

Ax /kxxGA/Gk

(2)

ntW,st,T('y,1,

w) = f

E"w,st,sUb+(`k,1, s,)A,.(w; s,)ds,.

Re(ar)= *1<0, 12(")>s

s,) is well defined as long as Re(s,ll) < 0. The distribution We consider the constant term. Let to = a2(µt1, µ-1t2) where µ E R+, tl, t2 E A1. It is well known that t 11 (3+)

Esub(1, tl t°l 8T)

(-Srll)Nl+sr11 ).

I12(sT) (µl-gr11

Let r' = (Ti, T2) be a Weyl group element such that Ti = 1 or (2, 3)Tl. We consider s,, as in (11.1.11) for T' also. If Ti = (2, 3)71, then s,,ll = -s,ll, STt12 = 8T11+ST12, and q(-S,ll)A,(w; S,)= A,, (w; s,,). Therefore, by the assumption on 1,

JRe(s)=r

Esub(l) t, t°, s,)A,(w; s,)ds, Itl1 (s,-,)N,1-s,,11 ,.,

T'=r,((2,3),1)T

Re(s,.,)=r'

12(8T')

A,, (w; S,, )ds,,.

Let B be the Borel subgroup of GL(1) x GL(2), b = (t,n2(u°)a2(ptl, P-%)), and db = dxl-tdxtdxtldxt2du°. We identify the maximal torus of GL(1) x GL(2) with GL(1)3. By a similar argument as in §4.2,

f

XWnBA/(k")3 ItI>1

12(ST')

OZ,W 0 (lY b)db =

Tw+(RW,°1k111(s,,)' 2(i - S,tll))

(8r'll - 1)12(8,') l

Part IV The quartic case

256

Let Tw+(Rw,o'L, 11(s"), 2(1 - ST'11))

(11.1.17) E W,st,sub+(LY, 1, sT')

(sT 11 - 1)12(sT') 1

F, w W,st,T OF, W) =

(2)7f Le(sr,)=r

W,st,sub+(`Y, 1, s7, )' ,, (w, ST' )dsT'.

Then (11.1.10)

fW,st,T (IF, 1, w) =

E

Q

w,st,T' (i , 1, w) + cl'w.t,T(w, 1, w).

T'=T,((2,3),1)T

§11.2 Technical lemmas Consider a Weyl group element T = (Ti,T2) such that T1 E 02. First we consider the distributions Styy T(gf,1, w) etc. in §11.1. We consider the following three possibilities for 12: (1) 12(ST) = -ST11 - 2sT12 + 2ST2 - 1, (2) 12(ST) = sT11 + 2sT12 + 2sr2 - 7, (3) 12(ST) = ST11 + 2ST12 + 2ST2 - 9.

If r2 = 1, we fix r1 = -2, r2 = 4, and choose r3 >> 0. Then L(r) < 8 - Cr3 << 0. w), S2'W'St,T(IQ,1, w) - 0 if TZ = 1. So we assume that r2 = Therefore, Q8 (1, 2).

Lemma (11.2.1) Suppose that T1 E 02 and T2 = (1, 2). Then by changing b if necessary, w) - 0 unless Ti = (1, 3, 2), (1) f2fv,T('y,1, w), (2) If 7-1 = (1, 3, 2), n'w,st,T('P,1, w) - 0, and f2ft,,T('p,1,w)

-CGA(w;P)Ew+((,ll(-1,)2,1))

Proof. If r2 = (1, 2), l2(PT) # 0 for all the cases. Therefore, by the passing principle (3.6.1), we can ignore the set 12(sT) = 0 when we move the contour. So for the case (1), we choose r so that 12(r) < 0. We move the contour so that r3 < 1 as follows. 1w,T('I`,1, w)

(2) 1

1rV-1

+0

3

fRe(s.)=(-2,6,2)

1

(2)7rVl'-'-l

3 £W,sub+('@, 1, ST)A'* ST)dST

2

JRe(.FW,sub+( ,1, ST1, 1)A1T1(w; ST1)ds

1.

*1+2r2 >7

Note that l2(-2, 6, 2) # 0 for all the cases. Since C > 100, Z(-2,6, < 12 + < 4 + C. So, we can ignore the first term. 2C 1, w) also. The same argument works for2)SE,,,St,T (IF,

257

11 Unstable distributions

Consider the second term. If T1 = 1 or (1, 2), we fix r1 = - 2, and choose r2 >> 0Then i(r) << 0 or i(r) = 3 + C. Therefore, we can ignore these cases. This proves (1).

Consider Tl = (1, 3, 2). We move the contour to Re(sT1) _ (-1 - 6, 2 + 6) where

6 > 0 is a small number. For (2), (3), we can use (3.6.1), and ignore the line l2(sri, 1) = 0 Therefore, in all the cases, 2

1

0

27r

=0

JR.(1)=(12) r1+2r2>7

1

EW,sub+('', 1, ST1, 1)A1T1 (w, ST1)dsrl

2

JRe(si)=(-i-6,2+5)

(2)7r 1

= [J

E

\

1, ST1, 1)A1T1(w, STi)dsTi

2

EW,sub+('lf, 1, ST1, 1)A1T1(w, STi)dsTl P

Res JRC(sr12)=2+6 Sr11=-1

27r V

EWsub+(I',1, ST1, 1)AiT1(wi -1, ST12)dsT12

Note that since we move the contour from the left to the right, the sign of the second term is negative.

Since L1(-2,2) = 3, we can ignore the first term. Since M-1(-1,5T12) _ 02(ST12 - 1), and

Res E Sub (l t , g°, s T1, 1)=-

s+ll=-1

,

VIt111(-i,s,12,1)

12(-1, ST12, 1)

we only have to move the contour to 1 < Re(sT12) < 2. Q.E.D.

Lemma (11.2.2) Let T' = (T1,T2) be a Weyl group element and 6 > 0 a small number. Then by changing 7/i if necessary, (1) S2v,st,,, ('I`,1, w) - 0 unless r' = TG,

(2) If r' = TG, QW,st,T,

('I`,1, w)

27rv -1 Re(sr,11)=1+6

EW,st,sub+('y, 1, ST,11, 1, 1)Yi2(sr,11)A1(w; ST,11, 1)dsT'ii.

Proof. Similarly as in (11.2.1), we can ignore the case T2 = 1, and assume that T2 = (1, 2). Then Qw,st,T,

_0

(

('I',1, w) 1

2

EW,St,sub+(W, 1, sT,i, 1)A 11 w, sT'1)dsT'i. 2 7r V1----1

*1+2,2>7

(We have to use the passing principle (3.6.1) for the case (1).)

258

Part IV The quartic case

If 1.1' = 1, (1, 2) or (1, 3), we choose rl, r2 >> 0. Then L(rl, r2,1) << 0. If Ti = (1, 2, 3), we fix r1 = 2, and choose r2 >> 0, and if Ti = (1, 3, 2), we fix r2 = 2 and choose r1 >> 0. In both cases, L(rl, r2, 1) = 3 + C < 4 + C. This proves (1).

Suppose rl = (1, 3). Then p, = (1,1,1). Clearly, 12(1,1,1) : 0 for all the cases. Therefore, we can move the contour crossing this line. So 1

2

) f('i)("i2) EW,st,sub+('I')1, 81.'1) 1)A1,'1(w; ST'1)ds1'1

(21rV

2

, 1+2r2>7

1

1

)'I

+ 2n V 1

Ew,st,sub+(W') 1, S7-'I) 1)A11.(w; ST'1)dsT'1

e(sT'1)=(1,5)

1) ST'll) 1) 1)

fRe(srii)=1+6

X 02(s1'11)A1(w)8T'11) 1)dsT,ll)

where 6 > 0 is a small number. We can ignore the first term by the usual argument. This proves the lemma. Q.E.D. The following lemma follows from these considerations.

Lemma (11.2.3) Suppose that r' = TG and b > 0 is a small number. Then by changing 7/i if necessary

E nw,st,T('I')l)w) *1EW2

r2=1,(1 2)

CGA(w; p) P2

+

27r

Ew+(`I',1, (-1, 2,1)) 12(-1, 2,1) JRe(s11)=i+8 EW,t,sub+( T, 1, sr'11, 1, 1)2(ST'11)A1(w; sr11, 1)ds111

Now we consider slightly different contour integrals. Let l = (2 (3,11 + 1), 12) where l2 is one of the following: (1) 12(s1.) = (-sT11 - 23,12 + 2s1.2 - 1)(s,12 + 1),

(2) 12(8,) = (-ST11 - 28,12 + 2ST2 - 1)(8,11 +'ST12 - 1), (3) 12(51.) _ (sr11 + 2sT12 - 2sT2 + 1)(-sill + 28,2 + 1), (4) 12(8,) = (ST11 + 231.12 - 2ST2 + 1)(3,11 + 28,2 - 3), (5) 12(8,) = (s1.11 + 281.12 + 281.2 - 7)(8T12 + 1), (6) 12(8,-) = (31.11 + 231.12 + 281.2 - 7)(sTll + 8,12 - 1))

(7) 12(ST) = (s1.11 + 251.12 + 2ST2 - 7)(5sr11 + 4s,-12 - 2ST2 + 1), (8) 12(51.) = (`81.11 + 2s,-12 + 2ST2 - 7)(s,-11 - 4sT12 + 2ST2 - 13),

(9) 12(ST) = (8,-11 + 2s,12 + 2ST2 - 7)(8,11 - 28,12 + 281.2 + 1), (10) 12(8,-) = (sill + 2s,12 + 25,2 - 7)(3s1.11 + 2s,-12 - 2sT2 - 9),

(11)

12(81.) = (5,-11 + 2s1.12 + 2ST2 - 7)(-s,11 + 2ST2 + 1),

(12) 12(3,-) = (5,-11 + 25,-12 + 281.2 - 7)(sT11 + 251.2 - 3),

(13) 12(81.) = (8,-11 + 28,-12 + 28,2 - 9)(5s,-11 + 43,12 - 2sT2 - 9),

(14)

12(8,-) = (5,11 + 28,-12 + 28,-2 - 9)(8,11 - 431.12 + 25,2 - 3),

11 Unstable distributions

259

(15) 12(ST) = (sill + 2sT12 + 28,2 - 9)(s711 - 2s.12 + 2ST2 - 5), (16) 12(87) = (s.11 + 2s.12 + 2sT2 - 9)(3sT11 + 26.12 - 28,2 - 3).

For these cases, let 121 (resp. 122) be the first factor (resp. second factor) of 12. Let 41 E 9(A). We consider 1

(2)3fR(S,- E1,sub(

, 1, s.)AT(w; sT)ds.,

where we choose the contour so that r = (rl, r2, r3) E DT, r2 > 1, and l21(r), 122(r) > 0.

Lemma (11.2.4) Suppose that T' = TG and 6 > 0 is a small number. Then by changing V7 if necessary, 1)31

1

2

N 27rV

E e(sr)=r

l,sub(, 1, sT)AT(wi 87)(187

1, sT'11, 1, 1)W2(sT'11)Al(w; 37'll, 1)dsr'ii.

-1 JRe(s11)=1+6

We devote the rest of this section to the proof of this lemma.

Lemma (11.2.5) Suppose that l(s.) = (2(s,11 + 1),12(57)) where l2(sT) is one of (1)-(16). Then by changing 7/J if necessary, (1) if T2 = 1,

)31

1

.(',)=r

27r

El,sub(i, 1, sT)AT(w; 5r)d5r - 0, and

(2) if T2 = (1, 2),

)3j 27rV -1

e(s,)=r

1

E1,sub(W, 1, sT)AT(w; sr)dsr

)21

e(s.,l)=(ri,r2)

27r

1,sub(I, 1, sTl, 1)AiT1(w; 8T1, 1)dsT1,

where we choose the contour so that (rl, r2) E DTl, r1 > 1, 121(rl, r2, 1) < 0, 122(rl, r2, 1) > 0 for cases (1), (2), and

(rl, r2) E DTJ, rl > 1, 12i(rl, r2, 1) > 0 for i = 1, 2

for cases (3)-(16). Proof. If r2 = 1, we can choose r3 >> 0, and Z(r) << 0. Therefore, (1) is clear.

260

Part IV The quartic case

We assume that r2 = (1, 2). Then P21-2 = 1. For cases (1) and (2), it is easy to check that the point pill is not on the lines 121(s1-i,1) = 0,122(81-1,1) = 0. Therefore, by the passing principle, we can move the contour crossing these lines. So we can assume that 121(ri, r2, 1) < 0,122(ri, r2, 1) > 0 for these cases. Consider the following values for (ri, r2).

(rl, r2)

(3,5)

(1), (2), (4) - (7), (10), (12), (13), (16),

(2, 20)

(3), (11), (8), (9), (14), (15).

(15,2)

Then for cases (1) and (2), 121(ri, r2, r3) < 0, 122(ri, r2) r3) > 0 for r3 = 2,1, 2, and for cases (3)-(16), 12i(ri, r2, r3) > 0 for i = 1, 2 and r3 = 2,1, 2. Therefore, 1

27f

)3f El,sub(W, 1, s1-)AT(w; ST)ds1-

e(sr)=(ri,T2,2)

)31

Re(s.,)=(ri,r2,2)

21r N/- 1

1,sub(W, 1, s1-)A1-(w; s1)ds1-

2

1

+N

E

( 27rV -1

J Re(srl)=(rl,r2)

E1,SUb(W, 1, s1-i, 1)X1,1 (w; srl)dsri.

In all the cases, L(rl, r2, 2) < 43 + 2C < 4 + C since C > 100. Therefore, we can ignore the first term. This proves statement (2). Q.E.D.

If r1 = 1, (1, 2), (2, 3), Li (ri, r2) < 0 for the above choices of ri, r2. Therefore,

)21

1

El,sub(W, 1, sT1, 1)AiT1 (w; 51-1, 1)ds1-i - 0

( 2r

e(s,l)=(rl,r2)

for these cases. Suppose that ri = (1, 2, 3), (1, 3, 2) or (1, 3). Let 6, 61 > 0 be small numbers such that 616-1 >> 0.

Lemma (11.2.6) By changing -i if necessary,

Fl,sub(W, 1,'STl) 1)A1T1 (w; sTl, 1)dsTl

( 27r N

)2j(w;

)21

1

e(sr1)=(r1,r2)

1

2r

sTl,1)ds1-t.

e(s,l)=(1+6,1+6)

Proof. For cases (1) and (2), the domain {(ri, r2) 1121(rl, T2, 1) < 0, l22(rl, r2, 1) > 0} contains (1+6,1+6). For cases (3) and (4), the domain {(ri, r2) 1121(ri,r2, 1) > 0,122(ri, r2, 1) > 0} contains (1 + 6,1 + 6). For ri = (1, 2, 3), (1, 3, 2), (1, 3), Pill = (2, -1), (-1, 2), (1, 1). For cases (6)-(11), (13)-(15), 12i(P1T1,1) # 0 for i = 1, 2. For cases (5), (12), (16), 121 (P1,1, l) # 0,

and the line segment joining (ri, r2) and (1 + 6, 1 + 6) does not meet the line

11 Unstable distributions

261

122(81.11, S,-12, 1) = 0. Therefore, the lemma follows from the passing principle (3.6.1)

for these cases. Q.E.D.

If Ti 54 (1, 3), Ll(1 + 6,1 + 6) < 4 as long as 6 > 0 is small. So

)2j

1

2iri

-1

e(s.,,)=(1+6,1 -6)

El,sub( ) 1, sr1, 1)Alrl (w; srl)dsrl - 0.

Suppose rl = (1, 3). Then

)21

1 271.

-1

)2

1 (2'i)71.

+

--

21r J

e(s.l)=(1+6,1+6)

E1,sub(W, 1, srl)

fRe(sj)=(i+5,1-5i)

(w; srl)dsrl

+l,sub(W, 1, srl, 1)Alrl (w, srl)dsrl

El,sub(W, 1, 81.11, 1, 1)02(sr11)Al(wi 81.11, 1)dsrll e (sr11)=1+6

27r fRe(sii)1+6

Fil,sub(W) 1, Sr11, 1, 1)02(sr11)Al(wi 81.11, 1)dsr11,

because the first term can be ignored by the assumption on 6, 61. This proves (11.2.4).

Q.E.D.

Chapter 12 Contributions from unstable strata §12.1 The case 0 = ((31) We now start the analysis of the constant terms 8p (4', w, w). We remind the reader that we are still assuming that 4' = Mv, 4, so if 6# (w) = 1, ID is K-invariant. In this section, we consider paths p = (a, s) whose ,3-sequences start with /31.

Since e((g°),w) = e(g°,w) and the characters epi do not depend on s(1), we only consider p = (0,s) such that s(1) = 0. For paths with s(1) = 1, all the results are valid replacing 4' by $ and w by w-1, and changing the sign. All the /3-sequences which start with Nl satisfy Condition (3.4.16)(1). Let = (ao,so) where 0 = (f31). Let pll = (01, 511), P12 = (D1, 512) where (i1 = (31, i1,1), and RRPo

511(2) = 0, 512(2) = 1. Let p21 = (02, 521), P22 = (D2, 522) where 02 = (/31,,31,2), and 521(2) = 0,522(2) = 1. Let 03 = ((31.,,(311,/31,1,1), and p3i = (03, 53i) for i = 1,2,3,4

where (s3i(2), S& (3)) = (0, 0), (0, 1), (1, 0), (1,1) for i = 1, 2, 3, 4 in that order. We consider contributions from these /3-sequences. Throughout this section, we assume that IF = Rao 4'po Let

(12.1.1)

do(A1) = a(1,1,1; ail, A1), d1(Al, A2) = a(A22, A2, A2; A11, A1), d2 (A,, A2) = a(A2 1, A2 1, A22; Al

1,

i),

d3(A1, A2, A3) = a(A2 2 -2'13 1+ 2 i-i 1, A1),

for A1, A2, As E R+.

Easy computations show the following two lemmas.

Lemma (12.1.2) A1a2. A1, (1) (2) ep121 (dl(A1,A2)) _X11, ep,22(dl(A1,A2)) = A1a2. (3) ep2i1(d2(Ai, A2)) = A1, ep221(d2(Ai, A2)) = Al 1. A2, A3)) = A1, ep3,2(d3(Ai, A2, A3)) = AjA2. (4) (5) ep321(d3(Ai, A2, A3)) = Al 1, ep322(d3(Ai, A2, A3)) _ (AiA2)-1 1 ep332(d3(A1, A2, A3)) = ala2. (6) ep331(d3(Ai, A2, A3)) _ A (7) ep341(d3(Ai, A2, A3)) = A1, ep342(d3(A1, A2, A3)) = (A1A2)-1

Lemma (12.1.3) A1-3 A2-6.

(1) op,,(dl(A,,A2)) = A2A2, o'12(d1(A1,.2)) = A'A', ,a,1(d1(A1,A2)) _ 21 (d2(Al, A2)) = A2A2, o' 22 (d2 (A,, A2)) = A4A2. (3) Qps, (d3(A1, A2, A3)) = A1'A3, 0P32 (d3(Al, A2, A3)) _ A1A3, spas (d3(A1, A2, A3)) = A1A1Am3, ap34 (d3(A1, A2, A3)) = Al 1A3

(2) o'

Let T, = {po, p11, P12}, P2 = {P21, P22, P31, P32, P33, P34}. By (3.5.9),

(12.1.4)

Epo

P. (4',

w,w) _

Ep

P+ (C w, w) + °p+OP, w, w)

PET,

pE`p,

+

Ep p(4',W,W) PEP2

12 Contributions from unstable strata

263

Po+N' w, w), w, w) The element do(A1) acts on ZaoA by multiplication by A1. It is easy to see that -po(do(A1)) = )s , Kao1(do(\1)) _ Al s Therefore, the following proposition follows from (3.5.13). (a)

Proposition (12.1.5) (1)

"po+(ID, w, w) - CGA(w; P)sao (w)Eao+(q', wao, 2).

(2)

=po+(1D, w, w) ' CGA(w; P)aao

(b) =po#(-1),w,w),

(w)Eao+(j;-ao W, wa 1, 4).

w)

Proposition (12.1.6) (1)

po# ( , w, w) - CGA(w; P)

(2)

po# ( , Lo, w)

2736# (w)'y(0) 2

CGA(w;P)2136#(w)9ao'F(0) 4

Proof. Let 0 = Do, p = po. Let go E Ma, 9a = do(A1)gao Then dga = d
cp#(',w,w) = `I'(0) f

w(go)A

p(ga,w)dga,

Ri. X MDA /Mak

p#('b,w, w) = a'0 T '(0)

f

ai <1

w(ga)A14e(ga,w)dga

R+x MOA /Mak 1<1

Let T = (1, T2) and r2 = 1 or (1, 2). Let sr2 be as in (11.1.11). By a similar consideration as in (3.5.20), 6#(w)

p#( D ,w, w) _

11

72

f

2vJ

1-e(8 T 2)=*' 3 >1

MT2(s-2)

A(w;P1,z2)

S T2

+1

d sTz,

where z2 and sit are related as in (11.1.11). If T2 = 1, L2(Re(z2)) = -r3 < -1 on the above contour. Therefore, we only have to consider the case 72 = (1, 2). We then move the contour to 0 < r3 < 1, and (1) follows. The proof of (2) is similar. Q.E.D.

w, w), E w, w) In (c), (d), let a = 01,p = p11, and W1 = Ra14)p,,. Let go = di(A1,A2)ga° where ga E MA is as in the second element of (11.1.3). Also in this case, M ,1A = MA. It is easy to see that dga = 2dxA1dXA2dga. Let p = A1A2. Then 2d \1dxA2 = (c)

dXA1dxp. Let T = (7-1iT2) be a Weyl group element, and sT as in (11.1.11).

Definition (12.1.7) (1) 1p11 = (1p11,1,1p11,2) where 1p11i1(ST) = 2(sT11+2s-12+3), lp,,,2(S) = -ST11 2sT12 + 28,2 - 1,

Part IV The quartic case

264

(2) "pll = (lpll,1, 1p11,2) where 1p11,1(ST) = 3 - lp11,1(ST), 1p11,2(ST) = 1p11,2(ST)

Note that 1,11 (-ST11, ST11 + ST12, ST2) = lpii (ST11, ST12, ST2) etc.

By (12.1.3)(1), \2)TZ+P = A1T2-1A2r11+28+12-3 _

(12.1.8)

o,p(d1(A1,A2))d1(A1,A2)TZ+P,a1(d1(A,,A2)) =

Ai`n2(Sr)µi

l(er)-3

The elements dl(A1, A2), 05 (d, (A,, A2)) act on ZDA by multiplication by µ, µ 1 respectively. Also the integrals defining =pll+(4), w, w), upll+(4), w, w) do not depend on tll, t21. Therefore, by (12.1.8), ll+(,D,w,w)=26#(w)

Qw,T(`I'1,1p111w)1 r1 EID2

r2=1,(1,2)

w, w) = 26#(w)

QW,T(gW'F1, 1p11, w). r1 E 2112

T2=1,(1,2)

The following proposition follows from (11.2.3).

Proposition (12.1.9) Suppose that r = 'rG and 6 > 0 is a small number. Then by changing 7/i if necessary,

- CGA(w; P)6#(w)Ew+('Y1, 3)

=pll+('D, w, w)

+ 2026#(w)J f 27rV -1

EW,st,sub+(``1) lpll, 5T11, 1, 1)

Re(sT11)=1+6

X 02(ST11)A1(w, ST11, 1)dST11,

w) - - CGA(w; p)6#(w)EW+(9W'F1, 0)

+ 2026#(w) 27rV

i

f

e(s,ii)=1+6

1W,st,sub+(gW 1, 1p11 , ST11, 1, 1 )

X g52(ST11)A1(w, si11, 1)dsi11.

(d)

We define w, w, ST12, ST2) =

W) W, ST12,

(0)

ST2)-

A T(w; -1, ST12, ST2) (ST12 + 1)(-ST12 + ST2) 1(O)

A. ( w; -1, ST12,ST2) (ST12 - 2)(-Sr12 +ST2) .

Then by (11.1.4), (11.1.5), (12.1.8), and the Mellin inversion formula, the distributions p11#(D,W,w),upll#(D W,W) are equal to 2

(12.1.10) E

CO,

T1

r2=1,(1,2)

T1E'ID2

r2=1,(1,2)

2 (_1 1r

1

/

(1\21) 27r v

fRe(, r12.ar2)=('2, r2>2, r3>r2

Ep11#,T((D, w, w, ST12, ST2)dsT12, ST2, r3)

f(arl2,ar2)=(r2,T3) r2>2, r3>"2

Gpll#.T O', W, w, ST 12, ST2)dST12, ST2

12 Contributions from unstable strata

265

Proposition (12.1.11) Suppose that T = TG and 6 > 0 is a small number. Then by changing ?,b if necessary, CGA('w;p)126#(w)`I'1(0)

(1) ( 2)

Ep11#(41,w,w)

(

"p11#

3

,W,w )

Jr

27r\ -1

2(ST11)Al(w; sr11, 1)

Re(srll)=1+6

s,.n(1 - srll)

d sr11

Proof. Consider the term in (12.1.10) which corresponds to r. If r2 = 1, we can ignore such a term by the usual argument. So, we assume that 7-2 = 1. If Ti = 1, (1, 2) PT = (-1, -1, 1), (-2, 1, 1). Therefore, by (3.6.1), we can move the contour crossing the line sri2 = 2. So in both cases, we can move the contour so that

r2 = 1 + 6, r3 = 1 + 26 where 6 > 0 is a small number. Then L(-1, r2i r3) < 2 + (1 + 26) C < 4 + C if 6 is sufficiently small. Therefore, we can ignore these cases.

Suppose Ti = (1, 3, 2). The point pr = (-1, 2,1) is not on the lines sri2 + 1 =

0, -sri2 + sr2 = 0. So we can move the contour crossing these lines by (3.6.1) so that r2 < r3. Then for (1), the term in (12.1.10) which corresponds to T = ((1, 3, 2), (1, 2)) is equal to

(2) 2 fR(.,22)=(2,3) rs>r2>2

)1

1

C2

+

Ep11 #,T

2

V -1

27r

W, w, 5T12) sr2)dsrl2dsr2

Ep11#,T(4), w) w, 8,12,sT2)ds,12dsT2

Re(srl2,sr2)=(3, 2 )

Res Ep11#,T(4,w,w,sr12,sr2)dsr12

Re(s,.12)=r2>2 sr2=1

We can ignore the first term as usual. If T1 = 1 or (1, 2), L(-1, 3,1) < 2 + C < 4 + C. Therefore, we can ignore both these terms. Suppose T1 = (1, 3, 2). Since M1.1 (-1, sr12) = 02(sr12-1) and this function has a simple pole at sr12 = 2, we only have to move the contour to 1 < Re(sTii) < 2. This proves (1). The proof of (2) is similar except that the integrand of the second term has a pole of order 2 at sr12 = 2,

and we leave as it is. If T' = TG, 5r'12 = sr12 - 1. We consider the substitution z -+ -TGZ. Then sr)11i Sr'12 are exchanged, and A1(w; sr)11i 1) = Al(w;1, sr'11) by the assumption 7,b(-TGz) = i,b(z). Thus, after these substitutions, we get the expression of (2) in terms of Q.E.D. (e) E 12+(4)w,'w), "p12+(4''w,w).

In (e)-(g), 0 = 01, P = P12, and W2 = Rat'I

12

Let go = d1(.X1, A2)9° where

goo E M OA is as in the second element of (11.1.3). By (12.1.2)(2), (12.1.3)(1),

"P12+ (D) w, w)

=2f R+x MaA /Mak

Al>1, ala2>1

w(9a)-1A14

A2

(`y2,9aA12(9a)w)dxAldxa2(1900)

Part IV The quartic case

266

W) w)

w(go)-1AlOz,(9W`Y2,00(ga))gp12(gD1w)d'AldxA2dga

=2 1.2 +x MDA/MO k

J.111, A1a2<1

Let µ = A1a2. Then A' A2 = a1µ3. The functions Oz, (W,go),Oz,(` 0Y,8s(ga)), and ep12 (gD, w) do not depend on tll, t21. Therefore, these distributions are 0 unless w is trivial. Let r, sr be as before. Easy computations show the following lemma. Lemma (12.1.12) Bp12(d2(A1, 2)) r=+v = Al 2(-srll-zsrl2-zsr2+s) µ 2(sr11+29,12-3)

Definition (12.1.13) (1)

1P12 = (lp12,1,lp1z,2)

where lp12,1 (Sr) = 2 (s111 + 2.s ,12 + 3),

lp12,2(Sr) = Sr11 +

2sr12 + 2sr2 - 7,

(2) 1P12 = (1p12,1,1p12,2) where lp12,1(sr) = 3 -1p12,1(Sr), 1p12,2(Sr) = lp12,2(sr). Note that 1p12(-sr11,sr11 + Sr12,sr2) =

lp12(sr11,Sr12,Sr2) etc.

By (11.1.4), (11.1.5), (12.1.12), SlW,r('1'2,1P12' w), T1 EQ2

*2=1,(1, 2)

p12+(4P,w,w) = 26# (w)

E

QWr(9WT2, lp121 W) -

r1E'h12

r2=1,(1, 2)

The following proposition follows from (11.2.3).

Proposition (12.1.14) Suppose that r = rG and b > 0 is a small number. Then by changing 0 if necessary, W, w) - -CGA(w; P)b#(w)EW+(%'2, 3)

f

2,o 2

+

JRe(sr11)=1+b

2a

EW,st,sub+(P2,1Piz) Sr11, 1, 1) X 02(Sr11)A1(w;1) sr12)dsr12,

p12+(41), W, w) " -CG A(w; P)b#(w)EW+(-g'-W'P 2, 0) 2g2 _ EW,st,sub+(`W W2, lp12, 5r11, 1, 1) + 27rV

1

fRe(sii)1+6

X 02(5r11)A1(W; 1,5r12)dsr12

(f) "p12#(4',w,w), up12#(4,w,w) These distributions are 0 unless w is trivial for a similar reason. We define A7(w; -1, Sr12, Sr2)

+sr2 -4)(Sr12+1)1 Ep12#,r(

,

w,w,sr12,sr2) = b#(w) `W1Y2(0)

Ar( w; -1, Sr12, Sr2) (Sriz + Sr2 - 4)(sriz - 2)

12 Contributions from unstable strata

267

Then by (11.1.4), (11.1.5), (12.1.12), and the Mellin inversion formula, the disWI w) are equal to

tributions ` P12# ("'I WI w)) "P12#

)2j..

1

(IF,

,1

-

2

r2=1,(1,2)

w) `ST121 S r2)ds,.12dsT2.)

(.j2,er2)=(*2,*3) r2>2, r2+r3>4

21 )

27r

r12

'

> P12#,T('f') W) W) ST12, Sr2)dSrl2dsr2 R<('r12,'r2)=(r2,r3)

r2>2, r2+'3 >4

,2=1,(1,2)

Proposition (12.1.16) Suppose that T = TG and S > 0 is a small number. Then by changing 0 if necessary,

-C G A (wp) 932S#( 3 )' 2(0)

( 1)

= p12#( p ,w,w)-

( 2)

2 p12#( 4 ,w,w ) ^' QS#(w)." w'F2(0) 27r

J

12(STII)AI(w; STIl, 1)d ST1I e(s,II)=1+6

(STIl - 1)(sTI1 - 2)

Proof. As usual we can ignore the case r2 = 1, and assume that T2 = (1, 2). We can ignore the cases rr1 = 1, (1, 2) by the same argument as in (12.1.11). Suppose Tl = (1, 3, 2). Consider (1). We move the contour in the following manner: 1

(2)2JR(l2.2)(2r3) Ep12#,

(WI) w, w, Sl2) 32)d12dS2

r2>2, r2+'3>4

2

p12#,T (TI) W, W, ST12, ST2)dsrl2dsT2

(2)7r-,/

+

fRe(si2,s2)=(4,)

f

1

27r

Res Epi2#,T( 1, W, W) ST12) 8T2)dsr12

e(sr12)=rz>3 Sr2=1

We can ignore the first term because L(r) < 8 + 1C < 4 + C. The point pT = (-1, 2, 1) does not satisfy the condition ST12 = 3. Therefore, we can ignore the pole of Ressr2=1 Ep12#,T('yl, w, w, sTl2, sT2) at ST12 = 3 by (3.6.1). Hence, Res E'p12#,T (W1, w, W, ST12, ST2)d5T12

1

JRe(si2)=r2>3 3,2=1

27r 1

27r

Res Ep12#,T(W1,w,w,5r12,sr2)dsT12

R'(' 12)=r2 8,2=1 1
- CGA(w; P) 02S#(w)y1(0) 3

We can ignore the first term in the usual manner. This proves (1), and the proof of (2) is similar to (12.1.11). Q.E.D.

Part IV The quartic case

268 (g) = p21

w, w), =P22 ( w, w)

Let 0 = 02. Let go = d2(A1, A2)9a where ga is as in the first element of (11.1.3). It is easy to see that dg, = 2dx )1dx )2dg°°. Let 'd = Rae Rp2i for i = 1, 2. In this case, Pa# = P1 x B2. By (12.1.2), (12.1.3),

P21 (D, w, w) = 2 f.2

w)d"Aldx.2dga,

w(9a)A2A2ez, (`I`1, 9sWp21(9a, x M ak /Mak

al<1

f

w(9a)-1A

R+x Mak /Mak

a1>1

Then A?.\ = Ai µz, a1 2 = Al µ2, and dxA1dxA2 = Let p = Let r = (r1, r2) be a Weyl group element such that rl E 01. Easy computations show the following lemma.

Lemma (12.1.17) Suppose Sr12 = -1. Then A2))Tz+P = A (-8x11+29r2)µ2(8,11-2)

(1) 0P21 (d2(\1,

(2) ep22 (d2(A1, a2))

TZ+P

-=

i (-s,11-28,2+4)µ2cw 11-2)

Definition (12.1.18) Let 42i = (2 (sill + 1), lp2i,2) for i = 1, 2 where (1) lpz1,2(Sr) = -s-11 + 2Sr2 + 1, (2) lp22,2(Sr)

= srll + 2ST2 - 9.

w) do not depend on g1,12, t21 for Note that the functions Oz, (lWe, ga), i = 1, 2. Therefore, the distributions .p21 (`Y, w) w), °p22 (I, w, w) are 0 unless w is trivial. We define Ep2i (D, W, Srll, Sr2) = [E1,sub(Wi, 42i, 8,)A,-(w; ST)I8,12=-1

for i = 1, 2. Then Ep2i ($, W, W) is equal to 1

2

6#(w) E (2ci) f

Ep2i ('P, w, sr)dsi11dsr2

R.e(s 11 ,8r2)=(r1 ,r3)

,1 E,d,l

for i = 1, 2, where we choose the contour so that r1 > 2, 2r3 > rl - 1 for i = 1 and

r1>2,2r3>9-r1 for i = 2. Proposition (12.1.19) P21(45,w,w),,, CaA(w;P)`V26#(w)E1(Ra2(Dp21,

(1) (2)

(4), w, w) - -CGA(w; P)

P22

2).

X1265 (w) E1(R02OPP221

2

Proof. Let ri = a, r3 = a for i = 1 and ri = 9, r3 = 2 for i = 2. Then 1

2

(2) fRe(sii,s2)(ri,r3) Ep2i (4), w, s,)dsT11dST2

(2) 1

=

7r V

+

i

2

fRe(aii,sr2)=(rl,r9)

4e

(s,11)=r1 sRes

Ep2i

(1)1 W1 Sr)dST11dgT2

E12i ('P, w, sT)dsrll

12 Contributions from unstable strata

269

Since L(ri, -1,r3') < 18+ 2C < wo for all the cases, we can ignore the first term. The rest of the argument is similar as before. Q.E.D.

(h) ,-p3i (41, w, w) for i = 1,

, 4.

Let a = D3. Let to = d3(A1,A2,A3)t°. Then dxt° = 2dxA1dxa2dxA3dxt°. By (12.1.2)(4)-(7) and (12.1.3)(3), / w(t°)A2A ®Z, (R1 4'p33) 1 2A 3

(-D, w, w) = j

(t°, w)dx to,

TO /Tk Al<1, ala2<1

P32

J

w, w) =

w(t°)-1AiA3OZa (R133 to)4p32 (t°)

w)dxto,

1

Tp/Tk

l>1, ala2>1

P33

w(t°)-'A4 A6 A3OZa

((D, w, w) =

J

(Ra4)p33) to)4p33 (t°, w)`"xt°>

TA /Tk

al>1, \1A2<1

paa (-D, w, w)

_f

w(t°)A1'A20Z,

(Ra.Dp34) t°Ap34 (t°, w)dxt°.

Tp/Tk

a1<1, N1),2>1

For a similar reason as before, these distributions are 0 unless w is trivial Let µl = Al A 2, and µ2 = ala2A3. Then A1A2A3

= A11µ1µ2, A1A3 = Alµl 1112)

2 A 4 A2A3

= A111i112, and Al 1A3 = Al 1µ11µ2

It is easy to see that dxto = 2dxAldxµldxµ2dxt o Easy computations show the following lemma.

Lemma (12.1.20) (1) Bp31

A2, A3))

(2) OP3z (d3(A1) A2, A3))

rz+P == 1A(-sr11-25,12-2Sr2+1)tts1 ,12-1µ2(s,11-1) Tz+P = A 2 (s -11+2s -12+2sr2-1) -s,11-x,12+2 z (sr11-1) 111 112

rz+p = z(-s,11-2s,12-2ST2+5) Sr12-1 z(srll-1) A1 µ1 112 (3) (s, 11 F2sr12+2s,z-5) -sr11-sr12+2 z (Sr11-1) (4) Bp3a (d3(A1, A2) A3))Tz+1 = A21 111 112

Definition (12.1.21) Let lp4i = (2 (sill +/ 1), lp4i,2) where (1) 1p31,2(ST) = (-STll - 2ST12 + 2ST2 - 1)(ST12 + 1), (2) 1p32,2(ST) = (-ST11 - 2sr12 + 2ST2 - 1)(ST11 + ST12 - 1)) (3) 1p33,2(ST) = (ST11 + 2sT12 + 2ST2 - 7)((ST12 + 1), (4)

1p34,2(ST) = (ST11 + 2ST12 + 2ST2 - 7)(sT11 + ST12 - 1).

By (12.1.20), (-D, w, w) = S#(w)

(2) 1

f

Psi

T T

3

E1,SUb(4),

1P30

ST)AT(w, ST)ds.r

for i = 1, , 4 where we choose the contour so that r1, r2, r3 > 1 and both factors of lp3i (r) are positive.

Part IV The quartic case

270

Let

i

E1(Ra3 p31, `Z (`SILL+1))

E P31 (4) , ST11 )

2(sTLL + 1)

EP32 (4) , S TL1 ) = ST 11) -

Ep33

E p39

(

, SILL) -

(ST11 +1)) sy11(8T11 + 1)

E1(Ra3 4)P32 , 2

E- 1 (Ra3 p33, 2 (3,11 + 1))

2(sTll - 3) E1(R53'p34, 2(s,11 + 1)) 3,11(3,11 - 3)

The following proposition follows from cases (1), (2), (5), (6) of (11.2.4).

Proposition (12.1.22) Suppose that 7- = TG and 6 > 0 is a small number. Then by changing b if necessary, 26#(W)

_p-3i (S , w, w) -

27rV1

fRe(sii)=i+5

8T11)02(8T11)A1(w; ST11, 1)dsrii

Ep3,

fori = 1,2,3,4. (i) Now we combine the computations in this section. Let i) = Do, P = Po. We define Jl((D,w) =sno(w) (Eao+(Ra0Pp.,wao,2)+Eao+(9aoR30 Po,wao ,4)) (0) - XTgs# (w) ( Rao ,Ppo 2

+

`_Do Raa -Ppo (0)

1

4

- X726#(w) (ElRD2P2l1 2) + 5E1(Ra24)p22, 2)) J2 (,D, w) = (Ew+(Ra1,Dp11 , 3) + Ew+(gwRa1(Dp11 , 0))

- (Ew+(Ra1IPp12, 3) + Ew+(,FwRa14 12, 0)) 02

+

(-Ra, lpii(0) + Rat p12 (0)) ,

Ta,1(P, W) = J1(', w) + 6#(w)J2(1D), 2Tw+(Rw,oRa1

(sill + 5), 2 (1 - s,11))

2 (sT11 - 1)(ST11 + 1)

2(1 - s,.11), 2(1 - ST11))

+ J(4D e T11 )_ 4

(STL1 - 1)(sT11 + 1)

2Tw+(Rw,oRa1 ,1Dp1, 2 (sTll + 5), 2 (1 - 3,11)) (sT11 - 1)(3,11 - 3)

2Tw+(Rw,o9wRa14'p12, 2(1

+

- ST11), 2(1 - S'11))

(3,11 - 1)(s,11 - 3)

12 Contributions from unstable strata

271

W, sill) = b#(W)(J3(I, s1-11) + J4('D, STll))

7

wRD, IDpii (0) + +Z26#(w)

+ +

s,ll(sTll - 1)

r

9wRa14)p12 (0)

l

(sT11 - 1)(s,-11 --2))

S1-ll) - 13P32(4)) ST11) 6#(W)(-E 33 (ID, ST11) + E 34

(4), STll)).

Then by (12.1.4)-(12.1.6), (12.1.9), (12.1.11), (12.1.14), (12.1.16), (12.1.19), (12.1.22), and by changing 0 if necessary, Epo (4),W, w) r., CGA(w;

W)

_

2 10

+ Let sion. We define

J5(4)) =

J6(D) =

J 21iV -1 Re(s,,,)=1+6

TO,2('D, W, ST 11) 02(ST 11)Al(w; S, 11, 1)dsT11.

W,1), J3,(2)(,D,1) etc. be the i-th coefficient of the Laurent expan-

-12(Tw+(Rw,oRo,

2-1

(Dpli, 3) +Tw+(Rw,oJIwRal4)pll, 0)),

(Tw+(Rw,o " wR51'Dp11, 3) + Tw+(Rw,o9wRa14P12' 0))

Then J3,(o)(4),1) = J5 ((P) + 2

+2

Tw+(Rw,oRa1 ,Pp11, 2(81-11 + 5), 0) d

sill + 1 srll=l Tw+(Rw,o9wR,i,Dp11, 2(1 - srll), 0)

ds1-11

d

ds,.11lsill-l

J o (4) 1) = J,(4)) + 2

d

sill +1

Tw+(Rw,oRal (Pp., 2(8.11 +5),0)

I

6,11 I3'11=1

d

81-11 -3

Tw+(RwoJFwRo1'Pp121

+2 dsTll

2(1- 8,11),0)

81-11 -3

3,-11=1

By the principal part formula for the standard L-function in one variable, if 4P is K-invariant, Tw+(Rw,oRo1 4)p1,, 2 (31-11 + 5), 0) + Tw+(Rw,o-q'w Ra1 (Ppll ) 2 (1 - sT11), 0) /

El Rw

1

2

(81-11 +3))+

2

D3

Ra3Ip31 (0)

sill + 3

2g'a3 Ra3 4p32 (0)

sill + 1

Tw+(Rw,oRa1 p12, 2(s1-ll + 5), 0) +Tw+(Rw,oFwRa1 Dp12, 2(1 - STIl), 0)

=

1

2(81-11 + 3)) +

2ga3 Ra3 -Dp33 (0)

2ga3 Ra3 p34 (0)

sT11 + 3

81-11 + 1

Part IV The quartic case

272

Therefore,

J3,(o)(D,1) = JA(D) + 2 (-Ei(Rw,2) + RD3 4)p31 (O) +

3

J4,(o)(,D, 1) = J6 (,D) 8

2

2))

R (O), 02 (0)

(E1(R'w,o-Dp12, 2) + E1,(1)(Rw,o41p12, 2))

FZ'3 RD3 4Pp33 (O)'

It is easy to see that _

1

ST11(ST11 - 1)

1

--

1

(s,11 - 1)(ST11 - 2)

- 1 + O(sl - 1)

,

ST11 - 1

- 1 + O(S'11 - 1).

1

sT11 - 1

Also 1)

Ep31,(o)

Ep32,(O)(4), 1) = 1)

Ep34,(o)( ), 1)

IE1,(-1)(Re3DP3111)IE1,(o)(Ra34p31,1), 4 4 2E1,(-1)(Ra3Dp321 1)

2E1,(o)(Rlo3'Dp32, 1),

4E1,(-1)(Ra3'Dp331 1)

1 E1,(o)(Ra3-pp33, 1),

1 E1,(o)(Ra34p34, 1).

1 El,(-1)(Ra34Pp3411) 2

2

We define J7((D)

-

1)

-2

4E1,(o)(Ra3'p31) 1)

El,(-,)(Ra3'Dp32, 1) + 2 E1,(o)(Ra31)p32, 1)

+ 8 E1,(-1)(Ra3N33, 1) + 4 E1,(O)(Ra3')p33, 1)

+ 2 E1,(-l)(Rb34N3411)

6

E 1,(o)(RO31)p34, 1),

J8(4D) = 2 (-3E1,(o)(R'wo4Dp11) 2)+El,(1)(R'w,o'Dp11,2))

- 2 (3E1,(o)(Rw,Ap12, 2) + E1,(1)(Rw,01)p1212)) Clearly, ga3 R534)13i (0) = E1,(-1)(Ra3443i 11)

for all i. Also if 4 is K-invariant, by (4.4.11), `323wRa143p11(0) = E1(R'w,o'1)p12, 2), 9329wRa1

(0) =

2).

12 Contributions from unstable strata

273

Therefore, S

w, 1) = 6#(w)

Ta,2,(o)

i=5

Ji(b)

By the principal part formula (4.2.15), J2(4') + J50) + J6(4) + J7(4) = EW,ad,(0)(Ra1,Pp11, 3) -

EW,ad,(o)(Ra1,Pp12, 3).

Hence, Ta,l (4),w) + Ta,2,(o) O, w, 1)

= Ji((D,w) +6#(w)Js(4) + 6#(w) (EW,ad,(o)(Ra14)p11, 3) - Ew,ad,(0)(Ra1'DP12, 3))

By Theorem (4.0.1), we get the following proposition.

Proposition (12.1.23) T,,i (,P, w) + T,,2,(o) (D, w,1) = ba (w)ED (RD gyp, wa, 2).

§12.2 The case a = (,32) We prove the following proposition in this section.

Proposition (12.2.1) Let p = (t, s) be a path such that Z = (,32). Then by changing V) if necessary, Ep (4), w, w) - 0.

Proof. Since .$((g°)`, w) = 6°(g°, w), we only consider p such that s(1) = 0. Let g° be as in the second element of (1.1.3). Let r = (T1,-r2) be a Weyl group element

such that r1 E 02Consider the situation in (3.5.17). In this case, we choose Aa = a(-1 4, _l, 1

i,

11

31

-1),

A,2) = a(A2 2, A2, A2, .2, A2 2),

where A1, A2 E 1R+ (A 3) = 1). Then ap(Aa)ATz+P

=

`26,11+4x.12+3s.2+9

ap(A((2))(A((2))TZ+P - A2r1+2x.12-2s,2+1

Therefore, LSp,r + h in (3.5.17) is Is, 1 + Sr11 + 2sT12 - 2sT2 = 0}. Let 91,23 = ka2(pt1i µ-1t2)n2(u) be the Iwasawa decomposition of 91,23 where p E l

1[8+, tl, t2

Then t(g°)urZ+P = p 8r11+1 if or = 1, and t(g5)aTZ+P = /-,S,11+1 if

v = (2, 3).

We choose rl = 3, r2 = 2, r3 = 4 if a = 1, and r1 = -3, r2 = 5, r3 = 4

if

o, = (2, 3). Then op(AD)A'Z+Pt(9a)aTZ+Pl

= A,A-2.

If q = Re(z), q E Dar for the above choice of r. Moreover, if T2 = 1, L(r) < 22 - 4C < 4+ C since C > 100.

Part IV The quartic case

274

It is easy to see that ) acts on ZDA by multiplication by ail. Hence, by (4.1.3)(2), the condition (3.5.16) is satisfied for all v, r, and p,T (4), w, w) - 0 unless r2 = (1, 2).

If T2 = (1, 2), pT = (-1, -1, 1), (-2,1,1) or (-1, 2, 1). These points do not belong to the set {sT 11 + sr11 + 2sr12 - 2sT2 = 0}. Therefore, by (3.5.19), we can replace V) if necessary and assume that "Ep,T (4D, w, w) = 0. This proves the proposition. Q.E.D.

§12.3 The case x = (03) We prove the following proposition in this section.

Proposition (12.3.1) Let p = (x, s) be a path such that x = (/33). Then by changing 0 if necessary, 8p (4), w, w) - 0. Proof. As in §2.2, we only consider p such that s(1) = 0. For )( = (A,, a2) E R+, let ' , ' , A ; 1 , 2 ) . Let g5 = d(A)g° where g°° is as in the first element of (1.1.3). Then dgo = 2d".\1dx\2dga. Let T = (T1iT2) be a Weyl group element such that Tl E 01. Let sT be as before. Consider the substitution p = A \2. Then d(X) = a (

P - 2s,.11+5,12+3 a1 2s,2+1 = 125t h +5,12-45,2-1µ8r2+1

(12.3.2)

Also dx,\1dxA2 = dxXldxp. Let W = Ra(Pp. The functions Oz,(`w,gD),ep(ga,w) do not depend on t13,t21. So E :7p (4P, w, w) = 0 unless w2 is trivial. By (11.1.4), (11.1.5), W, W)

E

E20 1

#(w) 27r

fRe(s2)=r3>i

E1(F, ST2 + 1)AT(w; 28T2 + 1, -1, 8T2)d5T2

,2=1,(1,2)

Let 1 w,

w) =

1)A (w; 2s+ 1, -1, ST2)dsr2. JRe(s2)=r3>1 E1( , 5T2 +

As usual, we can ignore the case r2 = 1. Suppose 72 = (1, 2). If T1 = 1, (2, 3), (1,2,3), pT = (-1,-1, 1), (1, -2,1), (2, -1,1). So in all the cases, pT does not belong to the set {Sr I sill = 2sT2+1}. Therefore, by (3.5.19), Ep,T (c, w, w) - 0 Q.E.D.

§12.4 The case x = (04) Let d1(A1) = dp1(A1) and d2(A1, A2, )3) = a(A 1A21, Ai 1A2, Ai; ', A3) for )11, )(2, A3 E R+ We consider paths po = (xo, so), pll = (x1, 521), P12 = (x1, 522)

such that Do = (i34),x1 = (/34,,34,1), and 521(2) = 0,x22(2) = 1. As in previous sections, we only consider the case so(1) = 521(1) = 522(1) = 0.

12 Contributions from unstable strata

275

Since YO = Za for 0 = 01, ,=pl. (,P, w, w) is well defined for Re(w) >> 0 for i = 1, 2.

Therefore, by (3.5.9), (12.4.1)

po

w, w)

w, w) =

po+(, w, w)

+ po# (D, w, w) - Epo# (4, w, w) + P12 (D) w, w) - PHi (4), w, w).

Easy computations show the following two lemmas. Lemma (12.4.2) ep111(d2(Ai, A2, A3)) = A1, ep,21(d2(A1, A2, A3)) = Al 1.

Lemma (12.4.3) (1) opo(dl(A1)) = Al 2, Kpo1(dl(A1)) = a1 (2) orp 11 (d2(A1, A2, A3)) = A1-2 A22

o'P12 (d2(A1, A2, A3)) = a1 2a3.

w, w), 2po+(4), w, w) In (a), (b), 0 = DDo,p = po, and T = Ra(Dp. Let ga = d1(A1)ga where go is as in the first element of (11.1.3). Then dgo = 2d'AldgO. The element d, (A,) acts on ZSA by multiplication by A1. Therefore, the following proposition follows from (3.4.14) and (12.4.3)(1).

(a)

Proposition (12.4.4) Po+(4b, w, w) - 2CGA(w; p)bao (w)EDo+(Roo'po, wao, -2).

(1)

°po+(4P, w, w) " 2CGA(w; P)bao (w)Eao+(31oo Roo 41po, w0-0 , 6).

(2) (b)

"po#(41,W,w)

Consider r = (r1,1) such that 'r1 E l1. It is easy to see that d1(A1)TZ+P =

A2s.,11+s.,12-3

By (11.1.3), (11.1.4), (12.4.3)(1), (12.4.5)

Epo#(4',w,w) = E T,E`Q,

b#(w)T(0)

27r1

A,T, (w; si11, -1)

fRe(s,ii)=rj>3

6#(w)-FOT(0)

dsTu,

sill - 3

A1T1(w ;8T11,1)dT1E1

JRe(sii )=r1>2

s'11-5

Note that A1T, (w; s,11, -1) 4-1,10; 5T111 -1, -1). Proposition (12.4.6) By changing zV if necessary, (1)

=po# (4), w, w) - -CGA(w; P)T2b#(w)Rao 4'po (0),

(2)

"po# (4), w, w) - -CGA(w;

Rao,D po (0)

3

Proof. Since the proof is similar, we only prove (1). If T1 = 1 or (2, 3), we choose

r1 >> 0 in (12.4.5). Then, L1(rl, -1) << 0 if 7-1 = 1, and Ll(rl, -1) = 2 < 4 if ri = (2, 3). Therefore, we can ignore these cases. If 7.1 = (1, 2, 3), MTl (sill, -1) =

Part IV The quartic case

276

02(sr11 - 1). The point pr = (2, -1, -1) does not satisfy the condition sill = 3. Therefore, we can move the contour crossing this point. So, A1T1 (w, sr11, -1)

/ Re ca,,

C1Sr11

sill -3

rl >3

=

f

J

Re

AiT1(w, sT11, -1) =''1

1<,l <2

dsTll

81-11 - 3

- CGA(w; p)X2

If rl < 2, Ll(rl, -1) < 4. So we can ignore the first term. Therefore, this proves the proposition. Q.E.D.

w, w),-'p12 _ (c) "pll w, w). Let Z) = D1, Ti = Rat 0p1, for i = 1, 2. Let to = d2(A1, A2, A3)t °. Then dxt° _ 2dxAldx)2dxA3dxt°. By (12.4.2), (12.4.3)(2), w(t°)a12A2A3Oz,

(12.4.7) Epll (-P, w, w) = f

(,@,, t°)ep11 (t°, w)dxt°,

TA/1-k

w, w) =

Pie

f

Aj<1

w(t°)-1A A A (`12, t°)ep12 (t°, w)dxt°. 2 OZ, 3

TA/Tk Al >1 2

Let tc = A1A2A3. Then

= A4µ2 and )16A2A =

A4/,t2

Also

dxAldxA2dxA3 = dxAldx.A2dxp. The functions Oz, (Ra4)p11 , t°), f°°p1, (t°, w) etc. do not depend on t11, t21. Therefore,

these distributions are 0 unless w is trivial. Easy computations show the following lemma.

Lemma (12.4.8) (1) ep11 (11201, A2, 3))

-1

2

Tz+p - 2sr11+s.12-sr2-2 Sr 12-3+2 M s.r2-1

A3))Tz+P = A 2s,11-sr12-sr2+4A2,12-8+21Ls,-2-l.

(2) 8P12(d2(A1, "2,

Definition (12.4.9) Let 42i = (lp2i,1, lp2i,2)

where

(1) 1p11,1(ST) = ST2 + 1, 1p11,2(8T) = 8711 - 3, (2) 1p12,1(81) = sT2 + 1, 1p12,2(s1) = ST11 + ST12 - 4.

The function Oz, (Ti, t°) does not depend on A,, A2. So by the Mellin inversion w, w) is equal to

formula, EP2i

(2)

b#(w)

7r

1

2

ffte(s1)j,2) r1,'2>>0

l,sub(

i, 1P2' S)ST12)AT(w; ST1)ST12)dsTl

for i = 1, 2.

Proposition (12.4.10) By changing % if necessary, P12 (4), w,

w) - Ep11(4), w, w)

- CGA(w; p)S#(w) (- E1,(0)(' 2, 2) + El,(l)(I'1, 2) _ El,(1)(T212) 4

2

2

1

J

.

12 Contributions from unstable strata

277

Proof. We define E(1)(4', 8,1) = E1,sub(Wi, 4221 Sr11 Sr12) - E1,sub(Wi) 1P21, 5r1, sT12), E(2)(4,, 5r12)

= E(1)(C 1, sr12)

Then E1(W1, 8,12 +

E(2)( 4i, 5r12) =

1)

+ E1(W2, sr12 + 1)

sr12-3

2

and P12

(), w, w)

=6#(w)

- :Pi, (4), W, w)

)

1

--

27r,/ -l

2

B.(er1)=(T1.'2)

r'(1) (.I,, sr1)Ar(w, 5r1, s-12)ds11.

11,'2>0

We can ignore the case r2 = 1 as usual, and we assume that r2 = (1, 2). If ri = 1, (1, 2), (2, 3), Then L(r) << 0. Therefore, we can ignore these cases, and we assume

that rl = (1, 2, 3), (1, 3, 2) or (1, 3). The point pr does not satisfy the conditions sT11 = 3, srll + ST12 = 4. Therefore, by (3.6.1), we can move the contour crossing these lines by replacing z,b if necessary so that (r1, r2) = (1 + 6,1 + 6) where 6 > 0 is

a small number. If r1 = (1, 2, 3), (1, 3, 2), L(rl, r2i r2) = (2+6)(1+C) < wo = 4+C if 6 is sufficiently small. Therefore, we can ignore these cases also. Suppose that Ti = (1, 3). Then

)21

1

21rV-1 1

e(,,l)=(1+a,1+a) r1,r2>0

ST1)AT (w, 5r1, 5r12)dsrl

E(1)

)21)=(1-61,1+6)

(2) +

ST1)AT(w, ST1, ST12)dsT1

C

27r

E(2) (,D, 5T12)0(ST12)02(Sr12)A(w11, sr12, Srl2)dsT12 Re(sr12)=1+6

JRe(sj2)=i+6

27r

E(2)sr12)O(Sr12)W2(Sr12)A(wi 1, Sr12, Sr12)dSrl2,

where we choose 6, 61 > 0 small and 616-1 >> 0. We proved that E1(W1i 2) = E1(W2i 2) in §3.8. But if w is trivial, 4) is K-invariant

by assumption. Therefore, this is a consequence of (4.4.7) also. This implies that E(1)(4),1) = 0. Since 1

1

8T12 - 3

= -1 -

1(512 4

- 1) + O((sr12 - 1)2),

E(2)(4), 5,12) = E(3)(')(sr12 - 1) + O((sT12 - 1)2),

where El,(0)( 2, 2)

E1,(1)Nf 212)

E1,(1)(IF 1, 2)

4

2

2

Therefore, "a12

W, w) -

p11 (D1 w, w)

^ CGA(w;

Q.E.D.

Part IV The quartic case

278

The following proposition follows from (12.4.4), (12.4.6), (12.4.10), (3.8.10).

Proposition (12.4.11) By changing z/b if necessary, upo (4), w, w) - 2CGA(w; p)sao (w)Eao (Rao 4)po , woo, -2).

§12.5 The case r) = (/35) We prove the following proposition in this section. Proposition (12.5.1) Let p = (Z,s) be a path such that 0 = (/35). Then by changing 0 if necessary, E p (4), w, w) - 0.

Proof. As in previous sections, we only consider a path such that s(1) = 0. Let

IF = R.

Suppose that f (q) is a function of q = (q1, q2) E (A1 /kx )2. Let FO be as before. Let dxq, dxt ° be the usual measures. An easy computation shows that

f

(w(t °)f (71,33(1 °), 72,23(1 °))dxt ° = #(w) fAi

A 1 /k x)5

/kx)2

For µ =0-11, µ2, p3) E R+, we define

d(µ) =d(µ1, 92,93) =a(µ 2 1µa, u1-12

u2/13-2

,µ1iµ3 ;

t,-1,

4

)

Let to = d(p)t°. Then dxt° = dxµldxµ2dxµ3dxt°, and µ1 = 171,33(t°)I, P2 = (t°)-p = 1 1 _ µ1 µ2µ3 , K02 (t°) = p11µ3 2 and I72,23(t°)J. Let r, sT = (ST12, sT11, sT2) be as before. It is easy to see that

(12.5.2)

(t0)Tz P

16

= µl

(5,11+1)L2

(sr11-2sr12+2sr2+1)

Since OZ, (WY, t°) does not depend on /13i by (12.5.2), w(t°)ia2(to)(to)TZ-POZ, (IF,

t°)dpldµ2dxt o

fnx(A1/kX)5 S#(w)µ3

2

-S-12+s,2 -3

E2(

,

2

(sT11 - 1), S,12 T12 + 1).

By the Mellin inversion formula, 1

21rV -1 JRe(8T2)=r3 >0

_

AT(w, ST12,'ST11,

µ3 5r11

2

- S r n+s+2 - 2

AT(w;sT)dsTz

3

+ ST12 + ^ ).

Let EON , w, 5T1) = E2 (x', 2 (ST11 - 1), ST12 + 1)AT(w, ST12) ST11,

5x11

2

3

+ ST12 + -).

12 Contributions from unstable strata

279

Then w,w)

(w)E (2)

2f ///

Eo(IF,w,srl)ds,l.

7r'

ll)=crl ,r2>

rl>3, r2>1, 2r2+1>rl

We can ignore the case r2 = 1 as usual, and assume that T2 = (1, 2).

If T1 = 1, we choose 2r1 = r2 >> 0. Then L(rl, r2i -1121 + r2 + 2) = -2(rl + r2) + 2C << 0, so we can ignore this case. If r1 1, pT is not on the hyperplane + 8r12 + 2. Therefore, we can ignore the rest of the cases by (3.5.19). ST2 = Q.E.D.

§12.6 The case a = (,36) Let Do = (N6), ai = (06, B6,i) for i = 1, ... , 4. Let po = (ao, so), pll = (a1i511)412 = (a1is12) be paths such that 511(2) = 0, 512(2) = 1. As in previous sections, we only consider such paths satisfying so(1) = s11(1) = 512(1) = 0. Throughout this section, T = Rao Dpo Let dl (A 1, A2) = a(A1 1A2, A1

1-2, -1-2

2,

d2(A1, n2, A3) = CL(Al l 2 13 1, Al

-1 3L-2 11 AI 2 ),

12 l)'1-2

for A1, A2, A3 E ]R+.

Let Zao = Vl ® V2 where Vlk = Sym2k2, V2k = k. Then dl(A1i A2) acts on VIA, V2A by multiplication by A1A2,A1A2 5 respectively. Let Xao, kvl etc. be as in §11.1. (a) Epo+(,D, w, w) etc.

Let p = po, a = Do. Let ga = d, (A,, A2)g° where g° is as in the first element of (11.1.3). Then dga = 8d" A1d" A2ds°, and dl(A1i A2)-2P = A12Az 4.

Easy computations show the following lemma.

Lemma (12.6.1) (1) Opo (dl(Al, A2)) _ (2) kaol (d1(X1, A2)) _ Al 4A2 4 "115"2 (3) kao2(d1(y1, X2)) _ "13"21.

The following proposition follows from (3.5.13).

Proposition (12.6.2) po+(,', w, w - 8CGA w p)bao(w)Eao+ tY woo) Xao, -3). "po+(qb, w, w) ^' 8CGA(w; p)Sao

IF, wao , kaolXao 7).

Next, we consider Ep,st+(-D, w, w).

Let to = d2(A1iA2,A3)t°. Then dxt° = 8dxAjdxA2d)A3dxt°. Let Ha,X5 be as in §11.1. Let ga = n'(uo)t°. Then ga E Xa if and only if A3 < a(u0)-2. Let dg-.o = dxt°du0.

Part IV The quartic case

280 By (12.6.1),

Zp,st+(4),w,w) JXo nBA/Tk

w(9a)A13A21ez;o(`y,9a)4 (9a,w)d9a,

=p,st+(4),w, w)= JXDflBA/Tk w()If f (q/) is a function of q = (ql, q2) E (A1 /kx )2,

J(Al/kX)a w(t °)f (72,12(1 °), 71,33(1 o))dxt ° = 6#(w) JAl/kx)2 f (q)dxq. Therefore, these distributions are 0 unless w is trivial. We will show that we can replace fp (go, w) by eN(t°, w) in the above integrals. It is easy to see that 60p(ge,w) is the constant term with respect to Pl x B2.

Lemma (12.6.3) 3A2 1e zo

(1) =p,st+N,w, w) - b#(w) LflBA/Tk

w, w) - 6#(w)

(2)

f

Xa fBA/Tk

(W, 9)N(t°, w)dxt°duo.

Al 7A2 5eza,o (go T, 6a (9a))6ON(t°, w)dxt°duo.

Proof. Fix a compact set C C Al so that C surjects to Al/kx. Clearly, I'y2,12(t°)I = A1A2, I71,33(1°)I = AlAz 5 Therefore, there exist Schwartz-Bruhat functions 0 < %Fj E .5"(A2), 0 < W2 E 9(A) such that W (9ax) << T1(A1A2X2,12, AJA2X2,12uo)'F2(A1A2 5x1,33),

W (ea(9a)x) << pl((AJA3)-1x2,12, (A1A2)-1x2,12uo)T2((AJA2

5)_1x1,33)

for x E Za ok, tip E C. Therefore, for any N > 1, Ozn.o (W, 9a) « (A1A2 5)-N E W1()1A2x, A1A3xuo), xEkX

,9z,,,, (go IF, 9 (go)) <<

5)N

(A1A2)-lxuo) xEkX

By the consideration as in §2.1,

Iep(9a,w) - 8N(t°,w)I «EiI,T,1(90,w), where the summation is over I = (11,12) such that I1 = {1}. So by (3.4.30), there exists a slowly increasing function h(t°) and a constant 6 > 0 such that if M > wo and l >> 0,

I $p(go, w) - 8N(t°, w)I < h(t°)As

for<wo-6
12 Contributions from unstable strata

281

Let X+ = {(A1) A2, A3, u0) E R+ x A I )13 < a(uo)

2

6}-

We substitute µ = A1 A2. Then the differences of the left hand side and the right hand side of (1), (2) are bounded by constant multiples of the following integrals h(t0)A (-N-1)µ3(5N-1)A3

fx+ al

T1(µx,µxuo)d"Ald"µdu0,

1

xEkx >1

`I'1(µ-'x, µ-lxuo)d"Ald µduo.

JX+

xEkx

al<1

These integrals converge absolutely if N >> I. This proves the lemma. Q.E.D.

Definition (12.6.4) (1) lpo,st(si) = (Sill + 2si12 + 3ST2 - 9, -ST11 - 2si12 + si2 + 1, 2 (1 - STll)), (2) lpo,st(sT) = (-sill - 2sT12 - 3ST2 + 13, sTll + 2sT12 - ST2 + 3, 2(1 - ST11))' We define T5+(R5,oIF, lpo,st(si)) ST11 - 1

1po,st(s'r))

sT) _

Sill - 1

It is easy to see that (t0)TZ+P - %2sr11+Sr12+3sr2-6 `-2sr11-sr12+sr2+2 \s,-12-1.

(12.6.5)

If

2

1

3

((1, 3),1)T((1, 3), 1),

A., (w; si),

AT (w; STl21 ST11) ST2)

because z/'(-TGz) _ V)(z). Therefore, by (12.6.5), and exchanging Sill, ST12, p,st+ ('P, W, w)

_

_ =

3 1

86#(W) E (2) frEDr, r1>1 1

86#(W) T

2 7r

Ep,st+(P, ST12) 5T11, si12)Ai(w, si)dsT

3

V) f

Ep,st+(,D)

(w; s1)ds,.

Re (
rEDr, rl>1

Similarly,

(2) f 3

p,st+(,D, w, w) - 86#(W)

Ep,st+((D, Si)Ai(w; si)dsi. Re(ar)=r

rEDr, rl>1

sT) are holomorphic for Re(si11) > 1, the following Since Ep,st+(-P, si), proposition follows by the usual argument.

Part IV The quartic case

282

Proposition (12.6.6) Suppose that r = TG and 6 > 0 is a small number. Then (1)

w) W)

8026#(w)

2xY-1 JRe(sjj)=1+5

Epp,st+(4'f ST11, 1, 1)02(STll)Al(wi srll) 1)dSTl1,

(2) =po,st+(4), w, w) 8p26#(w)

post+(

Le(sii )=1+6

s T11) 1f 1 )2 T11 (s)A

f

(w s T11) 1)ds T11 1

)

(b) uPli (,P, w, w)

Let i) = D1, and Wi =

Ra145P1i for i = 1, 2. Let

32

Al = I-Y2,22(d2(Ai, A2, A3)I = A1A2A3, 12 = I-Y1,33(d2(Ai, A2, A3)I = A1A2

-4

1

5.

3

Then A2 = Al it2 5) A3 = a1 bµi µ2 and dxAldxA2dxA3 = 110 Easy computations show the following lemma.

Lemma (12.6.7)

_ 24

-d"Ald"µld"µ2.

4

(1) opll (d2(A1, A2) A3)) = Al 3A2 'A3 =32Al 1 µ1u2 2 2

(2) op12 (d2 (A,, `A22, `3A3)) = A7A2A3 = A /p1p2 5. (8,12-1) p (43,11+58,12-28,2-7) (3) Boll (d2(^) A2, ^))TZ+P = ^51 (s,11+28,2-3) Al2 µ2 (-3,11 -sr12-23,2+4) 2 (8,12-1) (4) 9P12 (d2(A1, A2f A3))TZ+P 1 µ Xµ1° (-4s,11+8,12+2s,2+1)

-

Definition (12.6.8) (1) lpii,l (sr) = z ST11 + 1),

L10

(5sr11 + 48-12 - 2ST2 + 1), ST12 + 2sT2 - 6).

(2) 1p12,1(ST) = (2(ST11 + 1),10(ST11 - 4ST12 + 2ST2 - 3))ST11 + ST12 + 2ST2 -8).

By (12.6.7), and exchanging STil, sT12,

(2)

Pli (4), w, w) = 6#2w)

3

R.(.,)=

2,sub( i) lpli) ST)AT(w, ST)dsT

rl>jr2, r3>1

for i = 1,2 where the first (resp. second) coordinate of Wi corresponds to

X2,22

(resp. X1,33)- We define 1

1

cE +'P11 (D, STll) = - E2(Ral Dpii) (Srll 6 2

LP12 (D) sill) =

E2(Ral'DP12)

2

+ 1),

1

10

(5sr11 + 3)),

(Srll + 1), !-(Srll - 5)) 2(sll - 5)

Proposition (12.6.9) Suppose that r = TG and 6 > 0 is a small number. Then by changing t/i if necessary, 26

w, w)

2x #

fRe(sii)=l+6

l

ST11, 1)dsi11

12 Contributions from unstable strata

283

fori=1,2. Proof. We can ignore the case T2 = 1 as usual, and assume that r2 = (1, 2). Let r = (rl, r2, r3) = (20, 2, 2). Then by the usual argument, 1

(2) fRe(s)=r 1

2,aub(

i,lplt+ ,)A(w;s,)ds,

2

fRe(si)=(rj,T2)

(2)7r

2,SUb(Wa) 1pli, sri, 1)AiT1 (w; sTl)dsT1.

If 7-1 = 1, (1, 2), (2, 3), we choose rl >> r2 >> 0. Then L(rl, r2, 1) << 0. Therefore, we can ignore these cases. The point p1,1 is (2, -1), (-1, 2), (1,1) for -rl = (1, 2, 3), (1, 3, 2), (1, 3). So p1T1 is not on the lines 5,12 = 4, 5sT11 + 4s,12 = 11 for these cases. Therefore, we can move the contour crossing these lines by (3.6.1) so that (ri, r2) = (1+6,1+6). Then we only have to move the contour so that (ri, r2) = (1 - 61i1 + 6) where 6, 61 > 0 are small numbers and 616-1 >> 0. Q.E.D. W, W)

(c)

Let Z = Zo,p = po, and ZZ' = c02. Let tll = .1a2. Then dxAldx.\2 = Easy computations show the following lemma.

Lemma (12.6.10) (1) d (\ \ Tz+P = 1\^ll /12) \\1

sdxAldxµl.

3(2s..ii+8-12+25-2-5)4 (-2sr11-sr12+s,.2+2)

-

1

(2) ap(d1(A1,A2)) = Al 3µl 3 (3) nD 1(dl(A1, )2)) = Al 3 /.t1 3 .

(4) tv1(dl(A,, A2))) = (A,A3)-3 = tbl 3. 5)-l = Al $tti (5) kv2(dl(Ai, A2)) = Definition (12.6.11) Let 12i = (12i, 1, 12i,2) for i = 1, 2, 3, 4 where (1) 121,1(8,) = g(-s,ll - 2sn12 + ST2 + 6), 121,2(ST) = sr11 + 28,12 + 2sr2 - 9, (2) 122,1(8,) = 3 (-ST11 - 28,12 + ST2 + 1), 122,2(ST) = ST11 + 28,12 + 2ST2 - 7, (3) 123,1(Sr) = 3 - 121,1(8,), 123,2(8,) = 121,2(8,), (4) 124,1(8,) = 3 - 122,1(8,), 124,2(5,) = 122,2(3,)-

Note that 12i(-8.11, sill + 5,12,',2) = 12i(5T11, 5,12, ST2) for i = 1, 2, 3, 4. Let W1

W,' IF,

IF, T4=Ro,9v1IY.

By (12.6.10), and exchanging ST11, sr12, S2w,12i,T('yi, w)

1=P'O"i ((.D+W 1 w) = 26#(w) T1 E!W2

*2=1,(1,2)

for i = 1,2,3,4. The following proposition follows from (11.2.3).

Part IV The quartic case

284

Proposition (12.6.12) Suppose that r = Tc and 6 > 0 is a small number. Then by changing zb if necessary, p,a,,i (41, w, w) - 2CcA(w, p)

6#(w)Ew+(W i, 12i,l(-1, 2,1)) 12i,2(-1, 2, 1)

27r+b

2 + 2P 6#(W)

fRe(srii)=1

Fw,st,sb+( 'P, 12ii

1) 1)

X Y'2(s-11)Al(w, sr11, 1)dsT11

for i = 1,2,3,4. (d) PO,03,i

W, W)

Let 0' = c03. For the cases i = 1, 2, we make the change of variable µ1 =

A1A2, µ2 = A1A2A3 = µ1A3 Then A3 = µ1 2µ2, and (12.6.13)

d2(A1,X2,A3)TZ+n

_ Al3(2sr11+sr12+2s,-2-5) Al

6(-4sr11-5sr12+2s,2+7) 1(sr12-1)

µ2

For the cases i = 3,4, we make the change of variable µi = \i)2, µ2 =

(A1.2)-'A2 3 = µ1'A3 . Then )3 = µi µz , and (12.6.14)

d2(A1,X2,X3)T=+a

(-4sr11+3712+2sr2+1) 1 (s, 12-1) g (28-11 +Sr 12+2972-5) P2 Pi6 I1

In both cases, dXA = 1dXAid"µidXµ2 It is easy to see from (12.6.1) that vp(d1(A1, )2))A3 = Al 3µi 3 µ2 for the cases 8

2

i = 1, 2, and 0 p (dl (A1, A2))A 3 = Ai 3µ1µ2 for the cases i = 3, 4.

Definition (12.6.15) Let 132 = (2 (8,11 + 1),131,2) where (1) 131,2(Sr) = (5,-11 + 2Sr12 + 2sr2 - 9)(5ST11 + 4sr12 - 2sr2 - 9), (2) 132,2(S,-) = (s,-11 + 25r12 + 2sr2 - 7)(55-11 + 45-12 - 2sr2 + 1), (3) 133,2(5,-) _ (s,-ll + 2Sr12 + 2sr2 - 9)(5.11 - 4sr12 + 2sr2 - 3), (4) 134,2(5,-) = (5-11 + 25r12 + 25-2 - 7)(5,-11 - 4sr12 + 2sr2 - 13).

Let

IF1 = Ra' Fv1 Fa'I','Y2 = R1,'Y, T3 = Ra,.`'a'IJ,'F4 = Ra,.rv1 T. By (12.6.1), (12.6.12), (12.6.13), and exchanging 8711, 8712, W, w)

_

- 66#(W) for i = 1,2,3,4.

(

\2n

1

3

/

f

Re(sr)r1>'2, r3>1

sr)AT(w, ST)dsT

12 Contributions from unstable strata

285

We define sT11)

EP31

Yyjga'1', 1(ST11 + 1)) - 6E1(Ra3 = (ST11 - 5)(5sT11 - 7)

6E1(Ra3'I,

(,`i', 7

ST11) -

EP32

2(sTll + 1))

(sTn - 3)(5sT11 + 3) 6El(Ra39-o 'I',1(sr11 + 1)) , EP33 (), ST11) (sT11 - 5)2

6E1(Ra3gyiIF, 2(sr11 + 1)) P34

,

T11 (sr11 - 3)(sT11 - 15)

The following proposition follows from cases (7), (8), (13), (14) of (11.2.4).

Proposition (12.6.16) Suppose that r = TG and 6 > 0 is a small number. Then by changing 0 if necessary, 26#(w)

(I, w, w) -

ko

21rN

1

3T11)02(3T11)A1(W, 8r11,1)dsT11

fRe(sii)=i+fi

fori = 1,2,3,4. (e) °Po,a4,i (4), w, w)

Let 0'= N. Let 112 = Al , p3 = (A,A )-3. Then dxAldxA2 = 40dx p2dx p3. Let 11 = (A2, A3), and dxµ = dxp2d"p3

Let T = (Tl, T2) be a Weyl group element such that Ti E 01. The following lemma is easy to prove and the proof is left to the reader.

Lemma (12.6.17) (1) ap(d1(A,, A2)) = 112

356

(2) ial(dl(A,,A2)) = µ21µ3 9

(3) kv1(dl(Ai,A2)) = 113 (4) kv2(dl(A1,A2)) = µz 1µ3

It is easy to see that (12.6.18)

[dl(Al, A2)

Tz+P

s,ll+s,2-3 113 1(2x,12-sr2-3)

13,12=-1 = 112

Let `Y1 = RD,goIF, W2 = Rio, 9v2IF, T3 = Ra,.Fv29a'y, W4 = RD'W. We define distributions Ep,a,,i (C w, s,.ll, 5T2)'S for i = 1,2,3,4 by the following integrals in that order. 36#(w) ,11+s,2-5113 (23,11-3,2+2)el(1y1, pdxt, µz

5

JR2 XAl/kx

-

A2!51, 113 <1

36#(w) 5

112,11+s,2-5P3(2s,11-s,2-7)e1('y2,(

1RXAh/kX

-23

µ2G1, µ3<1

36#(w) 5

112,11+8,2-4113(2x,11-s,2+7)61(W3,µ

fRxA1/kX

11 t)dxpdxt, -2-3

µ2G1, µ3>1

36#(w) 5

112,11+s,2-4113(2x,11-s,z-2)01(4,112µ3t)d"11d"t.

fRxAh/kX 112<1, 113>1

-

µ)-lt)dxpdxt,

Part IV The quartic case

286

We define w, w, ST11, sT2) _ p,a',i

F'p,a',i

w, ST11f ST2)AT(w; sill, -1f sT2)

Let s'. = (STii, ST2) and dsT = dsr11dsT2. By (12.6.17), the Mellin inversion formula, rY-1)2f

p,e',i (4), w, w) _

e(s)=(r,.r2)

27

T1 E 9]7 1

40"i

w, wf

s/T)dsITf

T2=1.(1,2)

where we choose the contour so that (rl, -1, r3) E DT, and r1 >> r3 >> 0. For A E ] and i = 1,2,3,4, we define via (x) = WYi (,fix). Clearly, 113 (resp. qf4) is the Fourier transform of 11 (resp. W2) with respect to the character < >. Therefore, for any c1i c2 E C, 1

0 0

1

1

Act-1Ej+(1f'ia-1,1 - c2)dXA

A`1E1+('Fi+2a,c2)dxA+1 0

1 C1 - C2

E 1 (Ti+2A, C2) +

'Pi+2(0)

_

q1i(0) (Cl - 1)(c2 -

C1C2

1).

Hence,

p,a',1

w, 8r11, ST2) + p,a',3 w, sill, ST2) = b#(w) E1(W3, 5 (2ST11 - ST2 + 7)) sri1 + 2sT2 - 9

3(0)

+3(J#(w)

(ST11 + ST2 - 4)(2sT11 - ST2 +

7)'

q/ (0)

- 36#(w)(ST11 + sT2 - 5)(2ST11 - 8T2 + 2)' p,a',2

w, ST11, ST2) + Ep,a',4 (4, w, ST11, ST2)

E1( 4, 1(2sT11 - sT2 - 2)) ST11 + 2sT2 - 6

'1'4(0) (ST11 + ST2 - 4)(2sT11 - 8-,2 -

- 3b#(w)

2)'

gi2(0) sT2 - 7).

(ST11 + ST2 - 5)(2sr11

Proposition (12.6.19) By changing 0 if necessary, (1)

w, w) + "p,a',3 (1), w, w) ^ -CGA(w; p)

2d#(w)E1(Ra' IF1, -1) 5

+ 3CGA(w; P)b#(w) (2)

ap a ,2 ($f w, w) +

p,a 4 ('Dww)

--CGA(w, P)

(_T3(0) 10

+'F

100)

IZ26#(w)E 1 (Ra,T4 , - g1) 2

- 3CGA(w; p)b#(w) (w4(0)

- _28(0)

I .

12 Contributions from unstable strata

287

Proof. As usual, we can ignore the case T2 = 1, and assume that r2 = (1, 2). By the usual argument, 1

27rV---i P

f

w, w, sT) + Ep,a',i+2 (4', w, w, ST)) dsT

e(sr)=(rl,r3)

Res (Fip,a',i (4p, w, w, sT) + Ep,a',i+2 ((D) w, w, sT)) dsT11 J e(s,ll)=r1 sr2=1

21r

for i = 1, 2. The poles of the function ST11, ST2) +

ST11) 81-2)18

2=1

for i = 1, 2 are si11 = 2, 2, 3, 4, 6, 7. But pi, does not satisfy these conditions. Therefore, by (3.6.1), we can move the contour so that r1 = 1 + 6 where S > 0 is a small number. Then L(1 + 8,1,1) < wo if ri = 1 or (2, 3). So we can ignore these cases. Suppose 7.1 = (1, 2, 3). Then by the usual argument, g

27r\ -1

Res (Ep,a',i J e(srll)=1+b a*z=1

^' CiGA(wi P)932

w) w) sT) + Fp,a',i+2 w) sT

w, sT) + Ep,a',i+2

w, w, sT)) ds111

=(2+1)

Q.E.D.

(f) Now we combine the computations in this section. Let Z = 00, p = po We define J1(4), w) = 8ba(w) (Ea+('1`)w,,X,-3)+Ea+('a4')wa 1,ka11Xa 1,7)) +'x126#(w) 8#2w)

J2(4', w)

(_E1(R4, -1) + 2E1(Ra4'Y)-5))

(Ew+(Rw 62

+ 8# (w)

)

3)

3

1

(Ew+(R2I' ) - 3) + Ew+(JlwRa2'1',

+3'2128# (w)

+3'x128#(w)

(D4R4W(o)

3

))

R04 ."DWY(0))

+

8

10

(gNRa4 -q 10 aW(0)

10

+ R14 'F(O)

TO,1(1))w) = Ji('P,w)+J2(4),w), J3(`) s,11) _

8Ta+(Ra,o'P, sill - 41 -sT11) 2 (1 - ST11)) ST11 - I

8Ta+(Ra,o-'a'I', 8 - si11) sTil + 4, 2(1 - ST11))

+

J,4 (4k 8T.l )_

ST11 - 1

2Tw+(Rw,oRa2ga'Y) 1(s,-11 + 4), 2 (1 -s,11 ))

)

(sill - 1)(sTll - 5)

+ 27'w+(RwogwRa2. 'a'Y) 3(5 - sill)) 2(1 - ST11)) (si11 - 1)(sii - 5)

Part IV The quartic case

288

J (ffi)- - - 2Tw+(Rw,oRa2'y, -1, a (1 - sr11)) s

(sr11 - 1)(sT11 - 3)

2 (1 - sTii)) (sT11 - 1)(sT11 - 3) 5 Ta,2(-D)w,ST11)

i=3

4 6#(w)\-1)z+1F'V3i

+

sT11) i=1

The following proposition follows from (9.1.10), (12.6.2), (12.6.6), (12.6.9), (12.6.12), (12.6.16), (12.6.19).

Proposition (12.6.20) Suppose that r = rG and S > 0 is a small number. Then by changing zb if necessary,

O (4i, w, w) -CGA(w; p)To,l (,D, w) N

+

27w-1 Re(srll)=1+6

Ts,2(4),w, sT11)02(sT11)A1(w; ST11, 1)dsT11

Let Ta,2,(i) N, w, 1) etc. be the i-th coefficient of the Laurent expansion. We define

J6(4i) = -4 (Ta+(Ra,o1F, -3, -1) + Ta+(Ra,oFF'I', 7,5)),

J7(f) =

4

4

(TW

5

owR52W, 3) + Tw+(Rw,o9sRa2`y, 3)

-1(

3-))-

2

It is easy to see that d

J3,(o)(c,1) = J6(4)) + 8

+8

d

dsr11 S.11=1

J4,(o)(4),1) = J7(4)) + +2

+2

To+(R1,o'I', sr11 - 4, -sr11) 0)

d d

d dsr11

J5,(o)(41,1) = J8(4)) - 2

To+(Ra,o9a1W,8-sr11,sr11+4)0)),

3(5 - s,-11), 0)

d

dsrii

sr11 - 5

s+11=1

Tw+(Rw,oRa2 ga `I', (8rii + 4), 0) s,11-5s sr11=1 d dsT11 sr11=1

d

- 2dsr11 Is+11=1

,

Tw+(Rw,oRa2'I'1 - iLu 3 ) 0)

s,,11-3 s (si11 + 9), 0)

si11 - 3

12 Contributions from unstable strata

289

The distributions To+(RD,o'I', s111 - 4, -sr11, 0), Tt+(Ro,o_IIo'1', 8 - 5rii, `ST11 + 4, 0)

are equal to the following integrals A ''1-5A2 `11-3e2(R',o',AlA241, A1A2

'Rx(A' /kX)2 al>1

ai-sr11 AZr11+1@2(R1,o

JRx(A1/kX)2

"o'y, 1A241, A1A2

542)d" A1dxA2d
a1>1

where the first (resp. second) coordinate of R',oxF,R',o.3'0' corresponds to x2,12 (resp. x1,33). For IF E .Y(Za oA), we define ,Fo,p'1'(x2,12, X1,33) =

JA, (y2,12, Y1,33) < x2,12y2,12 + X1,301,33 > 42,1241,33-

Then if (b is K-invariant, R',ogoW = 9,',oR',oY. For W E 9'(Za oA), let R1, R2 be the restrictions to the first coordinate and the second coordinate respectively. Also let (12.6.21)

FMx2,12) = f 'y (y2,12, 0) < X2,12Y2,12 > dy2,12, A

F2'y(x1,33) = f i (0, y1,33) < x1,33y1,33 > dy1,33 A

Lemma (12.6.22) Suppose that 4D is K-invariant. Then 8To+(Ro,o'I', 5.u - 4, -sr11, 0) + 8To+(Ro,oA-o'1', 8 - sr11, sr11 + 4,0) = E2(R',o'1'+ 2- 2 (s,11 - 7), 2(5.11 - 3))

2E1(R1Ra,oAoIF,3(3.11+1)) 2E1(92Rao 0IF,s(s.ll+6)) 8Th - 9 3.11 - 5 2E1(FiRa,o'I', 3(s.11 + 6)) 2E1(R2Ra0o'1`, 5(srll + 3)) + s,11-7 S,11-3 + Proof. By (6.1.2), e2(R',o'I', A1A241,

542) - Al 2A2e2('` a,0Ra,0W, Al 1A2 391 1, Al 1A242 1)

_ A1 1Aa01(F1R',o9o'1', A1A291) + a1 2A201(R2R',o` oW, A11'X242 1)

- O1(RlRa0o'I', a1a241) - A11A581(92R',o'f','1 1Az42 1)

=A 2A2e1(R1R',o9oW,'l 1'2 3q1 1) + Al 1A2 391(92R',o.o'It, A1A2 542) A-'A 3e1(91R',o'I', A'A2 3q1 -1) - e1(R2Ra,o'I', A1A2 5q2),

-

Part IV The quartic case

290

We divide the integral according as A2 > 1 or A2 < 1. If A2 > 1, we use the first equation, and if A2 < 1, we use the second equation. Then (12.6.22) is equal to the summation of 1 1 E2(Ra,o'y, 2 (s1-u - 7), 2 (8,11 - 3))

and the integral of the following function . sj'3 -6 asT1 -7

-81-3 +2)

El+

1Ra,o

(

,

E1+ (R2Ra,o9a)a1 1'

-s1-5 - 11

(

+ ai'31-5

J

-sin - 3al

E1+

-s.-

,

___ -6

+2

5

81-7

11

A31T1

s'11

E1+ ((R1Ra o

3

s,11+6)

_____6

1

E1+

5

+

_11-3 -6 E1

5 I

s113+ 6\

91R' T

sT15+3\

over Al < 1.

By the principal part formula for the standard L-function in one variable, _8,11-6

-81-11 +2)

E1+

3

3

s 11-7

+

A1T3

E1+ (R1Ra o9a1')a 1

81-13+ 1)

J

4er1 ] -20

81-13+ 1)

E1

(R1Ra 0'Qa 3 A r" 691Ra 0. 5 '(6)

-8r11+2

+

A1,1-7R1Ra 0g5W(0)

+

8r11+1

-sr 5 - 1) srl1-6 1

+

5

8r11 + 6l

E1+

S

4sr1, -36

Al

5

/

(2R,0w,81-15+6)

E1

girl'

Airy'-7R2Ra ova (6) 81-11 + 1

+

69J2Ra ST11 +6

12 Contributions from unstable strata A81'11-5

291

-sT 1-3

Ei+ (RlR,o'')A1

J ST113+ 6)

A4s,11-12 1

sT11 +6 E1

3

3

As-"-SRiR'

A1iR',o'y(o)

oW(0)

+

s,11+3

sTii +6

-sT5 +2)

5

s,ll-5 1

+

E1+

+

srii +3

(R2Ra,o`I'),\,

A4sr11-28

i

l/

E1 (R2Ra o'M, sT11 5

5

+3

5

o'y(o) +

A

A1r1l-5R2Ra oy(0)

-s,,11+2

sT11 + 3

is the Fourier transform of Note that standard bilinear form on A etc.

with respect to the

If 4) is K-invariant, 1Ra,0JF,1@(0) = 92Ra,oY(0),

R1Ra,o9a''(0), RiRa,o'I'(0) = R2Ra,o'I'(0),

o(0). Now (12.6.22) easily follows from these considerations. Q.E.D.

If P is K-invariant, by the principal part formula for the standard L-function in one variable, Tw+(Rw,oJFwRa2 JFa iY, 3 (5 - sy11), 0) + Tw+(Rw,oRa2 9-a`y,

3

(8T1i +4),0)

+ 1)) + 3E1,(-1)(Ra3YwRa2.F'a'I',1) = E1(R1Ra,o9a'Y, 1(s-r11 3 2 - sill

+ 3E1,(-i)(Ra3ga'I',1)

3

8,11 + 1

Tw+(Rw,oRa2'I', 1

= E1(J91Ra,o`I',

3

s

0) + Tw+(Rw,ogwRa2 `F, 3 (srii + 9), 0)

(s,11 + 6)) +

3E1,(-1)(Ra3W,1)

s,11+3

,

3E,,(-,)(R D39wRD2 'F' 1) ST11 +6

Part IV The quartic case

292

Note that if oD is K-invariant, 5(ST11 + 6)) = Res E2(Ra, (P12) S') S'=1

--(Sr11 + 1)),

5

5(Sru + 3)) = Res E2(Ra,,kp,,, s', 5(s,11+3)).

E1(R2Ra,o4',

Therefore, 9

Ji,(0)(b) _ i=3

Ji(.D) i=6

-

E

1, 5)

5) +

E2,(-l,o)(Rs, p,2' 1, -5)

32

- 3El,(-1)(Ra3IF,1) + 98 15

E1,(-1)(Ra3..WRa2'y,1)

15

+

3E1,(-1)(Ra3-gra4',1) -

E1,(-1)(Ra39wRa2ga'y,1),

8

where 2E2,(l,o)(R'a OWY, -3, -1) + 1 E2,(o,1)(Ra,OWY, -3, -1).

Also,

1, 5) - 6E2,(o,o)(Ra,4'p,,5), Ep,,,(0)(41,1) = - 1E2,(-1,1)(Ra,Dp,,) 6 40E2,(-1,1)(Ra,4Dp,211) --)

EP12,(0)(D,1) =

- 8E2,(o,o)(Ra,

16E2,(-1,o)(Rol iDP121 1, -5)

11 -5),

EP31,(0)(C 1) =38 El,(-1)(Ra3J"wR02ga`I',1) + 4E1,(o)(Ra:,-FWRa2g0I',1),

EP32,(0)(4,1) =3E1,(-1)(Ra34',1) - 3El,(o)(R53W,1), 3

Ep33,(o)(p, 1) _ 8E1,(-1)(Raa.

3

W, 1) + 8 E 1,(0)(Ra3 ga'1', 1),

E133,(0)(b, 1) =42 E1,(-1)(R039wRa2'y, 1) +

4 E1,(o)(Ra39wRa2'1', 1).

We define 5) - 8E2,(o,o)(Ra,4p12, 1, -5)

+ 1E2,(-1,1)(Ra,4tp,,) 1,4)

40E2,(-1,1)(Ra,obp,2'1,-5

- 8E2,(-l,o)(Ra,4)p,,,1, 5) + 3E2,(-l)o)(Rs, P12,11-5),

12 Contributions from unstable strata 9

3

J11(I)) =16

+

293

E1,(-1) (Ra3 J'a `I', 1) + 8 E1,(o) (Ra3 -11'a T, 1) 9E1,(-1)(Ra3J"'wRa2.a`I',1) + 3E1,(o)(Ro39w Ro2goT,1) 4

4

- 2 E1,(-1) (Ra3 W,1) +8 E 1,(o) (Ra3 IF, 1) - 98E1'(-1)(R039wRa2I1`,1) - 14 E1,(o)(Ro3.` wRa2W,1). By the above considerations, we get 11

JJ(-D)

Ta,2,(o)(4),w,1) = 6#(w) -s

By the principal part formula (4.2.15), J2(4)) + J7(4)) + J8 (,D) +

Ew,ad(Ra2 IF, - 3) -

-Ew,ad(Ra29a IF, 3 ).

Therefore, Ta,l (b, w) + Ta,2,(o) (4), w,1)

= J1(4, w) + 6#(w)(Js('D) + JA (f) + +6#(W)(EW,ad(Ra2IF,-3) - 2EW,ad(Ra29 W,

3))

By (6.3.11), we get the following proposition.

Proposition (12.6.23) Ta,1('', W) + Ta,2,(o) (4),w) = 86a (W)Ea,ad(RaIp, wa, Xa, -3) + 6#(w)J9((D)

§12.7 The case t = (37) Let p = (0, s) be a path where 0 = (37). As in previous sections, we only consider such path such that s(1) = 0. Let W = Ra,Dp throughout this section. Suppose that f (q) is a function of q = (ql, q2, q3) E (Al /kx )3 An easy consideration shows that

I

A1/kx)5

w(1 0)f (71,33(1 °), 72,13(1 °), 72,22(1 0))dxlo

= 6a(w) f

A1 1kX)3

w(q)f(q)dxq.

Let

d(p) =d(µ1, µ2, 93) =

-3

z , µ2

1

µ2 µ3 u2

Let to = d(µ)1°. Then dxto = sdxµdxl0 , and Al = 171,33(t°)I, µ2 = I72,13(t°)I, µ3 = h2,22(t°)I

2

)

Part IV The quartic case

294

Let r, sT be as before. Easy computations show the following lemma.

Lemma (12.7.1) (1) K02(t°) = P1 2 µ2 213 2. (srll+sr12+2)A2 (srii+2sr2+3)µ3 (srii+3sr12+2sr2+6)

(2) (to)-'-P = p1

Definition (12.7.2) Let lp = (lp,1i lp,2i lp,3i 1) where 2(ST11 + ST12 - 1), lp,2(Sr) = 3(srll + 2ST2 - 3), lp,3(ST) = 6(Sr11 + lp,l(ST) 3s112 + 2ST2 - 3).

By (12.7.1),

(12.7.3) f

w(to)IGa2(to)(to)TZ-peZ, (,I, to)dxto =

ba(w) 6

TA /Tk

LetD={rIr1+r2>3,rl+2r3>4, s + 2 + 3 >

( ap (`I', w, w) = 6a (w) E

1

'-

By(12.7.3),

3

27r v"- 1

T

2}.

E3,sub(1I') lp, wa, s.)

JRe(s)=r

E3,sub(WI', lp, LID, ST)AT (wi ST)dsT,

where we choose the contour so that r E D fl DT. We define Ta,2(4>5 w, ST11) =

ba (w) STll E3(`I', WO) , 1 6 2 3

(S,11+2)). (Srll - 1), 1 6

Proposition (12.7.4) Suppose that r = TG and 6 > 0 is a small number. Then by changing V) if necessary,

P26a(w) f

U), W)

27r

Ta,2((D,w,ST11)02(STl0Al(w,8T11,1)ds,11.

Re(s,ii)=1+6

Proof. We can ignore the case r2 = 1 as usual, and assume that rl = (1, 2). The pole of E3,sub(`P, lp, wa, Sr) in the set is, I Re(srl) = (6, 3), Re(s12) > 0} is Sr2 = 1. So 1

27

e(sr)=(6,3,2)

)3j

1

F+3,sub(W, lp, wa, ST)AT(w, s1)ds1

27r-1 +0

E3,sub('I')lp, wa, s1)AT(wi ST)ds,

e(sr)=(6,3,2)

)21 E3,sub(lI, lp, wa, sr1, 1)A1T1(w; s11)ds1l.

(2) 1

Since C > 100, Z(6,3, < 18 + C < 4 + C. Therefore, we can ignore the first 2) a term. If rl = 1, (1, 2), or (1, 3), we choose r1i r2 >> 0. If Ti = (1, 2, 3), we fix r1 = z and choose r2 >> 0. If rl = (1, 3, 2), we fix r2 = z and choose rl >> 0. For these cases, L(rl, r2, 1) < wo. Therefore, we only have to consider the case r = TG.

12 Contributions from unstable strata

295

The pole structure of E3,sub('F, lp, wD, sTl,1) is as follows.

The point plr, is not on the lines sill = 4, 3sr12 + Srl1 = 1, 7, 8T12 + 8T11 = 3. Therefore, by (1.2.1), we can move the contour crossing these lines so that (ri, r2) = (1+6,1+S). Then we only have to move the contour so that Re(sT1) _ (1+6,1-61) where S, 6l > 0 are small and 616-1 >> 0. Since E3,sub(W, lp, wD, Sr11) 1, 1) = T0,2

w, ST11),

the proposition follows. Q.E.D. Let T8,2,(i) (b, w, 1) be the i-th coefficient of the Laurent expansion. The following proposition is clear.

Proposition (12.7.5) 2,0, 2)

T0,2,(o)(4,w)

+ +

S(w1)6(w2) 4

S(wl)S(w2) 12

1

E3,(l,-i,o)(`1', (w2, 1, w2), Z, 0, 1

1

2 1

E3,(o,-l,l)(F,(w2,1,W2), 2,0, 2).

§12.8 The case 0 = (,138)

Let Do = (08), and Di = (,38,,38,i) for i = 1, ... , 6. Let Po = (Do, 5o). Let Pil = (Zli, 5il ),Pit = Oi, si2) where sil (2) = 0, si2(2) = 1 for i = 1,2,3. As in previous sections, we only consider such paths such that so(1) = sil(1) = Si2(1) = 0. Let' = RDo4)po throughout this section. Let -1-2i l dl(Al,.A2) = a(A1-4A2 2 2) -1-2,

-13-2,

Cl2(A1, A2, A3) = Cl(A1

4-2 2) -1-2-3 1, -i'2L13i

a1A22).

Part IV The quartic case

296

Let Z-0 = V1 ® V2 where Vlk = Sym2k2, V2k = k2. Then d1 (A1, A2) acts on VlA, V2A by multiplication by A1 A2, A1A2 3 respectively. Let Xa,'w1 etc. be as in §11.1. (a) Epo (41, w, w)

Let ga = dl(A1, X2)ga where ga is as in the second element of (11.1.3). It is easy to see that dga = 14d
Let r, sT be as before. An easy computation shows that (12.8.1)

d2(Al,A2,X3)

TZ+P

2s*11+4sr12+33,-2-9X23,-11+23x12-23,-2-183"' -1 a

= Al

Easy computations show the following lemma.

Lemma (12.8.2) (1) o,,. (d, (A,, A2)) Al A2 A15A26. (2) Kao1(d1(A1, X2)) = A121 (3) kao2(d1(A,, A2)) _

Since Za'k = Zak, the following proposition follows from (3.5.13) and (12.8.1).

Proposition (12.8.3) (1)

=p,,+ (1), w, w) - 14CGA(w; p)6,0 (w)Eao+('1', wao, Xao, -3),

(2)

=p,)+ ((P, w, w) - 14CGA(w; P)bao (w)Eao+(9ao'I', wao , Kaa1x

(b) '

, 8).

p1t (1D, w, w)

Let Z = D1. In (b)-(d), t° = d2(A1,'2, A3)t °. Let µl = I'Y1,33(t°)I = A1A2 A3 , µ2 = I72,12(t°)I = A1A23A-'. 3

Then A2 = Al µl 2µ2

A3 = X1 2µiµ2 It is easy to see that

d"t° =

7dxAjdx121dx122d"to.

Easy computations show the following two lemmas.

Lemma (12.8.4) (1) ep;1(d2(X1, A2, A3)) = A1, ep:2 (d2(A1, A2, A3)) = Al 1 for i = 1, 2, 3.

Lemma (12.8.5) (1) ap11 (d2(A1, A2, A3)) = Al 4A A3 = 121 2 (2) ap12 (d2 (A,, A2, A3)) = A1A2A3 = A IL1 2122 5 A3))TZ+P= \73,-12-7 s.ll-S,-12+S,-2-1 3x11-23,-12+2sT2-1. (3) O,,, (d2(A1, A2, 122 121 1 (4) eplz

A2, 3)) Tz+P

3s,-11+2x,-12-2s,-2-3 = 1-78T11-73,12+14 2s,-11+s,-12-s,-2-2 122 121

Definition (12.8.6) Letlpli =(lpli,1ilp1i,2,lp1i,3) where (1) 1p11,1(ST) = -ST11 + ST12 + ST2 - 2, lpi,,2(ST) = -2ST11 + ST12 + 2sT2 4, 1p11,3(ST) = ST11 - 1,

(2) 'p12,1(ST) = ST11+2ST12-ST2-4, lp12,2(ST) = 2s111+3sT12-2sT2-8, lp12,3(s1) = ST11

+ ST12 - 4.

12 Contributions from unstable strata

297

By (12.8.1), (12.8.4), (12.8.5), and exchanging sr11, sr12, pli(4, W, w)

3

- 6#(W)

1

27rV

R.(-)=r

1

_ E2,sub(Ral-Ppli) lpii, sT)A'r(wi sT)dsr,

'1>'2, r3>1

where the first (resp. second) coordinate of R51 4)p1i corresponds to x1,33 (resp. x2,12) We define E2(Ra1,Dpii, -51-11, -2sr11 - 1) Sr11 - 1

Srll)

Ep11

where the first (resp. second) coordinate of Rat qDpll corresponds to x1,33 (resp. x2,12)

Proposition (12.8.7) Suppose that T = TG and 6 > 0 is a small number. Then by changing z/i if necessary, (1) =p11 (p, W, W) V (2) =p12

10 26#(W)

271-/

,

/Epi1 Re(s.,11)=1+6

(D, srll)02(sr11)A1(wi 8,11, 1)dsr11,

6#(W)£'2(Ro1'Pp12, -2, -5)

w, w) ^' -CGA(w; P)

2

where in (2) the first (resp. second) coordinate of Rat dopii corresponds to x1,33 (resp. x2,12).

Proof. The proof of this proposition is similar to (12.6.9). The only difference is that we have to check that the points (2, -1), (-1,2), (1, 1) are not on the lines 5T11-5712 = 1, 2, ST11-2ST12 = 2, 3, 25r11+5T12 = 6, 3Sr11+25x12 = 10, 11, 5x11 = 3

in order to use (3.6.1). Q.E.D.

(c) 'P2i ('P, W, w)

Let Z = 02. Let Al = I71,23(t°)I = .\i-4, µ2 = I72,13(t°)I =.1)23A3.

dxto = 7dxAldxµldxµ2dxto. 2

Easy computations show the following lemma.

Lemma (12.8.8) (1) up., (d2(\1, A2, \3)) = Al 4A2 2 = al 2µl 2 (2) up. (d2 (A,, )2, .\3)) = A1 (3) OP21 (d2(A1, A2, A3))

TZ+P = A z (5,.12+s, 2-2) z (2sr11+S, 12-8.1-2-2) 31-11-1 P2 1 N1

Part IV The quartic case

298

(4) Bp22 (d2 (A,, A2, A3)) T2+p == Al

2(-ST11 -8,12-S,-2+3) 21 (3,-11-3,12+3,2-1)

Al

3,-11-1

µ2

Definition (12.8.9) Let 42i = (lp2i,li 42i,2i lp2i,3) where (1) lpzi,1(ST) = 2 (2ST11 + ST12 -sT2 - 3)f lp21,2(ST) = ST11 - 1) lp21,3(sr) = S,-12+

S,2-3, (2) lpzz,l(ST) = z(ST11 - ST12 + ST2 - 1)f lpzz,2(ST) =sill - 1, lp2z,3(ST) = ST11 5T12 + ST2 - 5-

By (12.8.8), and a similar argument as in previous sections,

)1

F(

_

3

1

= 6#(W)

lp2i'sr)AT(w; sr)dsr,

>'2, ,-) 1

T3>1

where the first (resp. second) coordinate of Ra2 4DP2i corresponds to x1,23 (resp. X2,13) for i = 1, 2

Let 4)(21 E 9'(A) be a function defined by the formula 4)p21 (x1,23) = f W(O) x1,23, x1,33, 0, x2,13)dx1,33dx2,13

If 4) is K-invariant, D(1) is the Fourier transform of e22 with respect to the character < >. We define 4)p22 similarly for Ra21DP22

We define 1

Epzi (4)f sT11) = E2(Ra2 (bp21) 2 (2sr11 EP2 2 1

-3),s,-11 - 1),

(,D, sTll) = r12(Ra2 ADp21 I ST11 - 1, ST11 - 1)) E2(Ra24)V2z) 2(sT11 - 1))sT11 - 1)

E P22 (4)7 ST11) =

sill -3

Let '.217' = {(1, 2, 3), (1, 3, 2), (1, 3)}.

Proposition (12.8.10) Suppose that T = TG and 6 > 0 is a small number. Then by changing V if necessary, (1)

:P21(rf WI w)

-

26# w

()

27rVw-1 JRe(sii)=1+b 6#

+ T'=(T1 .(1.2))

Ep21 (D, ST11)02(ST11)A1(w; sf11, 1)dsr11

W

2

fRe(s,11)=ri

21r V'

X121 ((b) ST'11)A17i (w) sr'11, 2)dsT'11

,-1 E'ID'

E1('Dp211 22.

06#(W)

+

T'=('-

1,2)) ,-i EW

2

1

fRe(s,12)=r2

5T'12 - 2

A1T1(w) 2, ST)12)d8r'12,

12 Contributions from unstable strata

299

(2) ''P22('P)W)w)fRe(sii)=1 X26#(W)

() s711 )2(sT11)A1(w) sll) 1)ds

11

+6 P221

+

2

2

A171(w;2,s7'12)ds7i12

rl en,

Proof. We can ignore the case r2 = 1 as usual, and assume that T2 = (1, 2). Let r = (rl, r2i r3) = (10, 2, 2). By the usual argument,

(2) f5= 1

-p

5u(2

2,T2) 1

P2,' 1P2i) 8T )AT (w; s7)ds7

'r

2

fRe(si)=(ri

E2,sub(R52

P2i)1P2iIsr1,1)A171(w;sr1)dsrl

fori=1,2. The pole structure of E2,5Ub (R52 -DP2i )1P2ti) 871,1) for i = 1, 2 is as follows.

(1+S,1+S)

H

am.

If Tl = 1, (1, 2), (2, 3), we choose rl >> r2 >> 0. Then L(rl, r2, 1) << 0. So we can ignore these cases. The point pi, is (2, -1), (-1, 2), (1,1) for Tl = (1, 2, 3), (1, 3, 2), (1, 3). The point p171 is not on the lines 2s711 + ST12 = 4, 6, srll = sr12 + 2, 8711 + 5712 = 4 for these cases. So we can move the contour crossing these lines by (3.6.1). We move the contour to Re(sr1) = (1+6,1+6) for i = 1, and Re(s71) = (1+6,1+ s ) for i = 2. In this process, we have to cross the lines s711 = 2, s712 = 2, and the condition of (3.6.1) is not satisfied for these lines. When we cross these lines, we pick up the third term of (1), and the second term of (2). If Tl = (1, 2, 3), (1, 3, 2), 1

(2)7r V

2

-1

JRe(si)(i+6,1+6)

E2,SUb(Ra2'1 P21 ) 1P 211

5T1)1)A1r, (w; ST1)dsT1 - 0,

Part IV The quartic case

300

because L(1 + 6,1 + 6, 1) < wo for these cases. Similarly, we can ignore the cases Ti = (1, 2, 3), (1, 3, 2) for i = 2. Then we only have to move the contour so that (ri, r2) = (1 + 6,1- 61) where 61 > 0 is a small number such that 616-1 >> 0. This proves the proposition. Q.E.D.

(d) yp3i (,D, w, w)

Let 0 = 03. Let Al = I71,33(t6)I =.1)1 23, 'X /22 = Iry2,13(t6)I = A1A23A3-

Then A2 = .1i µr /22 g, A3 = "1 °/2 °/22 It is easy to see that dxto = ?dx.Xldxµld"122dxto. 5

Easy computations show the following lemma.

Lemma (12.8.11)

-21 4 2 2 (1) 0p31 (d2(A1, A2, A3)) _ Al 3A 22 A 3 =A 1 5/21 /25

(2) op32(d2(A1,A2,A3)) = AgA A3 = 11/21 µ (2sr11+8.12-8,2-2) Tz+{)-- A (8T31+38,12+2sr2-6)i (3) Op., (d2(A1, A2, A3)) Xµ2

(s,l1 -28,12+2sr2 -1)

(4) ep32 (d2N, A2, A3))

5(8,11-8,12+8,2-1)

Tz+ F

= All -28,11-38,12-2sr2+7) Al

Xµ2 (38,11+28,12-28,2-3)

Definition (12.8.12) 43i = (lp3i,1, Ip3i,2i lp3i,3) where (1) 1p31,1(Sr) = 5(2ST11 + ST12 - 8r2 + 2), 1p31,2(Sr) = 1(s,11 - 2sr12 + 2ST2 + 1), 1p31,3(ST) = si11 + 381-12 + 2ST2 - 9, ST2 + 4), lp32,2 (S r) = 5 (3sT11 + 2s112 - 2ST2 -3 ), (2) 432,1(ST) = 5 (81-11 - 51-12 1p32,3(ST) = 281-11 + 381-12 + 2ST2 - 12.

By (12.8.11), and a similar argument as in previous sections, w, w)

=p3i

3

6#(w) T

( 21rV1

1

)

Re(sr)=r

F12,SU11(Ra3

b13i 1

lp3i,

_ ST)AT(w; ST)dsT,

where the first (resp. second) coordinate of Raqbp3i corresponds to x1,33 (resp. X2,23)

for i = 1, 2. Let

srll) _

Ep32

F.,2(Ra341p32, 1(s111 + 4), b (3sT11 - 3))

28,11-7

Proposition (12.8.13) Suppose that T = TG and 6 > 0 is a small number. Then by changing 0 if necessary, PM ($, w, w) - -CGA(w; p) ` P32

w, w) '.,

2 6#(w)E2(Ra3'p31, 45 , 5)

3

X26#(w) 27f

fe(s,11)=1+6

Ep32 (-F, s111)02(s111)A1(w; sT11, 1)dsrij.

12 Contributions from unstable strata

301

Proof. The proof of this proposition is similar to (12.6.9). The only difference is that we have to check that the points (2, -1), (-1, 2), (1, 1) are not on lines s,11 = 2sr12+2, sill+3sr12 = 7, sT11+2s,12 = 4, 2sr11+3sr12 = 10, 3sT11+2s,12 = 10 in order to use (3.6.1). Q.E.D.

(e)

(b, w, w)

Let0=cDo,a'=c04,andp=po Let 9a = dl(A1) Az)dga, and µl = A1A 2. Then dg, = 2d
A 1'µi, and dl(A1) A2)-.+p = A1

(12.8.14)

(6.11+26.12+2s+2-5)

(5,ii+26,12-2sT2-1)

Easy computations show the following lemma.

Lemma (12.8.15) (1) vpl (d1(A1, A2)) = A 3A = Al 2µi I

_2

(2) Ka1(dl(A1, A2)) = A15A2 6 = Al 2µl 2

- /213. (4) Kv2(dl(A1,A2)) = (A1A = Al 2µi Definition (12.8.16) Let l4i = (14i,1,14i,2) where (A1A2)-3

(3) kvl(dl(A1,A2)) =

3)_2

(1) 141,1(8,) = 4(6,11 + 28,12 - 2sT2 + 7), 141,2(6,) = sr11 + 26,12 + 28T2 - 9, (2) 142,1(6,) = 4(sT11 + 2ST12 - 2sT2 + 1), 142,2(6,) = 8,11 + 28,12 + 2ST2 - 7, (3) 143,1(ST) = 3 - 141,1(8,), 143,2(ST) = 141,2(6,),

(4) 144,1(5,) = 3 - 142,1(6,)) 144,2(6,) = 142,2(8,).

Note that 14i(-ST11, S,11 + ST12, 8,2) = 142(6,11) 6,121 s,2) for i = 1, 2, 3, 4. Let

IF1=R14-Fv1 'e W,' 1'2=RD,T, `f'3=R54', 'F4=R149v1q,. By (12.8.14), (12.8.15), and a similar argument as before, =p,aI,i (,P, w, w) = 26#(w)

S2W,141,r(Pi, W)

for i = 1, 2,3,4. The following proposition follows from (11.2.3).

Proposition (12.8.17) Suppose that r = TG and 6 > 0 is a small number. Then by changing 0 if necessary, w) w)

2Ew+( i,14i,1(-1, 2,1)) 141,2(-1, 2, 1)

+ 2v2b#(w) 27fV

1

f

Re(s,jj)=1+6

CGA(w; p) 1W,st,sub-{-(Ti, 14i) sill, 1, 1 ) X 02(sTll)Al(w) sT11, 1)dsT11

for i = 1, 2, 3, 4.

Part IV The quartic case

302

w, w)

(f) up,a5,i

Let 0' = c05. Let to be as in (b). When we consider the cases i = 1, 2, we make the change of variable p, = A1A2, P2 = a,a2a3 = P1A3. Then (srii-1)

A1(s.,ii+2sriz+2srz-5)P

d3(,)--+P =

(12.8.18)

K v, (d2(A1, A2, A3)) etc. are the same as in (12.8.15). When we consider the cases i = 3,4, we make the change of variable P1 (A1a2)-'A3 2

A1A2, P2 =

(12.8.19)

2

= Pll\3. Then

dg(A)TZ+P = Al

4(sri,+2sriz+2srz-5) 16(3s,ii+2s,12-2S,2-3) )

Pi

P2

i(srii-1)

It is easy to see that 2

i = 1, 2,

r A l 2 Pl 2 P2

Qp(d2(A1,A2,A3))A 3 = Sl

(12.8.20)

r

3

i=3,4.

A 12P192

In both cases, dxto = 4d>A1d)p1d>P2dxto Definition (12.8.21) Let l5i = (2(s:1i + 1),15i,2) where (1) 151,1(8,) _ (ST11 + 28,12 + 2ST2 - 9)(8-11 - 26,12 + 28-2 - 5), (2) 152,1(8-) = (s-11 + 25:12 + 2ST2 - 7)(8,11 - 25:12 + 25,2 + 1), (3) 153,1(5,) _ (8,11 + 2sT12 + 28T2 - 9)(3s,11 + 28,12 - 2sT2 - 3), (4) 154,1(5,) = (5-11 + 2s-12 + 25,2 - 7)(38,11 + 25-12 - 2sT2 - 9).

Let Wi be as in (e). Then by (12.8.18)-(12.8.20), and a similar argument as before, w, w) \113

415#(w) 1

/

E1,sub('Fx, 152, 5:)A:(w, S,)dST r3>1, *1 Ar2

for i = 1,2,3,4. We define

Em"i (4), 5,11) -

4E1(R,5

4E1(RD,IF, 2(s-11 + 1 ))

Ep ,a5,2

(s-u - 3)(5:11 + 1) 4E1(Ra5.

p

s

E p,55,4

2(5,11 + 1))

(s-11 -5)2

'I',

2(5,11 + 1))

3(Srll - 5)(s,11 - 1)

(

,

8 ,11

)=

4E1(R,S9y,IF, 3

2(5:11 + 1))

(s,ll - 3 ) 2

The following proposition follows from cases (9), (10), (15), (16) of (11.2.4).

12 Contributions from unstable strata

303

Proposition (12.8.22) Suppose that T = TG and b > 0 is a small number. Then by changing 0 if necessary,

27rb

026#(W) OD, W, w)

JRe(s,-ji)=1+ V()S111)dST11

for i = 1, 2, 3, 4. (g) "P,a6,i (`b, W, W)

Let to be as in (b). We make the change of variable µi = A142, {12 = (A1A2 3)-'A3 = Al 1a2 A3 1

-.1

for the cases i = 1, 2. Then A3 = AI µ'1 ' 92, and (12.8.23)

d3(A)Tz+P = Al

(S,ii+s,iz+s,z-3)µi

-12

We make the change of variable µ1 = A1A2, µ2 = A1A23A3 = A1A23A3 1

4

for the cases i = 3, 4. Then A3 = Al ' µ1 11'2, and d3(A)Tz+P = A (s,i2+s,z-2)i (2s,11+s,12-e,z-2)µ2,h-1

(12.8.24)

It is easy to see that ( 12.8.25 )

o p(d2( 1, A2, A3))A 3 =

A

i=1 '2

2 2

-7 2 1

2

µ1µ'2

,

i = 3,4.

In both cases dxto = 7d)Aid1µldxµ2d1t1 Definition (12.8.26) Let 16i = (STii + 1,16i,2) where (1) 161,1(ST) = (ST11 + ST12 + ST2 - 4)(ST11 - ST12 + ST2 + 4), (2) 162,1(ST) = (ST11 + ST12 + ST2 - 4)(ST11 - ST12 + ST2 - 2), (3) 163,1(ST) = (ST12 + ST2 - 4)(2s,11 + ST12 - ST2 - 4), (4) 164,1(ST) = (ST12 + ST2 - 4)(2ST11 + ST12 - ST2 + 2).

Let

'113 = Ra69v29aIF, T4 = Ra6T.

IF1 = Rasga`y, W2 =

Then by (12.8.23)-(12.8.25), and a similar argument as before,

MO (41), W, W) 3

f

E1,sub('Pi, 16i, S)AT(w; S,r)dsr 2b#(w) E (2) f Re(,-)=r »''2. T3>2

for i = 1,2,3,4.

Part IV The quartic case

304

Proposition (12.8.27) By changing 0 if necessary, p,ae,i (1), w, w) - 2CGA(w; p)6#(w)E1,sub('Fi, 16i, 1, 1, 1)

for i = 1,2,3,4. Proof. We can ignore the case T2 = 1 as usual, and assume that r2 = 1. Let r = (ri, r2i r3) = (10, 5, 3). By the usual argument,

)31

1

27r V -1

(27r 1 0

31

V- 1

e(sr)=r

El,sub( i, 16i, s -)AT(w; Sr)C1Sr

) JRe(sri)=(ri,r2) E1,sub(Wi, 16i, Srl, 1)Alrl (w; sr1)dsT1. 2

The points (2, -1), (-1, 2), (1, 1) are not on the lines srii + Sr12 = 3, Sr12 = 5111 = Sr12 + 1, 2srii + Sr12 = 5. Therefore, by (3.6.1), we can move the

contour crossing these lines so that (rl, r2) = (1 + S,1 + S). Since El,sub(%Fi, lsi, sr) is holomorphic at (1, 1, 1), the proposition follows by a similar argument as before. Q.E.D.

(h) Now we combine the computations in this section. Let 0 = Do, p = P0. We define Jl(,D, w) = 148, (w) (Ea+(W, Loo, Xo, -3) + Eo+(-Fo'I', Lob 1, rIai Xa 1 8))

-

b#2w)

E2(Ro,'p12, -2, -5) +

6#3W)

E2(Ro.Dp31 5, ), 5

J2('D) = -2Ew+(Ro,-91o4', 1) - 1

2)

+ Ew+(Ro4AF, 2) + Ew+(JFwRa4IF, 2)

+

E1(J"a6Rae_R'a11,2) 2

+

E1(Ra6W,2) 4

2

- 2 El (RD, 9a IF, 2) -

RD, IF, 2),

Ta,l ((D, W) = J1 OD, w) + 6#(w)J2(4'),

2Tw+(Rw,oRo4-'a', 4(5 - sr11), z(1 - 8111)) (8111 - 1)(s,11 - 5) 2Tw+(Rw,o9wRa4g04', 4(8111 + 7), 2(1 - srll)) (sr11 - 1)(s111 - 5) + 2Tw+(Rw,oRa4`y, 4(8111 + 1), 2(1 - sril)) J4 (a

,

S r11 )

(8111 - 1)(srll - 3)

2Tw+(Rw,0 wRo4`W,4(11-8111),2(1-8111)) (Sr11 - 1)(sr11 - 3) TD,2 (D, w, 3,11) = b#(w)(-Ep11(4', sri1) + EP(12)1 (b, srii))

+ 6#(w)(Ep22 ('P, Srii) + Ep32 (4, Srii)) + 6#(w)(J3(4, 8111) + J4(D, 8111)) 4

+ 6#(W)

(-1)i+1Ep,oe,i (b, Sr11). i=1

12 Contributions from unstable strata

305

Since the third term of (12.8.10)(1) and the second term of (12.8.10)(2) cancel out, we get the following proposition by (9.1.10), (12.8.3), (12.8.7), (12.8.10), (12.8.13), (12.8.17), (12.8.22), (12.8.27).

Proposition (12.8.28) Suppose that 7 = TG and 6 > 0 is a small number. Then by changing 0 if necessary, Po (4, W, w)

,., Ta,1(,D,w)CGA(w; p)

+

P

2

27rV -1

f

_ e (srll)=1+6

ra,2(4), W, sTii)ni(w; sTii,1)dsT11

s#(W) f J

+

1

l z)) 27rV

2

Re01r1 l)=r2

A )"'1Ti (w; sr'11, 2)ds T'11

p21

r1 EM,

i=1,2

Note that we have proved in Chapter 10 that Mp;j (4), w, w) is well defined for

i,j=1,2and pi2(4),W,w)-0fori=1,2. Let Ta,2,(i) ((b, w,1) etc. be the i-th coefficient of the Laurent expansion as before.

It is easy to see that E2,(1,o)(R51,Ppll1 -1, -3) - 2E2,(o,1)(Ro1 cpil' -1, -3),

Ep11,(o)(1',1) Ep21 (o)

(4), 1) =E2,(1,-1) (Ro2 0P21 1 - 2, 0) + E2,(o,0) (Ra2 Dp211 -

Ep22,(0)(4),1)

2 2,(o,0)(Ra2'DP22, E 0,

0)_ 1

2, 0),

E2,(-l,o)(Ra24'P22, 0, 0)

4E2,(o,-1)(Ra2'P22,0,0) - 4E2,(-1,-1)(Ra2'DP22'01 0) E2,(1,-1)(Ro2'P22, 0, 0) -

Ep32,(0)(4), 1) _ -

-E2,(o,o) (Ra3

,1, 0) 5E2,(0,-1)(Ra34DP32'

15 E

2,(1,-1)

(Ra3 41) p32 ,

5

E2,(-1,1)(Ro2(DP22, 0, 0),

E2,(-1,0) (Ra3 4)P32

1,0) 1,

E

,1, 0)

52,(-1,-1)(Ra34DP32'

0) - 5 E 2,(-1,1) (Ra3

P32'

1 ' 0 ))

We define

4(Tw+(Rwo9wRN "o'p,2)+Tw+(RwoRa4go IF,1)), J6(4,)

2 (Tw+(Rw,oRa4'I', 2) + Tw+(Rw,oJ'wRo4'I',

2

))

1,0) .

Part IV The quartic case

306

Then

J3,(o)

,

1) =J5

2

4(srll + 7),0)

d I

dsr11

sr11 -5

=1

d Tw+(RwoRo4 JFo lF, (5 - sr11), 0) +2 sr11 -54 dsr11 Is.,,,=1 I

Tw+(Rw oRa4 `y , (sll + 1) , 0)

d

J4,(o)(4),1) =J6(41) - 2dsrll

,

srll - 43 Tw+(RwogwR04q,4(11-srll),0) d - 2dsru s,11=1 sr11 -3 s,

=1

Easy computations show that

= 4E1,(-1)(RD,

fa `y, 4 (srll + 7),0) + Tw+(Rw,oR04 -1'-O'Y, 4 (5 - srll ), 0) o%F,1) + 4E1,(-1)(Ro5JFwRo4JF,`F,1) sr11 +3

1-s- ll

- RessE2(Ra2,Dp22,

Tw+(Rw,oR04 +,

1

4(1 - sr11), s'),

1(sr11 + 1), 0) + Tw+(Rw,o. 'wRa4 WY, 4 (11 - srll), 0)

4E(Ra5'F,1)

4E1,(-1)(Ra59wR04W,1) 7-s,11

s,11-3 - Ress E2 (Ra2 4)p,,,

1

4

(sr11 - 3), s').

It is easy to see that

- Res

4(1

- srll),s) + 4E1,(-1) Ra5 ,a D 1) Ill

_ -E2,(o,-l)

+ O((srll -

(Ra24)p22, 0,

0) +

4E2,(o,-l)(Re2Dp220, 0)(srll - 1)

1))2).

Therefore,

J3,(o) (4),1) = J s (4) + 8 E2,(0,-1) (Ra2 DP22, 0, 0) - 8 E2,(1,-1) (Ra2 41) p22, 0, 0),

J4,(o)(4),1) = J6(4)) -

2

E2,(o,-l) (Ra2(l)p21, -

2

,

0) - 4E2,(l,-l)(Ra2Dp21, - 2 , 0) 4

4

- 2E1,(-1)(Ra5`T',1) +

4E1,(-l)(Rb59WRa4W,1)

12 Contributions from unstable strata

307

Also it is easy to see that 1

1

4E1,(-1)(Ra5

Ep,a5,1,(0)

4E1,(o)(Ra5

wRa4

Ep,a5,2,(0) (4),1) = E1,(o)(Ra5 `F,1),

Ep,a5,3,(0)(b,1) = 124E1,(-1)(RaS"''a`I',1) - 1 12 E1,(o)(Ra59a1y,1)

Ep,a5,4,(0)(4',1) = 3(RD,9wRa4I`,1) +

3E1,(o)(Ra59WRa4W,1).

We define 3

1

J7(P)

1

4E2,(1,-1)(Ra24p21,-2,0)

=E2,(o,o)(Ra2'Dp21,-2, 0)+

- 2E2,(o,-1)(RaAp21) -1- ,0) - 2E2,(o,0)(Ra2")P22,0,0)-

8E2,(1,-1)(R02"P22,0,0)

-5 - 5 E2,(o,o)(Ra3 'Dp32 ,1, 0) -

2E2,(-1,1)(Ra24'p22,0,0)-

1E2,(o,-1)(Ra24'p22,0,0)

8

0E2,(-1,0)(Ra2")P22,0,0) -

20

J8(f)

4E2,(-1,-1)(Ra2')p22,0,0)

20 E

2,(1,-1) (Ra3 4)p32, 1, 0)

E2,(O,-1) (Ra3 4)p32 , 1, O),

=14E1,(o)(RaS,FwRa4ga'',1)

1

+ 1E1,(-1)(RD,9wRa4gbW,1)

+ E1,(o)(Ra5 W,1) - 2E1,(-1)(Ra5'y,1)

-

2El,(-1)(RD,9wRa41Y,1).

1E1,(o)(Ra59WRa4IF,1)-

If 4) is K-invariant, E2,(-1,-1)(Ra2qDp22) 0, 0) = E1,(-1)(Ra5

z

,1),

E2,(i,-1)(Ra3'PP32) 1, 0) = -E1,(i)(Ra59a'P,1), E2,(-1,i)(Ra34'p32, 1, 0) = -E2,(-1,i)(Ra2'bp22, 010)

for all i. Therefore, these considerations show that T ,2,(0)(4), W,1) =6#(W)(J7(D) + J8(P) - 1 E1,(1)(Ra5,Fa`I',1))

+

-1, -3)

+ 2b#(W)E2,(0,1)(Ra1 (Pp11, -1, -3).

Part IV The quartic case

308

By the principal part formula (4.2.15) for E W,ad,(o) (Ra4'a IF, 1) etc.,

J2(D) + Js(D) = -2EW,ad,(o)(Ra4. aIF,1) + EW,ad(Ra4IF, 2

Note that by (4.4.7), if 4D is K-invariant,

E1(go,Ra69aW',2) ='b2"aF (0), E1(Ra6 `I', 2) = T2----wR14 A'F(0) EI(Ra69a`Y, 2) = tZ729WR14'Y(0) EI(9a6Ra61Y, 2) = 932T(0). So

T0,1('P,w) +Ta,2,(o)(D,w) =J1(4),w) +6#(W)(J7(f) - 4E1,(1)(Ra59a`y,1)) 6# (W)

2

EW,ad,(0)(Ra4'q'aW,1)

+ 6#(W)EW,ad(Ra4'I', 2 ) + 6#(W)E2,(1,o)(Ra1 lbpii, -1, -3) + 26#(W)E2,(o,l)(Ra1,Dp11, -1, -3)).

By (7.3.7), we get the following proposition.

Proposition (12.8.29) Ta, l (,D, w) + Ta,2,(o) (OD, w) =146a (w) Es,(o) (Ra 4Dp'b, wa, Xa, -3)

+ 6#(w)E2,(l,o)(Ra, 4)p11, -1, -3)

+ 26#(w)E2,(o,l)(Ra1Ip11, -1, -3).

§12.9 The case 0 = (09) Let p = (z, s) be a path such that a = (,39). As in previous sections, we only consider such path such that s(1) = 0. Let IP = Ra-Dp throughout this section. Let 223' be as in§12.8. Suppose that f (q) is a function of q = (qi, q2, q3) E (A' /kI)'. An easy consideration shows that w(t 0)f 0'1,23(1 0), 72,13(1 0), -Y2,22(t

0))dxt 0

=6a (w) fA' /kx)3 wa(t0)f(q)dxq. Let 1

d(µ) = d(µ1,µ2, µ3) = a(p 11%2 31,33 ,

-i3 AA , A1µ2 µ2-23

2

16

3,

µ3

µ2

0 ).

12 Contributions from unstable strata

309

Let to = d(µ)1°. Then dxto = 3dxµdxto. It is easy to see that 91 = 171,23(t°)I, 92 = 172,13(0°)I, P3 = I72,22(t°)I

Let T, sr be as before. Then r-02 (to) =µ1Sµ2'µ3 3, and

d(µ)" =

µlrii+sri2+2µ2 (3x.,11+2sr12+2sr2+7)µ3(s,.12+s,.2+2) 3

Definition (12.9.2) Let 1p = (lp,li 1p,2, 1p,3i 1) where 1

lp,l(Sr) = srll + Sr12 - 3, lp,2(sr) = 3 (3sr11 + 2sr12 + 2Sr2 - 9), lp,3(Sr) =

3

(8r12 + 8r2)

Let D = {r I r1+r2 > 4, rl+2(32+r3) -3, r 3r > 1}. Then a similar consideration as in §12.7 shows that

(2)

W ,W)

7rV

3

1

_

fRe(s)=r

E3,sb(

W,sr)Ar*sr)dsr,

where we choose r from D (1 Dr. As usual, we can ignore the case T2 = 1, and assume that T2 = (1, 2). The point (3, 3, 2) belongs to the domain D (1 Dr. By moving the contour to Re(sr) = (3, 3, ), a 1

)

3

'-

(,-,)=(3,3,2)

C2 7r V- 1

E3,sub(W, lp, Wa, Sr)Ar(w, Sr)dsr

3

1

2

+

E3,sub(T, 1p, Wa, sr)Ar(w; sr)dsr

Re(s*)=(3,3,

(2)7r

27r

2

)

fRe(si)=(3,3)

E3,sub(, lp, Wa, Srl, 1)nlrl(wi srl)dSr1.

Since C > 100, Lr (3, 3, _< 12 + C < 4 + C. Therefore, we can ignore the first 2) z term.

If Ti = 1, (1, 2) or (2, 3), we choose rl, r2 >> 0. Then L1(r1i r2) < 0 for these cases. The function E3,sub(W, 1p) We, Sri, 1) has the following pole structure.

Part IV The quartic case

310

The points (-1, 2), (1, 1) are not on any of the lines except for the line sT12 = 2. Therefore, by the passing principle (3.6.1), we can move the contour from the point (rl, r2) = (3, 3) to (1 + 6,1 + 6) where 6 > 0 is a small number crossing other lines. We define Cpl) E .9(A) by the integral '(x1,23, x2,13, x2,22)dx2,22

(x1,23, x2,13) = J

(Dp

A

Let f (C W, ST'11) = E2('D p

1

, sr'll - 1, s -'11 - 1)A,,,, (w; sT'11, 2),

and r = (1 + 6,1 + 6). Then by the above consideration,

P6a(w )

2

1

r

fRe(srj)=r E3,sb( , lp) Wi, s-rl, 1)Ai 1(w; .rl)d- j

2

P6#(W)

+

w, s

fRe(s11)=ri>1

*'=(,-' (1,z))

11)dS

'll

T1' Effi'

If r1 = (1, 2, 3), (1, 3, 2), L1(1 + 6,1 + 6) < 4. Therefore, we can ignore these cases in the first term. Since the function E3,sub(F, 1P, WD, S,1, 1) is holomorphic at

s,j = (1, 1), we get the following proposition.

Proposition (12.9.3) By changing V) if necessary, (4), w, w) ^' CGA(w; P)

6a

3W) E3(RaDp, Wa,

e6#(W)

+

*'=(+1,(l,z)) 27r V -1 ,. E'm'

-1, -3, 3)

f Re(sr'ii)=ri>1

f

w, sr'11)dsT'11.

12 Contributions from unstable strata

311

§12.10 The case Z = (,31o) In this section, 00 = (31o), 01 = (/31o, /310,1), 02 = (010, /310,2), a3 = (010, /310,3), and 04 = (31o, 810,2, N10,2,1). Let po = (ao, so), pll = (al, 511), p12 = (Z1, 512), p21 = ('02,s21),P22 = 02,522)) p31 = (03,531),P32 = (D3,s32) where (5il(2),si2(2)) = (0,1)

for i = 1,

Let Ni = 04,540 for i = 1,

, 4.

, 4 where (54i(2), 54i(3)) =

(0, 0), (0,1), (1, 0), (1,1) for i = 1,2,3,4 in that order. As in previous sections, we only consider such paths such that po(1) = 0 etc. Let tY = Ra04Dp0 throughout this section. Let di (A,) = dp2(Ai), d2(Al, A2) = a(A1 2, A1, Ali

d3(Al, A2, A3) =

A2), A

-A-',

Easy computations show the following two lemmas.

Lemma (12.10.1) (1) epi, l (d3(A1, )'2, X3)) = A1, epi21(d3(Al, A2, A3)) = Al 1 for i = 1, 3. (2) ep2,1(d2(A1,A2)) = A1, ep221(d2(A,,A2)) = A11. (3) epa11(d3(Al, A2, A3)) = A1, epa,2(d3(A1, A2, A3)) = A A2. epa21(d3(Al, A2, A3)) =All, ep4,2(d3(Al, A2, A3)) = (A A2)-1 1 (5) (5) epasl (d3(Al, A2, A3)) _ A1 epa12(d0l, A2, A3)) = A A2. (a1A2)-1

(6) epa41(d3(Al,A2,A3) _ A1, epa12(d3(Ai,A2,A3)) =

Lemma (12.10.2) (1) opo(d1(Al)) = A6 (2) op11 (d3(A1, A2, A3)) = Qp12 (d3(A1, A2, A3)) = A A2. (3) ap21 (d2(Al, A2)) = O'p22 (d2(Al,, A2)) = A1A2.

/

(4) 0P31(d3(A1,A2,A3)) = 0P32(d3(A1,A2,A3)) = Al A2.

(5) opal (d3(Ai, A2, A3)) = o 43 (d3(A1, A2, A3)) = A?A2A3. (6) o 42 (d3(Al, A2, A3)) =ttapama (d3(Al, A2, A3)) = A2A3. Let Pi = {po,p21,p22J, P2 = {P11,P12}, T3 = 431,P32,P41,P42,P43,P44} The

paths Po, p21, p22, p4i for i = 1,2,3,4 satisfy Condition (3.4.16)(1). The paths P11412 satisfy Condition (3.4.16)(2). For the paths p = P31432, =p ((D, w, w) = 0. Therefore, by (3.5.9), (12.10.3)

EP(='p+(41,w,w)+C.p+(-D, w, w))

Ep0E 0(-',w,w) = PET,

+ E EP 0-'P# (.D, W, W) - =P# (41, W, W)) PEP,

+ pE12

+ PEP3

and all the distributions in the above formula are well defined for Re(w) >> 0.

Part IV The quartic case

312

(a) _po(,D,W,w)

Let a = Do,p = po. Let ao = dl(A1)g°° where go E M°A. Then dg-0 = 2d
Proposition (12.10.4) so+(4,, w, w) - CGA(w; P)bao (w)Eao+(IF, Loa., 3).

(1)

w, w) - CGA(w; P)boo (w)Eao+(,Foo T, wao , 3).

(2)

Let to = d3(.X1, )'2,.\3)t °, and ga = n"(uo)t°nB2 (u4) for uo, u4 E A. It is easy to see that dxt° = 2dxA1dxA2dx.3dxto Let dg, = A32dxt°duodu4. Clearly,

olp(g,,w) is the constant term with respect to P2. Let OON(t°, w) be the constant term with respect to the Borel subgroup as before. We show that we can replace .'p (go, w) by N (to, w).

Lemma (12.10.5) w(t°)ap(to)8Zn,o ('F, 9o)eN(t°, w)dga

(1) Ep,st+("D, W, w) r., f Xa fBA/Tk

(2) =p,st+(ID,w,w) -

f

A1>1

(9a))8N(to,w)dgo.

X1 n2A/Tk Al<1

Proof. Consider (1). By the consideration as in §2.1, -'p (9a, w) - 4°0(t°, w)I < E 4,r,1(t°, w),

where the summation is over I = (Il, 12) such that Il = {2} or 12 = {1}. By (3.4.30), there exist a slowly increasing function h(t°) and a constant b > 0 such that if M > Wo, l1i 12 >> 0,

1,,1(t°, w) << h(t°)(A2 + A3 )

for wo - 6 < Re(w) < M. We choose Schwartz-Bruhat functions 0 < 411 E 9(A), 0 < W12 E 9(A2) so that

fI

o

(IF, 9o)Idu4

I'Y2,33(t°)I-1e1(W1,'Y1,33(t°)) E 'F2(_Y2,22(t°)x, y2,22(t°)xuo). xEkx

Then by (1.2.6), for any N > 1, 11i 12 >> 0, the difference of the left hand side and the right hand side of (1) is bounded by a constant multiple of the following integral

f

h'(t°)(AA 'A 3)-N(A + A

)

TAO /Tk xA

Al>1, a2<

a(up)-1

T 2('2,22(1°)x, -Y2,22(t°)xuo)dxt°duo,

X

xEkx

12 Contributions from unstable strata

313

where h'(t°) is a slowly increasing function independent of N, 11, 12.

Let p = ai,2A-2. Then 3 (A2A2

2)-N (A1 +

12)

= ANA

4N((/-'AI

2X2)11

12).

+ A3

If N >> 11, 12 >> 0, the above integral converges absolutely. This proves (1). Since ea(d3(Al, A2,,A3)) = d3(Al 1,.2iA3), and the integral is for Al < 1, the proof of (2) is similar. Q.E.D.

Let T, sT be as before. Let p = A2A3 2. Then (12.10.6) (t0)Tz-p = AsT1+2sT12+3A3.2+1As,11+1 = )S-11+2s.12+3/,,S.2 +1A8.11+25,2+3 l 2 1 3 1 3

Definition (12.10.7) (1) lst,l(ST) = (2 (ST11 + 2s112 + 1), Sr2,

2

(-sill - 2sr2 + 3).

(2) 13t,1(ST) = 21(7 - STll - 2ST12)) ST2) 2 (-sT11 - 2ST2 + 3)).

(3) 13t,2(sT) = lst,2(sT) = sTll + 2ST2 - 3.

We define Ep,st+(OD) W, ST) = 8a,5t(w)

wf ST) = sa,st(w)

7a+ (Ro,o xF, wa,st ) lst, l (s,) ) 13t,2(8T) 1 7a+(RaoJFaW, wa,st, lst,2(ST)) ,

lst,2(ST)

Since I12,33(t°)I = A A2A2 = A2/tA3, by (12.10.6) and a similar consideration as before, _

3

s)A(w;s)ds,

p,st+(4>,w,w) - E (2) JRe(s)=r T

w, w) r TE T

( (2)7f

3

1

__

p,st+(, w, s)A(w, s)ds

fRe(s)=r

,

where we choose the contour so that r E DT, r1 + r2 > 3. w, 5r1, 1) are holomorphic for Re(sr11) > 1, Since Ep w, STI) 1), we get the following proposition by the usual argument.

Proposition (12.10.8) Suppose that T = TG and 8 > 0 is a small number. Then (1)

W, W)

P28a,5t(w) Ep,st+( >w) sTll, 1, 1)02(sT71)A1(w srll, 1)dTll) 27rVr-1 Je(sr11)=1+b (2)

P28a,st(w)

fRe(s,,)=1+5

() w

1 1)2(s II )AI (w sTII1)ds f II' f

)

Part IV The quartic case

314

(b) Epo#(,P,W,w),

Let r = (Ti, (1,2)). Let sT be as before. It is easy to see that

d 1( A 1) Tz+P ]9x11=-1,9rz=1 = 29r12-4 fcao(dl(A1)) is as in (a). By (11.1.4), (11.1.5), (12.10.2), and the Mellin inversion formula,

upo# (

, W,

b#(W) IY ( 0 )

w) =

Z-1

[

-(T1.(1,2))

A1T1 (w; -1, ST12)

2ir/

-

dST2,

ST12 + 1

e(9r1z)-T2> 2

+i E'ID2

('D' W, w)

PO

=

A11( w,1,ST12)

S,12-5

fRe(s .1 z)=''2>2

=(,(12)) r1 E'N2

ds2.

Proposition (12.10.9) By changing t/' if necessary, qj2S#(W)T(0)

po#(4', w, w) - CGA(w; P)

3

CGA(w; p) Z2d#(W)3

po# ((D, w, w)

,

so

(0)

Proof. If 7-1 = 1, (1, 2), we choose r2 >> 0, and we can ignore these cases. Suppose Tl = (1, 3, 2). Then 9rz+1, 9rx-4 are holomorphic at p1T1 = (-1, 2). Therefore, the proposition follows from the passing principle (3.6.1) and the usual argument. Q.E.D.

w, w) P12 Let 0 = (l l. Let t° = d3(A1, A2, A40 . We make the change of variables

(C) ":'P11 (), W, w),

2

2

/11 = I'y1,33(t°)I = A1 A21A3, /A2 = I72,23(t°)I = ).X2.

Then it is easy to see that dxt° = dxAjdXµldxp2dxt°. Also A4A2 = Aiµ2. Easy computations show the following lemma.

Lemma (12.10.1 0) (1) ep11

\ A2, 13))TZ+P = -9T31+23x12-2sT2+1 1

(2) 9P12

A2,

2 (9*11-1)

A3))Tz+P = A_38.11_2312_28.2+7 i

P22

(9x11+2sT2-3)

(9x11+2sT2-3)

Definition (12.10.11) Let 42i = (2 (8111 - 1), 2(ST11 + 2S,-2 - 1), lpxi 3) where (1) lp11,3(ST) = -ST11 + 2ST12 - 28T2 + 3. (2) 1p12,3(S1) = 3ST11 + 2ST12 + 2sT2 - 9.

By (12.10.1), (12.10.2), (12.10.10) and a similar consideration as before, P. (', W, w) 3

b#(W)

1 T

2 -7r

_ F+2,sub(Ra1,Dp1., lpli, s1)AT(w; s1)dST,

12 Contributions from unstable strata

315

where we choose the contour so that r E DT, r1 > 3, rl + 2r3 > 3, and lpl; (r) > 0 for i = 1, 2. Also the first (resp. second) coordinate of Ra1,kp1i corresponds to x1,33 (resp. X2,23). We define E p11 (

f

E2(R61Dp11,

8T11) =

2(8'11 - 1), (8,11 + 1)) f

3 -sill

/ / _ E2(Ra4)p12 f 2(s'11 - 1), 2(s'11 + 1))

Ep12 OD, 3'11) -

3sT11 -5

Proposition (12.10.12) Suppose that r = TG and 6 > 0 is a small number. Then by changing V) if necessary, 926#(W)

(1) "p11 ('D,Wf w) ,v (2)

p12

(Df W f w) `

I

Ep11

5'11, 1)d3T11,

926# (W) Epi2 27rV 1 fRe(s111)=1+6

sT11)I2(sT11)A1(wf s'11f 1)dsT11

21rV -1

Re(srll)=1+6

Proof. The proof of this proposition is similar to (12.6.9). The only difference is that we have to check that the points (2, -1), (-1, 2), (1, 1) are not on lines sT11 = 2sr12 + 1, 2sr12 + 3sT11 = 7 in order to use (3.6.1). Q.E.D.

(d)

P21+(1'1W f w), '='p21+(4), w, w) etc.

In (d), (e), 0 = (l2. Let ga = d2 (A,, A2)90. Then dga = 2d' Ald' A2dga . Let A _ d2(A1, A2), and dxA = dxAid)
P22+(4)1L"1W)1

P22+

Lo' W)

are equal to

IRxM

W (9a )A1A2oz, (Ra P21 f 9a)"'p21 (go, w)d" Adga, /Msk

Al <1, ),2 X2>1

W (9a )A210z, (ga Ro 4)p22 f ea (9a) )p21 (9a, w)d" Adga f

IRxMA/Msk a1<1, a1 a2<1

fRXM/Mk

w(9a)-'A 6A2ez,

(Ra4)p21

f go)-422 (go, w)dx Adga f

X1>1, ).1)2>1

fRxM

w(9a)-1A 'ez,

(9aRa'Dp22f 00

/Msk

al>1, X2P2:51

respectively. Easy computations show the following lemma.

(90Ap22(901

w)d"Adgaf

Part IV The quartic case

316

Lemma (12.10.13)

-

Tz+p = As,"' +2s,12-2s,2-1µs,2-1 (1) Bp21 (d2(A1) A2)) )2))Tz+p = Al1 ,11-2x,12-2sr2+5µs,2-1. (2) OP22 (d2()1,

Let

Definition (12.10.14) Let 102, = (lp2i,l) lp2i,2), 1P2i = (lp2i,l,lp2i,2) be linear functions for i = 1, 2 where (1) 1p21,1(ST) = ST2 + 1, 1p21,2(Sr) = 3r11 + 2ST12 - 2Sr2 + 1, (2) 1p21,1(Sr) = 3 - lp21,1(ST), 1p21,2 (ST) = lp21,2(ST), (3) lp22,1(ST) = ST2 + 1, lp22,2(ST) = sT11 + 2sT12 + 2ST2 - 7, (4) 1022,1(ST) = 3 - 1P22,1(Sr)1 1022,2(ST) = 1022,2(S,)-

Note that 102i (-ST11) ST11 + ST121 ST2) = 1021 (ST11) ST12, ST2) etc.

By (12.10.1), (12.10.2) and a similar consideration as before, =p2i+((D, w, w) = 6#(w)

E

j,,, ,T(Ra'Pp2i) w),

r1EW2 r2=1,(1,2)

"p2i+('D, w, w) = 6#(w)

1:

W142{,T(9-0 Ra4)p2;,w),

,1 E202

,2=1,(1,2)

for i = 1, 2. We define

E2+(, ST11) =

)102,, ST11)1)1)1

E2+(, sill) = 2F+W,st,sub+(ga2 R02 4'p2i I 42i 1 ST11) 1, 1)

for i = 1, 2. The following proposition follows from (11.2.3).

Proposition (12.10.15) Suppose that r = TG and S > 0 is a small number. Then by changing z// if necessary, (1)

=p2i+(4),w,w)

2CcA(w; P)b#(w)

C # fRe(si (2)

Ew+(RDDp21 , lp2i,1(-1) 2, 1)) lp2i,2 (-112) 1)

1)=1+b

1)dSTll)

p2+ (,

021+(4),w,W)

^ 2CGA(w; P)s#(w) Ew+(ga2

Rat 'Dp21 ,

lp2i,1(-1, 2,1))

lp2;,2(-1, 2,1)

+

P26#(W #() JRe(sii)=1+5

p2i+(, sT1 1)2(s111)Al(w; s111,1)ds111

fori=1,2. (e) "p21#(')'w,w), ' 022#(4),w, w) etc. Let p = A 2. Then ai. 2, a21 =dip-1

12 Contributions from unstable strata

317

By (12.10.1), (12.10.2), p21 # (4D) w, w) = Ra2

JR2

X MO A /MDk

w(g0

1

2S,21(ga, w)d" A1dx /.cdga,

a<1, µ(1

"p21#(41fwfw) =

J

w(g0')A21421(go, w)d"A1dx µdga f + X MIA/Mak

A1<1, µ<1

'p22#(,D7 wf w) = R021)p22 (0)1

µdga°,

w(ga)-1A1 26'22 (go,

R+x MIA /Mak

al>1, µ<1

=P22#(4>, w, w) = aRa2 p22 (°) f

22

R2 x MIA / Ma k R+

(go, w)d" Ald"pdga

A1>1, A<1

Easy computations show the following lemma.

Lemma (12.10.16) A2))Tz+P =

(1) 0P21 (d2(A1,

(2) 0P22 (d2(A1, A2))T +P = a1

= \ sr11+2sriz-2sr2-1µ5r2-1 sr11-2sr12+3As -2-1 = Al srll-2sT12-2872+5µw2-1. A8T11+25712-3A3T2-1

Let sT = (S,12,8,2) and ds'T = dsT12ds,2 We define AT(w; 8712, -1, ST2)

ST) = 26

Ep21#,7(`Y

(ST12 - ST2)(ST2 + 1)

Ep21#,7 ((D, w, w, sT) = 2b#(w)9-DRa2 4)p2, (0)

Ep22#,T(

f

wf

Ep22#, (Yfwf

f

W,

S'7 = 26## ((w)Ra2

p22 (

0)

AT (w; ST12, -1, Sr 2) (8712 + ST2 - 4 `)('ST2 T 1)

Ra2 P22( 0 )

8T') =

, 8T2) ' A. ( w; sr12, (ST12 - ST2)(ST2 - 2)'

A7

f

(w; w; ST12f -1, ST2)

(8712 + ST2

- 4)(872 - 2)'

Then by (12.10.15) and a similar consideration as before,

)21E2#,T((A),

'-'p2i#(-'f w, w)

w,

'1E2

,'r3)

T2=1,(1,2)

P2i# (4), w, w)

E(

rl EW2

)2f 2%rV1

1

.e(sr)-(T2,'r3)

Ep2i#,T (4k, w, w, sT)d4T,

T2=1,(1,2)

where we choose the contour so that (-1, r2, r3) E DT and r2 > r3 etc.

Proposition (12.10.17) By changing O if necessary,

(2)

p21#((D,w,w) - CGA(w;P)T2b#(w)Ra2`'p21(0), -2CGA(w; P)`Z126#(w)gaRa2 `DP21(0)1 ' jP21# ((D, w, w)

(3)

E P22# (ID, w, w)

(4)

Ep22#(ID, w, w) - 2CGA(w; P)Z2b#(w)9bRa2 DP22 (0)

(1)

-CGA(w; P) 2b#(w)Ra2 DP22 (0)f

Part IV The quartic case

318

Proof. We can ignore the case T2 = 1 as usual, and assume that T2 = (1, 2). If TI = 1, (1, 2), we choose r2 » 0, r3 = 1 + 6 where S > 0 is a small number. Then L(-1, r2, r3) < 2 + (1 + 6)C < w° if 6 is sufficiently small. Therefore, we can ignore these cases. Suppose TI = (1, 3, 2). Then the point pr = (-1, 2, 1) does not satisfy the conditions ST12 = sr2) sr12 + sr2 = 4, sr2 = 2, -1. Therefore, we can move the contour crossing these lines by the passing principle (3.6.1). Since Ep21#,rN, w, w, sT) etc. are holomorphic at s' = (2, 1), the proposition follows. Q.E.D.

(f)

w, w).

w, w), '^--'P32

= 03. Let to be as in (c). Let is = Iy2 33(t°)l = A1.A2A3. Then A1A2A3 =

Let

AIA2 t, and d1t° = dxA1dxA2dxµdxt°. By (12.10.1), (12.10.2), P31 (,D, w, W)

w(t°)-'A

1A2A28Z.

t°Ap3 (t°) w)dxt°)

T2 /Tk a1<1 P32

(4)1 w, w)

w(t°)A1A2A29Z, (R5DP32 f t°)4032 (t°, w)dxt°. Tp /Tk Al >1

Easy computations show the following lemma.

Lemma (12.10.18) A2, 3)) Tz+P =

(1) ep31

(2) 9

32

(d3( 1, A2,

3))Tz

1s*

-1

12-1

z (-s*11 +2srz+1)

A2

µ 2 (s.,ll-1)

+P - -13+11 -sr12+4 2A2(-8T11+2sr2+1) 16 (srll -1) 2

Let

Ep31,7

(4if w) wf sT12) sT2)

Ep32,T(4i)

w)

W, sT12)

sT2)

2S# (w)E1 (RZI 4> p31 f

sT2 + 1)

AT(w; 2sr2 + 1, Sr12, ST2) ,

ST12+1 AT(w; 28,-2 + 1, ST12, ST2)

28#(w)E1(Ra

4?p32) sT2 +

1)

sT12 + 2sr2 - 3

Then by (12.10.7) and a similar consideration as before, =p3: (4>, w) w)

_

(

1

)21 e(sr12,s.2)=(rl,r3)

w, w) ST12, Sr2)dsr12dsr2,

where we choose the contour so that (2r2 + 1, r2, r3) E Dr, r2 > 0, and r2 > -1, r2 + 2r3 > 3 for i = 1, 2 respectively. Proposition (12.10.19) By changing t/i if necessary, 8p3i (4>, w, w)

0 for i = 1, 2.

Proof. We can ignore the case r2 = 1 as usual and assume that r2 = (1, 2). Then the point pT is not on the line sril = 2sr2 + 1 for all the cases. Therefore, the proposition follows from (3.5.19). Q.E.D.

12 Contributions from unstable strata

(g) Ep4i (c, w, w) for i = 1,

319

, 4.

Let 0 = 04. Let to be as in (c). Let pi = .1\2i and p2 = A A2A1. Then A1 2 3 = ip1p2, A2A3 = Al 2µ2 It is easy to see that dxt° = dxA,dxpldx p2dxt °.

By (12.10.1), (12.10.2),

t°)1041 (to W)d"t°

w(t°)A1A2A3eZ,

w, w) _

p41

TA/Tk a1 2 <1 w(t°)-1)t2A3OZa

p42 ('D' w, w) _

(RD,P042, t°)4042

(to w)dx x1dx t°,

TA/Tk f)`1>1, X2P2>1

w, w) =

.1043

w(t°)-'A 6A2A3eZa (Ra4'p43, t°)4pa3 (t°, w)dxt0,

J

TA/Tk 1 a2<1

II/a1>1,

044 (4), w, w)

=

w(t°)A2.eZ,

J

t°)4044 (t°, w)dxA,dxt°.

TA / Tk

a1<1, aia2>1

Easy computations show the following lemma.

Lemma (12.10.20) (1) 0P41

(2) 9P42

(3) ep43 (4) ep44

(t0)TZ+P

= 1r11+2sT12-2s,z-lpl (-3T11+2srz-1)02 (3T11-1)

(t0)TZ+P = Al 8111-28112+2srz+lp (-8111-2srz+3)02 (8111-1) (t0)TZ+P = al 8111-23x12-23x2+spi (-8111+2srz-1)02(3.,11-1) As,"' +28,12+28.,2-5Alz(2

0 TZ+P

(-sT11-23.,2+3)

z (s,11-1) 02

Definition (12.10.21) Let 44i = (2 (sTi1 + 1), lp4;,2) where (1) lp41,2(ST) = (sT11 + 2ST12 - 2ST2 + 1)(-ST11 + 2ST2 + 1), (2) 1p42i2(&T) = (sT11 + 2sT12 - 2sT2 + 1)(sT11 + 2ST2 - 3),

(3) 1043,2(ST) = (sT11 + 2sT12 + 2ST2 -

7)(-ST11 + 2ST2 + 1),

(4) 1p44,2(ST) = (ST11 + 2ST12 + 2ST2 - )(ST11 + 2Sr2 - 3).

By (12.10.20) and a similar consideration as before,

Ni (D, w, w) _

3

=2b#(w) T

(2)1r

fRe(s)r

E1,sub(RDipa1,1p4:>ST)AT(w;sT)dsr,

where we choose the contour so that r E DT, rl > 1 and both factors of lp4.,2 are positive for i = 1, 2, 3, 4. We define Epal

STll) _

EP42

(D, ST11) -

E043 Nf ST11) = E044 (Df ST11) =

2E1(R14'PP41' 2(s,11 + 1)) (ST11 + 1)(3 - sT11) + 1)) 2E1(R04'DP42' 2(ST11

(ST11 + 1)(ST11 - 1) 2E1(R14 Dp43, 2 (ST11 + 1))

(sT11 - 3)(3 - 8,11) 2E1(RD4 4044 , 2 (ST11 + 1))

(ST11 - 3)(ST11 - 1)

Part IV The quartic case

320

The following proposition follows from cases (3), (4), (11), (12) of (11.2.4).

Proposition (12.10.22) By changing 7/i if necessary, 2826#(W)

p4i (, LO, w) -

21r

JRe(s,ii=1+6 Epi (, ST11)AiT1(w; Sr111 1)dsrll

fori = 1,2,3,4. (h) Now we combine the computations in this section. Let 0 = 00, p = po We define

Ji(,W) =ba(W) (ED+(`F,WD,3)+Ea+( ,IF,Wa if 3)) 26#(W) 3

(RD 41p (0) + 9, RD -Pp (0)),

-

J2(4))

(Ew+(R,24'p21,

2) + Ew+ (-9'wRa2-Pp21 f 1))

- (Ew+(Ra2 P22 f 2) + Ew+(9wR02 4P22' 1)) + Q32 (Ra2 (I)P21 (0) + 2.rwR,2 (I)p21 (0)) + 02 (Ra2 4)}f22

(0) + 29w Ra2 DP22 (0)) f

TL, 1(4),W) =Ji(,W)+5#(W)J2(4), J3(4f W) sT11) =

To+(RD,oIF,wo,3t, 2(srll + 3), 1, 2(1 - sT11)) ST11 - 1

Wa t, 2(5 - ST11)f 1, 2(1 - ST11))

+

J(4) s T11 )_4

ST11 - 1

2Tw+(Rw,oRa24)P21, 2, 2 (1 - 8T11))

f

(sT11 + 1)(s"11 - 1)

2Tw+(Rw,o9wRa24 211 1, 2(1 - ST11)) (S T11 + 1)(STll - 1)

J5(, Srll)

=2Tu +(RK,ORaz'Dp22 ) 2f 2(1 - ST11))

(sTll - 3)(sTil - 1) +

2Tw+(Rw,o9wRa24'p22f 1, 2(1 - ST1l)) (STll - 3)(sTll - 1)

7a,2(, Wf ST11) -ba,st(W)J3(, Wf ST11) + 6#(W)(J4(, ST11) + J5(, ST11))

+ b#(W) E (-1)iEpli (, si11) i=1,2

+ 8#(W)(Ep41 (, STll) - EP42 (, ST11)) + 5#(W)(-Ep43 (, Sl1) + EP44 (4)1 S"11))

By Theorem (10.0.1) and (12.10.3), (12.10.4), (12.10.8), (12.10.9), (12.10.12), (12.10.15), (12.10.17), (12.10.19), (12.10.22), we get the following proposition.

Proposition (12.10.23) Suppose that T = TG and 6 > 0 is a small number. Then by changing 7/i if necessary,

'po (, w, w) -CGA(w, p)Ti,l (, w, w)

+

g

_

2

JRe(sii)=h+5

W,STl 1)2(sT1l)A1(wfST11,1)dsii.

12 Contributions from unstable strata

321

Let Ta,2,(o) (P, w, 1) etc. be the i-th coefficient of the Laurent expansion as before. We define

J6(P, w) = -1(Ta+(Ra,o`F, wa,st, 2,1) + ZTa+(Ra,o9-aW, wa, t, 4,1)).

Then it is easy to see that d Ta+(RD,0W, wa,st, ds-11 Sri1=1

J3, (0) (ID, w,1) =Js(4 , w) +

Ta+( Ra,o

a IF, wa,9t, -1

1(sill + 3), 1, 0)

2 (5 - sTll),1, 0).

S, 11=1

Also, Ta+(Ro,o'W, wo,st,

2

(sT11 + 3),1, 0),

ar11+1 2

e2(Ra,ow,1L1E2 141, µ1µ242)d µ1d",L2dXg1d"g2,

2 x (A1 /kx )2 µ1 >1

7a+(Ro,oAoIP, wat, 1(5 - ST11),1, 0) 3-erll p1 62(Ra,o9-a`I',t1Ft2141,µ1µ24z)d"p1d"µzd 4id"42 2 .2 x (A' /k X )2 c it

For IF E Y'(Za,oA), We define 9a,0T(x1,33, x2,22) _ f (y1,33, Y2,12) < x2,12y2,12 + x1,33y1,33 > dy2,12dy1,33 A2

Then if (k is K-invariant, Ro,O-5FaRa(D(x1,33, x2,22) =.a,oR',oRa-P(X2,22, x1,33)

For T E 9(Za oA), let R1, R2 be the restrictions to the first coordinate and the second coordinate respectively. Also let (12.10.24)

91(x1,33) = f (yi,a, 0) < X1,301,33 > dy1,33,

.2(X2,22) =

//

J

'F (0, Y2,22) < x2,22Y2,22 > dy2,22

The proof of the following lemma is easy, and is left to the reader.

Lemma (12.10.25) Suppose that 4) is K-invariant. Then -QriRa,0W(0), 92R0 ',09a`y(0) = F2Ra,o`y(0), E2,(-1,-1)(Ra1 p11, 0, 1) = -E1,(-1) (Ra445P42' 1), E2,(-1,-1)(Ra1 DP12) 0,1) =

-E1,(-1)(Ra4N44)

1),

E1,(i)(91Ra,o`W,1) = (-1)tE2,(i,-1)(Ra1 iDp11 , 0,1) for all i, E11(i)(91Ra o a1Y,1) _ (-1)iE2,(i,-1)(Ra1 4(Dp12, 0,1) for all i.

Part IV The quartic case

322

Lemma (12.10.26) bo,st (w)T1+(Ra,o4y, wo,st, 2 (S'11 + 3), 1, 0)

+ ba,st (w)T1+(Ra,o9-a IF, wa ba,st(w)

=

2

+ b#(W)

+ b#(w)

/

Ez(Ra,o'I`,wa,st,

sill + l

ST11 + 1 4

4

,

(4E2(_l_l)(Rll4'p11,0, 1)

+

2E2,(o,-1)(Ra1P1201)) 2E1,(o)(Ra44p42, 1)

+

(ST11 + 1)2

4E1,(-1)(Ra4'P1441 1)

2

/

ST11 -3

4E1,(-1)(Ra4')p42, 1)

Ta+(Ro,o41, wa,st,

)

ST11 + 1

(4E2(_l_l)(RDi p12,0,1) (sT11 - 3)2

(srll - 3)2

)

2E2,(o,-1)(Ra14)p11,0,1)

(ST11 + 1)2

+ b#(w)

- 6#(w)

t,12 (5 - ST11),1, 0)

)

sT11 + 1

2E1,(o)(Ra4'bp44, 1)

s,11-3

+

(ST11 + 3), 1, 0), Ta+(Ra,oJFa 4y, wa,5t, (5 - ST11), 1, 0)) 2

are equal to the following integrals "" 2+1

fRx(A1/kx)2

wa,st(41,4z)A1

_

z(Ra,o`I`,-

1,

1 A2)d"Ald A 2dxgldxg2,

a1>1

fRX(A1/kx)2

wa,1St(41,42)A

z .. Q2(Ra,O-9,4I`,AjA 1>A1A2)dxA dxA2dx41dxg2.

a1>1

Note that wa,st(41,g2) = we,st(42,41) By (6.1.2),

A12

A1r12

O2(Ra,0W, -A1-A2 141,

E)2(ga,ORa,OF,A 1241

erll-1 = Al 2 A2

-

A1A2 141) + Al 2 A1r,2-1

1O1(R1Ra,oq,,

A1",2+1

A1'a 141) -

A2

+li-3

= Al

1)

A21e1(R2Ra,o9a4Y,

2

- Al

191(R1R/0,0

W, Al 1'2 142

1)

1)

A-1e1(92Ra,o`I',A 1A 2'g2 1) srll-1 2 261 (9R',09, T, A1 Az4z) + Al °r11+1

er 11-1 2

1,

a+11

A2E)1(91Ra,oqf, Al 1' 241 1) - Al

2

A201(R2Ra,04y, 1A242)

Note that each term is independent of either ql or 42. We divide the integral according to A2 < 1 or A2 > 1. If A2 < 1, we use the first equation, and if A2 > 1, we use the second equation. Then (12.10.26) is equal to the sum of the function ba,st(w) 2

/

Ez(Rao4y,wa,st,

ST11 + 1 ST11 + 1 4 , 4

12 Contributions from unstable strata

323

and the integral of the function

-

2

E1+((92Ra,o9a1If)a1,1) + Al

+ 6#(w)(A1 2 E1+((R1R',O1F)a1, 0) + Al r 1,-3

- 6#(w)(A

2

r -1

+ 6#(w)(A1

2

E1+((RlRa,o9a'F)a 1-1, 0))

2

E1+((1F2Ra,o'F),-1,1))

2

r11 2

E1+((R2Ra,o9a`F),\ 1, 0) +.

1))

D

"'+1

E1+((91Ra,o'F),\ 1, 1) + Al

E1+((R2Ra,o'F)A1, 0))

2

over 0
r -3

.,, -1 Al

2

= Al

E1+((" 2Ra,0JFa 2

2

E1+((R2Ra,oFa)a 1, 0)

2

E1,(o)((92Ra,o9a'F)a1,1) + Al rl I -1

rl 1 -3 A1

1) + Al

El+((RlR',oca'F)a 1, 0) + Al

E1+((91Ra,o9a'F)A 1,1)

2

."'-1

r11-1

Al A1

2

Al Al

2

_ Al

2

E1,(o)((FlR0',o9a'F)A1,1) + Al

EI+((RIR',o'F)A1, 0) + Al 2

2

`'

1R1,O9o1F(0)

E1+((" 1Ra,o1Y)a 1,1

2

E1,(o)((91Ra,o'F),\ 1, 1) + Al

E1+((JT2Ra,o'F), 1,1) + Al 2

92Ra,oJgaW(0),

2

2

E1,(o)((92Ra,o'F)a 1, 1) + Al

2

91Ra,o'If(0),

E1+((R2Ra,o1F)al, O) 2

Y2Ra,o'I'(0).

is the Fourier transform of R2Ra o i a'F with respect to the standard bilinear form on A etc. r11-1 By (12.10.25), terms like Al 22,9Ka1F(0) cancel out. It is easy to see that Note that _0'2 R'

A-' (log Aj)Ej,(_j)(92R',O9-o IF, 1) + Aj 1E1,(o)(92Ra,o `' a1F,1),

E1,(o)((FlRa,o9a'F)A1,1) = - Aj 1(log +A1 1E1,(o)(91Ra,o9a'F,1), E1,(o)((91Ra,o`F)a-1,1) =Ai(logA1)E1,(-1)(_F1Ra,o'F,1) + AlE1,(o)(91Ra,o'F,1),

E1,(o)((92Ra,o'F)a1) =A,(log Al)E1,(o)(92Ra,o'F,1) +AlE1,(o)(F2R'a ,OF,1).

If P is K-invariant, 92Ra,o9a'F = Ra4 n44, 92Ra,o'F = Ra4Dd42

Part IV The quartic case

324

Therefore, integrating over Al < 1, and using (12.10.25), we get the proposition. Q.E.D. We define

J7('F, W) =8(E2,(l,o)(Ra,o4,ws,st) 2, 2) + E2,(o,1)(Ra,oI,wa,st, 2, 2)), Js((D) = - E2,(-1,-1)(Ra1 p11, 0,1)

1

- 1 E2,(o,-1)(Ra1

0,1)

1

0,1)

+ E2,(-1,-1)(Ra, 4'P12, 0, 1) + 1 E2,(o,-1)(Ra1 1

+ E1,(-1)(R044)p42,1) - 21,(o)(RaJp42,1) E E 21,(o)(Ra4'Pp441 1),

Js(,P) =2(Tw+(Rw,oRa2()P21,2) +Tw+(Rw,o9wRa2'DP21,1)),

J1o(p) = 2 (Tw+(Rw,oRa2

4p22,

2) + Tw+(Rw,o9wRb2'bP22, 1))

Then by (12.10.26), oa,3t(w)J3,(o)('P, w,1) = sa,st(w)(JO), w) + J7(4,w)) + 5#(w)J8('D).

also, J4,(o)(4',1)

= J9(.') +

1(Tw+(Rw,oRa2'PP21, 2, 0) +

Tw+(Rw,o9wRa24)P211 11 0)),

J5,(o)(4),1) 1

= J10(-P) -

2(Tw+(Rw,oR024P22, 2, 0) +Tw+(Rw,o-11wRa2'DP22,1, 0))

By the principal part formula for the standard L-function in one variable, Tw+(Rw,oR021'P21, 2, 0) + Tw+(RwoJFwR02'DP21 , 1, 0) = -E2,(-1,0) (Ra1 p11, 0,1) + E1,(-1) (Ra4 'bp41 , 1),

Tw+(Rw,oR02 4P22, 2, 0) +

Tw+(Rw,o-11wRa2'kP22, 110)

= -E2,(-1,o)(Ra1 4DP12, 0, 1) +

E1,(-1)(R544)p43,1).

Therefore,

J4,(o)('D, 1) = 6#(w)(Js('P) - 2E2,(-l,o)(Ra1,Dp11, 0,1) +

A'(0)(4), 1) = b#(w)(J1o(1k) +

0,1) - 2E1,(-1)(R041)p43,1))

12 Contributions from unstable strata

325

It is easy to see that Ep11,(o) (1p, 1) =1 E2,(o,o) (Ra, 4)p1, , 0, 1) + 1 E2,(1,-1) (Ra1 2 2

0,1)

+ 2E2,(-1,1)(Ra,(Dp11,0,1)+ 2E2,(o,-1)(Ra1,Dp11,0,1) 1

1

+ 2E2,(-1,o)(Ra1 4)p1,, 0, 1) + 2E2,(-1,-1)(Ra1 4p11, 0, 1),

0,1) - 2E2,(1,-1)(Ra1

2E2,(o,o)(Ra1

0,1)

- 2E2,(-1,1)(Ra14)p12,0,1) 9

3

- 2E2,(-1,0)(RD, 4DP12, 0,1) - 2E2,(-1,-1)(Ra1 qtP12, 0,1).

Also 1

Ep42,(o)((,), 1) = 2 E1,(1)(Ra4 Dp421 1) - 2 E1,(o)(Ra4-pp42, 1)

+ Ep43,(0) ('p , 1) = 2- E 1)

1 E1,(1)

l,(o)(R544)p43,

1) - El,(-1)(Ra4 p43, 1), 2 E1,(o)(Ra4'Dp44, 1)

(Ra4Dp4411)

-

1).

We define J11(4)

2E2,(o,o)(Ra,-Dp11,0,1) 0, 1) - E2,(o,-1)(Ra14bp11, 0, 1)

2E2,(-1,1)(Ra1

- E2,(-l,o)(Ra1 4bp11, 0, 1) - 2E2,(-1,-1)(Ra1 bp11, 0, 1)

-1

2E2,(1,-1)(Re,4DP1210,1) E2,(-1,1)(Ra1 Pp, 0, 1) - E2,(o,-1)(Ra1 DP12, 0, 1)

- E2,(-1,o)(Ra1 Dp12, 0, 1) - 2E2,(-1,-1)(Ra1 Dp12, 0, 1)

-

2

E1,(1) (Ra4 OPP421 1) -

2

E1,(1)(Ra4'Dp44, 1),

1

J12(1) =2E1,(o)(Ra4'p41,1) +

+

2E1,(-1)(Ra4'Dp41, 1)

1

1

2E1,(-1)(Ra4'Dp43,1).

Then by the above considerations, 6# (W)

Ji(4),W)

To,2,00D, W) = ba,.t(W) E i=6,7

+

12

E Ji(.D) i=9

Part IV The quartic case

326

By the principal part formula (4.2.15), Ji(41) = -Ew,ad,(0) (Ra2,Dp2i , 2) - EW,ad,(o) (Ra2 4)P22, 2)* i=2,9,10,12 Therefore,

T ,,1(4', W) + Ta,2,o (4 , W )

= SD,st(w) T, Ji(4),W) + 6#(w)Jii(-D) i=6,7

- 6#(w)(Ew,ad,(0)(Ra24)P21, 2) + EW,ad,(o)(Ra2'DP22, 2)).

By (5.6.4), we get the following proposition.

Proposition (12.10.27) T

w) + Te,2,(o) (4),w)

= bb (w)Ea,ad (RD Pp, wa, 3)

+ sa,8t(w) (E2(lO)(ROwwO,St,

) + E2,(o,

2,

2

Wa,st,

2

,

2)

Chapter 13 The main theorem §13.1 The cancellations of distributions In this chapter, 0 = (/3i) for i = 1,

,10, Ds,i = and 88,2 = (,38,08,2)Let P8,2 = 08,2, $8,2), p9 = (D9, s9) where 88,2 = (138, 08,2), a9 = (09) and 58,2(1) _ 59(1), 58,2(2) = 0. Then EP8 2 = -Ep9. An easy consideration shows that R814Dpl = qDp9) Therefore, the third term of (12.8.28) and the second term of (12.9.2) cancel out. We define

i=1,6,8,10

+ 2604 (w) (Ea4 (R54 $, wa 1, -2) - Ea4 (R04 , wa, -2))

+

sa93w)

I E3(R89

22

,wa91, -1, -3

-

+ 9329J36#(w)(4(0)

3) -

-1, -3, 3)

I

4)(0)),

and

E

T2(4), w, s-11)

w-1) sT11) - TO ,2(4D, w, sT11)).

i=1,6,7,8,1°

Suppose that r = TG and b > 0 is a small number. Then by (10.1.1), (10.1.2), and the results in Chapter 12, by changing if necessary, I(4), w, w) '.'Ccn(w; p)T1(4),w) p2

+

-

fRe(ii)=1+S

siii)2(sT11)A1(w, sTll, 1)ds,11.

We get the following lemma by Wright's principle.

Lemma (13.1.1) The distribution T20), LO, ST11) is holomorphic at si11 = 1. Therefore, the cancellations of higher order terms of T2((D,w,si11) are guaranteed, and we are not obliged to check them. Hence, we get the following proposition.

Proposition (13.1.2) I°(4), w) =

w) + T2('D, w,1).

Clearly,

T2(`I', w,1) = E (Ta.,2,(0) ($, w-1, 1) -

(D, W,1))

i=1,6,7,8,10

Let ps,1 = 08,1,55,1), p6 = (x6,56) be paths such that 68,1(1) = 56(1), 58,1(2) = 0.

Then it is easy to see that E2,(i,9)(Ra8.1 `Dp8,1) -1, -3) = E2,(7,i)(Ras

-3, -1)

Part IV The quartic case

328

for all i, j. Also if p7 = (D7,57)410 = (aio, sio) are paths such that 57(1) = slo(t), E3,(1,-1,0)(Ra7I 7,w57, (w2, 1) W2), E3,(0,-1,1)(Ra7Op7, Wa7, (W2,1) W2),

2

0,

1

2) =

21

),

1, 0, 1) _ -E2,(o,1)(Ra1o,o4'p1o,wo,st, 1, 1)

22

2

2

We define (13.1.3)

f(4), w) =6a, (w)Ea1(R51'p, wa1, 2) + 2654 (w)Ea4 (R5, 4), wa4, -2) + 8656 (W)Ea6,ad(Ro6'D, was, X56, -3)

-3, -1)

+ +

+

26#(w)E2,(0,1)(Ra6io-b, -3, -1) 1 1 657(w) 6 E3,(0,o,o)(R574,w57, 2,0' 2)

+ 1465, (w)Eas,(o) (Ro,,P, woe, Xas, -3)

22 33

bag (w) 3

+ 65io (W)F'alo,ad (R51o 4), W,,o, 3)

_ +

balo,8 (w) E2

(1,o)(Ralo,o,D,walo,st)

balo,st (w) 24

E2,(0,1)(Ra1o,O

2, 2)

, walo,st, 12 , 21 )

By (13.1.2), we get the following proposition.

Proposition (13.1.4) I0(t1 w) = f * w-1) - f (C w)

§13.2 The principal part formula We are finally ready to prove the principal part formula for the zeta function for the space of pairs of ternary quadratic forms. For 4D E .'(VA), X E lam, let 4)5(x) = D(Ax) as before. The following relations are easy to verify, and the proof is left to the reader.

Lemma (13.2.1) 4)),(0) _ (D(o).

(1) (2)

(3) (4) (5)

E51(R51,Da,w51,2) = E54 (R54

)' , Wa4, -2) = E54 (R54

, w54, -2).

Ea6,ad(RD6(1) A,W56,X56,-3) = E56,ad(Ro64),w56,X56,-3). E2,(1,o)(Ra6 O4)A, -3, -1) = E2,(1,0)(Ra6,04), -3, -1)

- (log A)E2(R'a6,o4), -3, -1). (6)

E2,(o,1)(Ra6i04)a, -3, -1) = E2,(0,1)(Ra6i0-D, -3, -1)

- (log A)E2(RD6,0 4)' -3,-1).

13 The main theorem

329

2, 0,21

(7) 6D,

wa7, 2, 0,2)

1, 1

+ A -3 tog (logXA)SD 1ost( W) E2(R'Dio,o cj, walost, 2, 2

(8) 6ae (W)ED8,(0) (Rae A, was, Xae, -3) = 8 x8 (w)ED8,(o) (Ras 4), WD8, X08, -3)

+

2A(log.)6#(w)E2(R0g,oRag,D, -3, -1).

2,2

(9)

2

3)

(10) (11)

3)

EDio,ad(RD1o4)a,WD1o,3) = A-3ED1o,ad (R,1oID,Wa1o,3). E2,(1,0)(R0'

X-3E2,(1,o)(Ra1o,oID,WD1o,st,1,1) 2 2

2 2

1

1

- A-3 (log.)E2(Ra1o,0-D, w,1o,st, 2, 2 )

Note that (8) of the above lemma follows from (7.3.7). Similar relations hold for also. We define F(-1) (1, w, 0) =9J29T36# (w)4)(0) + 26D, (w)ED, (Ra4 D, W041 -2)) + 86Dg (W )Eag,aa (Rag

, was, Xo6 , -3)

+ 146x8 (W)EaB,(o) (Rae,D, Loa, XD81 -3)

+

6#(U)) (5E21(1,0)(Ra,,,0

, -3, -1) + 3E2,(o,1)(Ra6,0 (1), -3, -1)),

36#(w)E2(R0' g,oD, -3, -1),

F(-2)(C W, 0)

6

F(-1)((D, w, 2) =6a1(w)ED,(RD,Ob,wD,,2)+ F(-1)(4), w, 3)

(w)E3(Ra94),wa9,-1,-3, 3),

Wa 2' 0, 2 + 601o (w)Ealo,ad(RD1o D) wD1o, 3))

_

(w)

6D 1o,

8

412,(1,0) (R'610'0 4) , W01o,st, 2,

+ halo,st(w)E2 24

6alo,st(w)

F(-2)(1,w,3)

4

,

(o ,

2)

1), 1)(R' 10, 04),wD lo,st, 1, 2 2 ,

1

E2(R1o,0(D,wojo,st, 2>

1

2)

We define

F(

w)s) = F(-1)(4),w,2) s-2 +

F(-j) 03 (s - i)i j=1,2

By replacing

Theorem (13.2.2) Zv(t1, w, s) = Zv+(-D, w, s) + Zv+($, w-1,12 - s) - F($, w-1,12 - s) - F(,D, w, s).

330

Part IV The quartic case

§13.3 Concluding remarks We used the passing principle (3.6.1) in various places to move contours. Instead of establishing cancellations of various distributions, it enabled us to ignore them as long as we can check some easy conditions. We also used Wright's principle to ignore higher order terms of certain distributions. However, it is possible to study the poles without these two methods, and in fact, the author's original approach was to establish cancellations explicitly. It is an interesting exercise to establish cancellations of various contour integrals without using the passing principle (3.6.1) or Wright's principle, because one can see that the cancellations between different paths are happening. They correspond to cancellations between divergent integrals, and our two methods and the choice of the constants in the definition of the smoothed Eisenstein series enabled us to handle most of them without establishing the cancellations. In Chapter 12, we computed contributions from unstable strata, and obtained more or less special values of the zeta functions for the corresponding representations. Those zeta functions have higher order poles, so we had to recover the higher order terms. When we analyzed the constant terms in Chapter 12, they came not only from the unstable strata, but from the adjusting terms also. The author does not know if this is a special case of something more general. But for this reason, we had to compute the contributions from unstable strata explicitly, and compare with principal part formulas of the corresponding zeta functions. This was rather annoying, and the author hopes to find a better interpretation of this phenomenon in the future. If the center of the stabilizer of a point in Zap contains a split torus, we could ignore jp (,D, w, w) in most cases. However, this is not always the case, for example, in the case G = GL(2) x GL(2), V = k2 ® k2. This is another reason why we still cannot handle all the cases simultaneously. We restricted ourselves to groups which are products of GL(n)'s, mainly because we do not have the uniform estimates of Whittaker functions. As far as the formulation of equivariant Morse theory is concerned, there does not seem to be any major problem in generalizing our formulation. However, for other groups, we still do not know how we can describe the Eisenstein series in terms of generalized Whittaker functions, and until we find out the answer to this problem, we cannot handle other groups. On the other hand, there is no doubt that the cases where the group is a product of GL(n)'s contain the most interesting cases (see [84]), and we hope our approach enables us to handle more cases. It is clear from the computations in this book that the complexity of our computations increases exponentially with the rank of the group. Therefore, the number of the distributions we have to handle can be quite large, and may require a computer calculation. However, we still do not know when the significant cancellations as in §10.3, §10.6 and §13.1 will occur. Also, in order to do such a task, our theory has to be generalized so that we can handle reducible representations. However, these are questions to be answered in the future.

Bibliography [1] Borel, A. Some finiteness properties of adele groups over number fields. Publ. Math. IHES, no 16:101-126, 1963.

[2] Borel, A. Linear algebraic groups, 2nd enlarged edition. Springer, Berlin, Heidelberg, New York, 1991.

[3] Borevich, Z. I, and I. R. Shafarevich. Number theory. Academic Press, 1966.

(4] Brioschi, F. Sulla risoluzione delle equazioni del quinto grado. volume I, pages 335-341. Ulrico Hoepi, Milano, 1904.

In Opere mathematiche,

[5] Bump, D. Automorphic forms on GL(3, R), volume 1083. Springer Lecture Notes in Mathematics, Berlin, Heidelberg, New York, 1984.

[6] Bump, D. The Rankin-Selberg method: a survey. In Number theory, trace formula and discrete groups (Oslo 1987), pages 49-109. Academic Press, Boston, 1989.

[7] Casselman, W., and J. Shalika. The unramified principal series of p-adic groups II: The Whittaker function. Compositio Math., 41:207-231, 1980.

[8] Cogdell, J. W. Congruence zeta functions for M2(Q) and their associated modular forms. Math. Annalen, 266:141-198, 1983.

[9] Datskovsky, B. A mean value theorem for class numbers of quadratic extensions. To appear in the memorial volume for Emil Grosswald, 1992. [10] Datskovsky, B., and D. J. Wright. The adelic zeta function associated with the space of binary cubic forms II: Local theory. J. Reine Angew. Math., 367:27-75, 1986.

[11] Datskovsky, B., and D. J. Wright. Density of discriminants of cubic extensions. J. Reine Angew. Math., 386:116-138, 1988.

[12] Davenport, H. On the class-number of binary cubic forms I and II. London Math. Soc., 26:183-198, 1951. Corrigendum: ibid., 27:512, 1952.

[13] Davenport, H., and H. Heilbronn. On the density of discriminants of cubic fields I. Bull. London Math. Soc., 1:345-348, 1961.

[14] Davenport, H. and H. Heilbronn. On the density of discriminants of cubic fields. II. Proc. Royal Soc., A322,:405-420, 1971.

[15] Denef, J. On the evaluation of certain p-adic integrals. In Progress in Math. No. 59, pages 25-47. Birkhauser, 1985. [16] Gauss, C. F. Disquisitiones arithmeticae. Yale University Press, New Haven, London, 1966.

[17] Goldfeld, D., and J. Hoffstein. Eisenstein series of 2-integral weight and the mean value of real Dirichlet series. Invent. Math., 80:185-208, 1985. [18] Gray, A., and G. B. Mathews. A treatise on Bessel functions and their applications to physics. Macmillan, London, 1952.

[19] Gyoja, A. Gauss sums of prehomogeneous vector spaces. Proc. Japan Acad., Math. Sci. 61 Ser. A no. 1:19-22, 1985. [20] Hermite, C. Sur la resolution de 1'equation du cinquieme degre. In Oevres, volume II, pages 5-12. Gauthier-Villars, Paris, 1908. [21] Hironaka, Y., and F. Sato. Spherical functions and local densities of alternating forms. Amer. J. Math., 110:473-512, 1988. [22] Igusa, J. Some results on p-adic complex powers. Amer. J. Math., 106:1013-1032, 1984.

[23] Igusa, J. On functional equations of complex powers. Invent. Math., 85:1-29, 1986.

Bibliography

332

[24] Igusa, J. On a certain class of prehomogeneous vector spaces. J. of Pure and Applied Algebra, 47:265-282, 1987.

[25] Igusa, J. Zeta distributions associated with some invarinats. Amer. J. Math., 109:1-34, 1987. [261 Igusa, J. b-functions and p-adic integrals. In Algebraic analysis, pages 231-241. Academic Press, 1988.

[27) Igusa, J. On the arithmetic of a singular invariant. Amer. J. Math., 110:197-233, 1988. [281 Kempf, G. Instability in invariant theory. Ann. of Math., 108:299-316, 1978. [29] Kempf, G., and L. Ness. The length of vectors in representation spaces. In Algebraic Geometry, Proceedings, Copenhagen, volume 732, pages 233-242. Springer Lecture Notes in Mathematics, Berlin, Heidelberg, New York, 1978.

[30] Kimura, T. Study of irreducible prehomogeneous vector spaces. Master thesis in Japanese, Univ. Tokyo, 1974.

[31] Kimura, T. The b-functions and holonomy diagrams of irreducible regular prehomogeneous vector spaces. Nagoya Math. J., 85:1-80, 1982. [32] Kimura, T., and I. Ozeki. On the micro-local structure of the regular prehomogeneous vector space associated with Spin(10) X GL(3). Proc. Japan Acad., Math. Sci. 58 Ser. A:18-21, 1980.

[33] Kimura, T., and M. Muro. On some series of regular irreducible prehomogeneous vector spaces. Proc. Japan Acad., Math. Sci. 55 Ser. A:384-389, 1979.

[34] Kimura, T., F. Sato, and X.-W. Zhu. On the poles of p-adic complex powers and the bfunctions of prehomogeneous vector spaces. Amer. J. Math., 112:423-437, 1990.

[35] Kirwan, F. C. Cohomology of quotients in symplectic and algebraic geometry. Mathematical Notes, Princeton University Press, 1984.

[36] Klein, F. Weitere Untersuchungen uber das Ikosaeder. In Abhandlungen, volume II, pages 321-384. Springer, Berlin, 1922. [37] Klein, F. Lectures on the icosahedron and the solution of equations of the fifth degree. Dover, New York, 1956.

[38] Kneser, M. Lectures on Galois cohomology of classical groups. Tata Lecture Note, Bombay, 1969.

[391 Kronecker, L. Sur la resolution de l'equation du cinquieme degre (extrait d'une lettre adressee a M. Hermite). In Werke, volume IV, pages 43-47. Teubner, Leipzig, Berlin, 1929.

[40] Langlands, R. The volume of the fundamental domain for some arithmetical subgroups of Chevalley groups. In Proceedings of symposium in pure mathematics, volume 9. 1966. [41] Langlands, R. On the functional equations satisfied by Eisenstein series, volume 544. Springer Lecture Notes in Mathematics, Berlin, Heidelberg, New York, 1976.

[42] Lipschitz, R. In Sitzungsber, pages 174-185. Akad. Berlin, 1865.

[43] Loeser, F. Fonctions d'Igusa p-adiques et polynomes de Bernstein. Amer. J. Math., 110:1-22, 1988.

[44] Milnor, J. Morse theory. Annals of Mathematical Studies, Princeton University Press, 1969.

[45] Muller, I. Theorie des groupes-decomposition orbitale des espaces prehomogenes reguliers de type parabolique commutatif at application. C. R. Acad. Sci. Paris, t. 303 Serie I, n° 11, 1986.

[46] Mumford, D., and J. Fogarty. Geometric invariant theory, 2nd edition. Springer, Berlin, Heidelberg, New York, 1982.

[47] Narkiewicz, W. Elementary and analytic theory of algebraic numbers. PWN, Warszawa, 1974.

[48] Ness, L. A stratification of the null cone via the moment map. Amer. J. Math., 106:1281-1325, 1984.

Bibliography

333

[49] Ozeki, I. On the micro-local structure of a regular prehomogeneous vector space associated with SL(5) x GL(4) I. Proc. Japan Acad., Math. Sci. 55 Ser. A:37-40, 1979. [50] Ozeki, I. On the micro-local structure of a regular prehomogeneous vector space associated with GL(8). Proc. Japan Acad., Math. Sci. 56 Ser. A:18-21, 1980. [51] Piatetski-Shapiro, I. I. Euler subgroups. In Lie Groups and their Representations, pages 597-620. John Wiley and Sons, New York, Toronto, 1975. [52] Rubenthaler, H. Espaces prehomogenes de type parabolique. These, Universite Strasbourg, 1982.

[53] Rubenthaler, H. Espaces prehomogenes de type parabolique.

Lect. Math. Kyoto Univ.,

14:189-221, 1982.

[54] Rubenthaler, H., and G. Schiffmann.

Operateurs differentiels de Shimura et espaces

prehomogenes. Invent. Math., 90:409-442, 1987.

[55] Sato, F. On zeta functions of ternary zero forms. J. Fac. Sci. Univ. Tokyo, Sect IA, 28:585604, 1981.

[56] Sato, F. Zeta functions in several variables associated with prehomogeneous vector spaces I: Functional equations. Tohoku Math. J., (2) 34:no. 3 437-483, 1982. [57] Sato, F. Zeta functions in several variables associated with prehomogeneous vector spaces III: Eisenstein series for indefinite quadratic forms. Ann. of Math., 116:177-212, 1982.

[58] Sato, F. Zeta functions in several variables associated with prehomogeneous vector spaces II: A convergence criterion. Tohoku Math. J., (2) 35 no. 1:77-99, 1983.

[59] Sato, M., and T. Kimura. A classification of irreducible prehomogeneous vector spaces and their relative invariants. Nagoya Math. J., 65:1-155, 1977. [60] Sato, M., and T. Shintani. On zeta functions associated with prehomogeneous vector spaces. Ann. of Math., 100:131-170, 1974. [61] Sato, M., M. Kashiwara, T. Kimura, and T. Oshima. Micro-local analysis of prehomogeneous vector spaces. Invent. Math., 62:117-179, 1980.

[62] Serre, J. P. Cohomologie galoisienne, volume 5. Springer Lecture Notes in Mathematics, Berlin, Heiderberg, New York, 1965.

[63] Serre, J. P. Extensions icosaedrique. In Oeuvres, volume III, pages 550-554, Springer, Berlin, Heiderberg, New York, 1986. [64] Shintani, T. On Dirichlet series whose coefficients are class-numbers of integral binary cubic forms. J. Math. Soc. Japan, 24:132-188, 1972.

[65] Shintani, T. On zeta-functions associated with vector spaces of quadratic forms. J. Fac. Sci. Univ. Tokyo, Sect IA, 22:25-66, 1975. [66] Shintani, T. On an explicit formula for class 1 Whittaker functions on GLn over p-adic fields. Proc. Japan Acad., 52:180-182, 1976. [67] Siegel, C. L. Uber die analytische Theorie der quadratischer Formen. Ann. of Math., 36:527606, 1935.

[68] Siegel, C. L. Uber die Zetafunctionen indefiniter quadratischer Formen. Math. Z., 43:682-708, 1938.

[69] Siegel, C. L. The average measure of quadratic forms with given discriminant and signature. Ann. of Math., 45:667-685, 1944.

[70] Siegel, C. L. On the theory of indefinite quadratic forms. Ann. of Math., 45:577-622, 1944.

[71] Stade, E. On explicit integral formulas for GL(n,R)-Whittaker functions. Duke Math. J., 60:313-362, 1990.

[72] Suzuki, M. Group theory, volume I. Springer, Berlin, Heiderberg, New York, 1982.

334

Bibliography

[73] Tate, J. Fourier analysis in number fields and Hecke's zeta function. Ph.D. thesis, Princeton University, 1950.

[74] Van der Waerden, B. History of algebra. Springer, Berlin, Heidelberg, New York, 1985. [75] Vinberg, E. B. On the classification of the nilpotent elements of graded Lie algebras. Soviet

Math. Dokl., 16:1517-1520, 1975.

[76] Vinogradov, I. M. Selected works. Springer, Berlin, Heidelberg, New York, 1985.

[77] Weil, A. Sur certain groupes d'operateurs unitaires. Acta Math., 111:143-211, 1964. [78] Weil, A. Sur la formule de Siegel dans la theorie des grouper classiques. Acta Math., 113:1-87, 1965.

[79] Weil, A. Basic number theory. Springer, Berlin, Heidelberg, New York, 1974.

[80] Weil, A. Adeles and algebraic groups. Birkhauser, Boston, Stuttgart, 1982. [81] Weyl, H. The classical groups. Princeton Univ. Press, 1946.

[82] Whittaker, E. T., and G. N. Watson. A course of modern analysis. Cambridge Univ. Press, 1927.

[83] Wright, D. J. The adelic zeta function associated to the space of binary cubic forms part I: Global theory. Math. Ann., 270:503-534, 1985. [84] Wright, D. J., and A. Yukie. Prehomogeneous vector spaces and field extensions. Invent. Math., 110:283-314, 1992. [85] Yukie, A. Prehomogeneous vector spaces Eisenstein series and invariant theory. preprint 1992.

[86] Yukie, A. On the Shintani zeta function for the space of binary quadratic forms. Math. Ann., 292:355-374, 1992.

List of symbols f (x) << g(x) §0.1 VR5 §0.2

o(x), oco(x), µ(x), µ,,,(x), b,,,,, §0.3 ZZ,,,(4p) s), Xx,,,(4D) s),

s) §0.3 b(s), b*(s) (0.3.4)

an(tl) ... , tn), nn (U) (1.1.1)

K=

§1.1

p (half the sum of positive weights) §1.1 O(s), On(s) (1.1.2) P after (1.1.2)

93n (1.1.3)

NT, NT (2.1.4) IT after (2.1.5) (2.2.1)

a(u), a

(2.2.2)

Ea(g°, z) (2.2.3) M(z) (2.2.8), (2.2.10) TG before (2.3.1), after (3.1.8)

Wn(a, z) etc. (2.3.1) An,,, (t, z) (2.3.7) Qn,,,(a, z) (2.3.12)

D(n, r) (2.3.16) K, (x) (2.3.18) 6(a,t) (2.3.25) W, (a, z) (2.4.2) El,,,,(g°, z) (2.4.6) Dr, DI,T etc. (2.4.7) T+ before (3.1.1) OL(4),g) (3.1.1) ZLN, W, X, S), ZL+(D, W, X, s) (3.1.2) VkS (3.1.3)

rdi,M(µ) (3.1.5) w(g°) (3.1.7) [x,y]v (3.1.9), (3.1.10)

g° after (3.1.10) Mv,u, (3.1.11) t, ( , ), 11 11 §3.2 (3.2.3)

0 = (01, ... , 13a) (3.2.6) Pa, Ma after (3.2.7) Ma the end of §3.2

go after (3.3.1) RD IF,RaI1 (3.3.2) ODp, Op (3.3.3), (3.3.4), (3.3.5) Po#, Ua#, Pa (3.3.6)

List of symbols

336

P'DA , M'A D

etc. (3.3.7)

S4, Tai after (3.3.7) epi (3.3.9), (3.3.10)

Apo etc. (3.3.11) ga, dgo after (3.3.11), before (3.3.13) ep (3.3.12) wp after (3.3.12) °p (3.3.13) to the beginning of §3.4 (, )o, II 11° after (3.4.1) L(z), A(w; z), A, (w; z) (3.4.2), (3.4.3), (3.4.6) MT(z) (3.4.5) e(g°, w, V)) (3.4.7) ep (ge, w), Sp' (ge' n(ua' ), w) (3.4.8), (3.4.9), (3.4.10)

8u(g°, w), eu,r(g°, w) (3.4.12), (3.4.14) rp etc. after (3.4.13) (A ga)T (3.4.17) C(A'oMM°A/M k, r) (3.4.18) eI,T,v(g°, w) etc. (3.4.27) CG (3.4.31)

fi(w)

12 (w) (3.4.32)

es, ('1'i, 9a) etc. (3.5.1) p (-D, w, w) etc. (3.5.2) Jo(41,go),Ja(`D,go) (3.5.6)

p(,P,w,w) (3.5.8) 4a _after (3.5.9)

s, A(w; sT) §3.6 pT §3.6

Ei('yi, S), Ei,(j .....

so)

(3.8.1), (3.8.2)

Sn,i etc. §4.0.

W (the space of binary quadratic forms) the beginning of Chapter 5. Xv (4.2.2), §5.5 Tv2 (W, w, s, sl) etc. (4.2.3) Ro, Ro, Rv,o, Ra2,o etc. (4.2.10), (5.3.2) ZV,ad(D, w, s) (4.2.14), (5.6.3) a,i etc. (4.3.1)

.fo, ho (4.3.2)

cp (4.3.3), (4.3.4) '131 etc. after (4.3.4) ZV2,ad,(i) ('I', W, so) etc. (4.4.1)

Ra,o, Ro,o (4.4.2)

Ip(I,w) etc. (4.4.5) ez;,o (Ro Dp, go) etc. (4.6.1)

Xo etc. after (4.6.2) t°, to (5.1.2), (9.2.4)

TV (c w, s, Si, s2) etc. (5.2.1), (6.3.1) Ozv,0 (wi)t', uo) etc. (5.2.4)

List of symbols

Xa2,Xa §5.4, (11.1.8) =a,i (4),w, X, w) (6.1.7), (6.1.8)

Zero(x) §8.2 =p,al,i (,', w, w) (9.1.7), (9.1.8) Ep,tot(4,W,W) (9.1.9) n'(u),n"(u) (9.2.3) 01,02 (11.1.2)

L(sT), A(w; sT) etc. (11.1.3) RW,o etc. before (11.1.6) ®W,st(W, 9) etc. (11.1.6), (11.1.7) OZ,,o (IF, h) (11.1.10) Ei,sab(W,1, sT) (11.1.14) Esub(1, t, 9°, Sr), 4sub(1, t, 9°, w) (11.1.15) cW,T(`y,1, w), EW,sub+(,P, 1, sT) etc. (11.1.15), (11.1.16), (11.1.17)

91' before (12.8.10)

337

Index

adjusting term, 91, 105, 118, 156, 162, 172, 179, 183,252 adjusted zeta function, 105, 118, 172, 179, 183 asymptotic formula, 1, 18 ,B-sequence, 69, 122, 154, 180, 209 Bessel function, 49, 50 b-function, 8 bilinear form, 12, 65, 71, 99, 103, 108, 155, 173, 180, 209

binary quadratic form, 1, 12, 18, 149, 180, 251 binary cubic forms, 2, 18 Borel subgroup, 23, 29, 34, 66, 149 Bump, D., 29, 49 Casselman, W., 42, 43 characteristic function, 12, 15, 106, 113 class number, 1 classification, 4, 14 Cogdell, J., 98, 104 complete type, 6, 8, 64, 91, 196 constant term, 34, 196, 262 contour, 80, 94, 96, 138, 146 contragredient representation, 7, 12 convex hull, 63, 67, 87, 109, 199, 224 cubic equation, 3

Gamma function, 49 Gauss, C. F., 1 geometric invariant theory, 21, 67 Goldfeld, D., 2, 18 Grassmann variety, 4 half the sum of positive weights, 23, 34, 62, 65, 72 Heilbronn, H., 18, 19 Hilbert, D., 67 Hoffstein, J., 2, 18 hyperfunction, 151 ideal class, 1 Igusa, J., 8, 13, 20 incomplete type, 6, 12, 20, 64, 91, 105 infinite place, 7, 29, 49, 50 integral orbits, 13 integral test, 24, 35 invariant measure, 23 involution, 12, 65, 71 irreducible, 4, 5, 20, 76, 85 Iwasawa decomposition, 23, 34, 65, 112, 224

Kempf, G., 21, 68

Khayyam, 0., 3 Kimura, T., 4, 10, 20, 21, 70 Kirwan, F., 21, 68

Datskovsky, B., 2, 15, 18, 107 Davenport, H., 6, 18, 19 density theorem, 2, 20 differential operator, 8 Dirichlet series, 12, 17, 104, 106, 149 discriminant, 1, 18, 19, 150

k-stable, 63, 67, 69, 179

Eisenstein series, 22, 29, 149 equivalence class, 1, 13, 107 equivariant Morse theory, 21, 66, 151 error term, 1, 13 Euler factor, 16, 19 explicit formula for p-adic Whittaker functions, 29, 42

Lipschitz, R., 1

Laurent expansion, 99, 123, 142, 152, 195 length, 31, 69, Levi component, 5, 68, 70, 75 Liouville's theorem, 12 Lindel5f's theorem, 12 local theory, 7, 15, 18, 20 longest element, 40, 65, 71, 76 Mass formula, 150 maximal compact subgroup, 29 (see 23 also),41, 65, 125

filtering process, 15, 20 finite place, 8, 29, 42 Fourier analysis, 1 Fourier expansion, 29, 30, 55 Fourier transform, 8, 65, 71, 108, 173, 209 functional equation, 5, 8, 119, 173, 194 fundamental domain, 167

measurable function, 24, 112, 133 measure (Haar measure), 5, 8, 12, 19, 23, 38, 65, 73, 95, 100, 122, 125, 158 Mellin transform, 49 meromorphic continuation, 5, 20, 114, 123, 156, 181 micro-local analysis, 151 minimal combination of weights, 67, 198 Mumford, D., 67

Calois cohomology, 13 Calois group, 13

Ness, L., 21, 68 non-constant term, 75, 196, 216

Index

1 PS, 23, 67, 72

orbit, 2, 7, 13, 18, 21, 72, 112 orbit decomposition, 21, 70 Ozeki, I., 10, 21, 70

pairs of ternary quadratic forms, 3, 11, 19, 196 parabola, 3 parabolic subgroup, 5, 62, 68, 70, 71, 75, 122, 203 parabolic type, 5 passing principle, 96 path, 71, 125, 131, 132, 158, 196, 216, 262 permutation, 14, 23, 31, 35, 55, 76, 250 Pjateckij-Shapiro, I., 29 Poisson summation formula, 9, 28, 89, 155, 180, 208 pole, 10, 18, 21, 97, 104, 142, 149, 185, 196 prehomogeneous vector space, 2, 29, 62, 67, 70, 85, 91, 98, 105, 152, 173 principal part, 11, 18, 20, 62, 91, 99, 107, 152, 169, 181, 191, 196, 273, 293, 308, 326 principal quasi-character, 62 product form, 8, 15 properly-stable (point), 66 quartic case, 93, 152, 196 quadratic field, 1 quadratic form, 5, 9, 14, 19, 105, 196 quartic equation, 3

rational character, 2, 23, 69, 73, 173, 197 rationality, 21, 67

rational orbits, 13 reduced, 4 reducible, 8, 91, 173, 196 reductive group, 2, 66 regular, 5 regular elements, 69 regulator, 2 relative invariant polynomial, 3, 5, 198 representation, 2, 18, 65, 69, 91, 152 residue, 15, 106 Riemann zeta function, 12 Rubenthaler, H., 5 Sato, M., 2, 4, 8 Sato, F., 8, 20, 152

339

semi-stable (point), 3, 66, 87, 180, 198 Shalika, J., 42 Shintani, T., 2, 6, 8, 20, 29, 42, 50, 98, 105, 118, 149 Shintani's lemma, 62, 73, 84, 124 Siegel, C. L., 1, 105, 149 Siegel domain, 24, 60, 63, 72, 82, 103, 135, 141, 174, 211

Siegel parabolic subgroup, 5 Siegel-Weil formula, 14, 149 single orbit cases, 14, 20 smoothed Eisenstein series, 22, 73, 75, 84, 100, 149, 152, 196, 212 split torus, 5, 23, 62, 66, 87 square free integer, 1 stabilizer, 6, 19, 67 Stade, E., 40 Stirling's formula, 49 (Morse) stratification, 21, 66, 153, 198

Tauberian theorem, 13, 17, 106 tempered distribution, 15 ternary quadratic form, 3 theta series, 21, 24

unipotent radical, 5, 68, 75 unstable (point), 66, 102, 122, 152, 186, 262 unstable distributions, 185, 250 vanishing principle, 95 Vinberg, E. B., 5 volume, 6, 12, 24

Watson, C. N., 49 Weil, A., 25, 104, 149 weighting, 75 Weyl chamber, 66, 69 Weyl group, 23, 34, 65, 69, 96, 99, 254

Whittaker, E. T., 49 Whittaker function, 29, 40, 50 Wright, D., 15, 97 Wright's principle, 97, 104, 121, 170, 192, 196, 327, 330

Zariski open, 2, 5, 13 zero set, 3, 198 zeta function, 5, 20, 62, 105, 155, 180, 196 Zhu, X: W., 20

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