THE UNIVERSITY SERIES IN MATHEMATICS Series Editor: Joseph J. Kohn
Princeton University INTRODUCTION TO PSEUDODIFFERENTIAL AND FOURIER INTEGRAL OPERATORS Fran~ois
Treves
VOLUME 1: PSEUDODIFFERENTIAL OPERATORS VOLUME 2: FOURIER INTEGRAL OPERATORS A SCRAPBOOK OF COMPLEX CURVE THEORY C. Herbert Clemens
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Complex Curve 'Ihcory C. I-Icrhcrt (]crncns I -/111'('
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R. PLENUM PRESS
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i· \\ 'l ( ·•F h. A' n I ON DON
Ck 1cns, Charles Herbert, 1939;crapbook of complex curve theory. University series in mathematics) P illiography: p. I .:ludes index. Curves, Algebraic. 2. Functions, Theta. 3. Jacobi varieties. l. Title. II. Series: Un crsity series in mathematics (New York, 1980-. ) Q,o ·h5.C55 516.3'5 80-20214 lSI -! 0-306-40536-9 (' he
@ 1980 Plenum Press, New York A Division of Plenum Publishing Corporation 22~ West 17th Street, New York, N.Y. 10011
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f'J•
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Preface
This is a book of "impressions ol a journey through the theory of con'· plex algebraic curves. It is neither self-contained. balanced, nor particularly tightly organized. As with any notebook made nn a journey, what appe::~.r~ is that which strikes the writer's fancy. Some topiCS ar,Jear because of their compelling intrinsic beauty. Others are left out hccause, for all their impor .. tance, the traveler found them boring or was too dull or lazy to give them their due. Looking back at the end of the journey. one can see that a common theme in fact does emerge, as is so often thr case: that theme is the thcorv of theta functions. In fact very much of the material in the book is prepara tion for our study of the final topic, the so-calkd Schottky problem. More than once, in fact, we tear ourselves away frum interesting topics lead ir_1~· elsewhere and return to our main route. -- -·Some of the subjects are extremely elementary. In fact, we begin wtth some musings in the vicinity of secondary school algebra. Later, on occasion and without much warning, we _jump into some fairly deep water. Our intent is to struggle with some deep topics in much the same way that a beginning researcher might, using whatever tools we have at hand or can grab somehow or other. Sometimes we use no background material and do everything in detail; sometimes we use some of the heaviest of modern machinery. We hope to motivate further study or, preferably, further discussion with an expert in the fkld. In short, our aim is to motivate and stimulate mathematical activity rather than to present a finished product. and our point of view is romantic rather than rigorous. The material treated here was originally brought together for a Summer Course of the Italian National Research Council held in Cortona, Italy. in 1976. It comes from so many sources that adequate acknowledgment would be difficult. The treatment of real two-dimensional geometries of
Preface
constant curvawr~ com.:s from Cartan's classic text on Riemanqian geometry; ;(·vera! items concerning the arithmetic of curves are borrowed from Serre's lovely book, A Course in Arithmetic; Manin's beautiful I heorem on rational points of elliptic curves given in Chapter Two was cxpla inel1 to the author hy A. Beau ville; some of the theta identities in Chapt~.::1 Thn~~~ an: lifted from the famous analysis text of Whittaker and Watson, and the construction of the level-two moduli space for elliptic curves was motivated by David Mumford'~ way of viewing the moduli space of curves of a Jlxed genus. The discussion of the Jacobian variety in Chapter Four leans heavily on work of Joseph Lewittes, and the discussion of the Schottky problem comes from work of Accola, Farkas, Igusa, and Rauch. But perhaps the author's greatest debt is to Phillip Griffiths, through whom he came to enjoy this subject. The author also wishes to thank Sylvia M. Morris, Mathematics Department of the University of Utah, for preparing the manuscript, and.Toni W Bunke-r, of. the-same department, for preparing the figures. Herbert Clemens Salt Lake City, Utah
Contents Notation
ix
Chapter One • Conics 1.1. Hyperbola Shadows 1.2. Real Projective Space, The "Unifier" 1.3. Complex Projective Space, The Great "Unifier" 1.4. Linear Families of Conics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5. The Mystic Hexagon... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6. The Cross Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . 1.7. Cayley's Way of Doing Geometries of ~onstant Curvature......... 1.8. Through the Looking Glass 1.9. The Polar· Curve ............................_._.,.................... 1.10. Perpendiculars in Hyperbolic Space 1.11. Circles in the K-Geometry 1.12. Rational Points on Conics
I 5 7 9 11 13 17 20 22 26 30 33
Chapter Two Cubics 2.1. Inflection Points 2.2. Normal Form for a Cubic 2.3. Cubics as Topological Groups .................................... . 2.4. The Group of Rational Points gn a Cubic ........................ . 2.5. A Thought about Complex Cofijugation 2.6. Some Meromorphic Functions on Cubics ........................ . 2.7. Cross Ratio Revisited, A Moduli Space for Cubics .•....•.......... 2.8. The Abelian Differential on a Cubic ............ ; ..... _............ . 2.9. The Elliptic Integral 2.10. The Picard-Fuchs Equation ....................................... . 2.11. Rational Points on Cubics over IFP ............................... . 2.12. Manin's Result: The Unity of Mathematics ...................... . 2.13. Some Remarks on Serre Duality ......................... ·........ .
37 39
42 45
50 51 52 53
55 58 62 ~s
69
Contents
Chapter l'hree .\.1 J.2. 3.3. .l.4. 3.5. 3.6. 3.7. 3.8. 3.9. 3.10. 3.11. 3.12. 3.13. 3.14. 3.15.
Chapter Four 4.1. 4.2. 4.3. 4.4. 4.5. 4.6.
4. 7. 4.8.
4.9. 4.10. 4.11. 4.12.
Theta
Fmu~tions
Hack to th~: Uroup Law on Cubics You Can't Parann~trize a Smooth Cubic Algebraically Meromorpluc Fun.:tiuns on Elliptic Curves Meromorphic Functions on Plane Cubics The Weierstra~s p-hmction Theta-Null Values Give Moduli of Elliptic Curves The Moduli Space of" Level-Two Structures" on Elliptic Curves Automorphisms of Elliptic Curves . . . . . . . . . . . . . . . . The Moduli Space of Elliptic Curves And So, By the Way, We Get Picard's Theorem The Complex Structure of..// The j-Invariant of an Elliptic Curve Theta-Nulls as Modular Forms A Fundamental Domain for r 2 Jacobi's Identity
95 96 98
100 102 106 109 111
• The Jacobian Variety
Cohomology of a Complex Curve Duality The Chern Class of a Holomorphic Line Bundle Abel's Theorem for Curves The Classical Version of Abei's Theorem The Jacobi Inversion Theorem Back to Theta Functions The Basic Computation Riemann's Th~:orem Linear Systems of D..:grec f/ Riemann's Constant Riemann's Singulariti~::s Theorem
Chapter Five
113 116 118 122 127 131 132 134 136 138 139 142
Quarries and Quintics
5.1. Topology of Plane Quartics .~.-"fhe Twenty-Eight Bitangents ,-- 5.3. Where Are the Hyperellipti,: Curves of Genus 3? 5.4. Quintics
Chapter Six
73 75 78 82 85 89 92
•
147 150 155 158
The Schottky Relation
6.1. Prym Varieties ..
161
6.2. Riemann's Theta Rclauon 6 3. Products of Pairs of Theta Functwus 6.4. A Proportionality Theorem Rdating Jat:obians and Pryms 6.5. The Proponionality Thwrem of Sd:utlky .lung 6.6. The Schollk:, Relation
164 167 168
173 174
HEI·I·.IU:NCES
181
1"--nE\
18.~
Notation Most of the notation used in this book is quite standard, for example. 1L = ring of integers,
Q =field of rational numbe~, IR = field of real numbers,
C =field of complex numbers. Each of the six chapters is divided into sections, for instance, Chu lLer Three has Sections 3.1, 3.2, etc. Equations are numbered consecut i 'ely within chapters-(3.1), (3.2), etc.-as are the figures. Square brackets will be used to enclose matrices and are also sed later in the book in expressions involving theta functions with chara, .eristic, for example, O[l](u; r) When there are complicated exponents, the exp form of the expo .:!ntial is used with the convention exp{x} = e".
CHAPTER ONE
Conics
1.1 HyperbOla Shadows Let's start our study of curves in a very elementary way, with the ct th<1f we learned about in school, and let's engage in some reasoning a ~'-'L vcs just as we might have done when we first met them. This very elementary, of course, but it will set the proper tone for follows---much less elementary material approached in a similar way. Sometime or other during his formal schooling, every student of 1; ematics has studied the set of solutions to the equation
Ax 2 +Bxy+Cy 2 +Dx+Ey+F=0,
rves 'out ; all :hat ath-
(1.1)
where A, B, C, D, E, and F are given constants. If these constants art real, then the set of solutions to equation (1.1) may form an ellipse a hyperbola a parabola
two parallel lines two· crossing lines one line "counted twice"
a point the empty set the whole plane
1.2)
in the real plane ~ x ~- The notions of ellipse, hyperbola, and par: bola can be unified. To do this, we place the (x, y) plane in three-dimem .)nal space by considering it to be the set of points {(x, y, z): z = 1} 3
in ~ (Figure 1.1). Suppose we start, say, with the hyperbola
xy= 1, z=l and (Figure 1.2) put a light at·(O, 0, 0). To see the shadow of our i bola, we take a movable plane, which we will call a movie screen, and
1.3)
Chapter 1
l
:-axis
Figure 1.1. Putting the l. x, J.) plane into IJil.
movie screen I
~
J..
.--~--
.. --7 ;;J,..:.~
----)
1 Ii.ghr I
Figure L2. 1'1'1•1' ~·t Ill!!, 1.""11 !h.'- onglll. .
3
Conics
it in various positions in space so that we can see the shadow o; our hyperbola from many different angles. If our movie screen is the plan
z= 2, then we see a shadow which is simply a bigger hyperbola, but if our r ovie screen is the plane
x=2, then what we see is no longer a hyperbola but a piece of a parabola (r gure 1.3). Let's check this computationally. Any poinLOf the shadow o the movie screen x = 2 must lie 01,1 some line passing through (0, 0, 0) nd a point of the hyperbola given in equation (1.3). So for each· such lOint (x, y, z) on the shadow, there must exist a real number Jl such that
= 1,
AZ
(1.4)
(Jlx)(Jly) = 1. In other words, 1
2z xy = 1,
or
z
and
=1
:~:o.
I
movie screen
I
I
'
I
I ----------+.,--+------~
\ y
'' '
= 1/Z \
''
''
''
. Figure 1.3. The shadow is now a parabola.
y-axb
Chapler I
4
But now if such a point (x, y, z) lies in the movie screen x = 2, it must lie the parabola
011
2y :.=
72
Finally, we are Lalking about a shadow, so the number ). in equation (1.4) must satisfy the inequality 0 <_A..:::;, l, tlu\t is,
z > I. In a similar way, if we put our movie screen in the position X
+.Y = 4,
we obtain as the shadow the set of points (x, y, z) such that
X+ J' = 4, xy =~ z2 , z 2: 1.
This ~;-pai.:e
~dow
(1.5)
is a piece of an ellipse. To see this, rotate three-dimensional
45° around the z axis by m~ans of the transformation
z 1--t z.
The plane x plane
+y
-=
4 has, as its preirnage under this transformation, the X=
2j2.
The set ( 1.5) has preimagc X=
Hx::
2fi,
y2) "'~ .::2' L
In other words, in our rotated
~.dup,
our rotated hyperbola has the shadow
:' ·1-Jyl
4,
s
Conics z-axis
,
,. , .
I
y-axis \
\
' ... ......... ....
.........
---
Figure 1.4. Now an elliptical shadow.
on our rotated movie screen x = 2j2. ellipse (Figure 1.4).
So
the shadow is a piece lf an
1.2 Real Projective Space, The "Unifier" Now it seems clea~ that we will have an easier time understandi1 ,~the behavior of the various projections of our solution set onto the me.. able screen if we do not allow ourselves to be limited by the "extran 1us." physical properties of our model. In particular, things will be easier to grasp if we let the "shu i ow" consist of all points of intersection of the cone of solutions xy'= z 2
with our movable screen
ax + by + cz + d = 0. In fact! if our line of thought is to equate any two solution sets (a) which are "slices" of the same cone, and (b) from which the original cone can be reconstructed, then we might as well forget altogether about making slices and .rnply consider the cones themselves. That is, instead of studying the solutil 1 sets to equation (1.1) in ~ 2 , we will study solution sets to the homoy. 11eous equation
Ax 2 + Bxy + Cy 2 + Dxz + Eyz + Fz 2
=0
(1.6)
Chapter I
6 111 IH:~
Written in (symmetric) matrix form, this last equation becomes
A
[x y :::] [ B/'2 D/2
B/2 C E/2
~~;] r.,;- [X] y F'
=0,
z
or simply
v M
'v=O,
{1.7)
where v = (x, y, .:::) is a vector in fi;g 3 and /11 is a symmetric 3 x 3 matrix. Suppose now we make a linear chang~ of the coordinates (x, y, z) as follows. We map 11-1 3 onto itself by the transformation
+ IllY + 113 z), Y t--> (/2 1 X + f2 2.1' + /2.3 Z ), zt---t (/ 31 x + 132 v + 133 z), x H (I J I x
(1.8)
where L = (Iii) is an invertible J x J matrix. The cone {1.7) then has, as prcimage under this mappmg, the cone P •
eLML) · 'v = 0.
In order to break cone~ 111to lypes in the same spirit in which we prepared the .list (1.2), we will say that two cones are equivalent if some equation for one of them can be obtained from some equation for the other by a linear rhange of coordinates (1.8). Now put
(v 1 , u2 ) = v1 where vi= (xi> Yi, zi)
E
or there is a vector v 1
E
M · 'v 2 ,
{1.9)
lf.gJ Either
U~ 1 such that
(v 1 ,
ud = ± 1.
By a (linear) change of coordinates we can find an equivalent cone for which l'l :.=
(1, 0, 0)
and for which the set of vectors ··perpendicular" to v 1 , thal is,
7
Conics
is simply the (y, z) plane. The equation (1.7) of this n.ew cone will be ol the furm ·
[x y z]
[+~0 E/2g £~2] F
[x.yl = 0.
(J 10)
·z
Again either C = E = F = 0 or there is a vector v2 in the (y, z) plane that for the cone (1.10),
1ch
After a further change of coordinates we can then obtain an equiv< ,;nt cone whose equation is of the form
[x y z]
[ci0 ~0 ~] [~]z
= 0,
83
where e1 ~ e2 ~ 0, e2 ~ 8 3 , and ei E {0, ± 1}. As shown in Table A, th list (1.2).of types of cones is considerably shortened.
1.3 Complex Projective Space, The Great "Unifier" Now suppose we begin our entire discussion again. Let A, B ... , FE IC, the complex numbers, let our solution set lie in IC 2 , etc. Agai' we pass to complex "cones," A. in equations (1.4) becomes a complex sc dar, etc. Again we equate cones if they can be transformed, one into the o her, by an invertible linear transformation of IC 3 • The list of possibilities, 1 :ven in Table B, is now even shorter (since every second-degree equatim has complex solutions).
Table A &I
I
0
&2
I I I 0 0 0
Description
&3
1
0 -1
0 -I 0
Point ~Line
"Usual" cone Plane counted twice Two intersecting planes 1Jl3
II
Chapter I
Table B tl
/;2
r.l
Description
I 0
I 0 0
I 0 0 0
" Usual" cone Two intersecting planes On<~ plane counted twice CJ
We will call the set of complex lines through the origin in C 3 the
complex projective plane and denote it by CIP 2 • It can be thought of as simply the usual complex plane with "ideal points" at infinity, one ideal point for each complex line through the origin. The relation between the usual (affine) coordinates (x, y) of the plane C 2 and the (homogeneous) coordinates (x, y, z) of the space of lines through the origin in C 3 is as follows: if the affine set of points is given by the equation '1 ~-ln\111 ' ,,n-"· .T
,~ L..,;
= 0
(1.12)
'
m, n sN
1he projective set of complex lmes 1s given hy \:' L
.1
,.mu"-
."'1m,.'\.-'~
+ n))
= 0•
(1.13)
m,n:SN
We can freely pass from the affine obJect to its projectivization, which is well determined as long as there is some Amn -=/= 0 with (m + n) = N. Then all but a finite number ( :s; N) of elements (lines) of the projective object correspond to elements (points) in the affine object. Conversely, we pass from the projective object {I.l3) to the affine object (1.12) by setting z = 1, that is, by intersecting the "cone" (1.13) with the complex plane z = 1 in C 3 •. Fr6111 now on it will be convenient to call the elements of C IP 2 -~points" rather than "lines." So now let us look at the solution set for the homogeneous equation (1.6) in ICIP 2 . First, we will always assume that A, 8, ... , Fare not all 0. Also notice that if we multiply all six of these numbers by the same nonzero complex scalar, the solution set docs not change. Thus the set of "conics" in CIP 2 , i.e., solution sets for equations of the form (1.6), has no more than tive parameters, the conic is determined by the line through 0 and (A, B, . , F) iu (;!•
Conics
1.4 Linear Families of Conics Let's move on now to a less elementary setting. We wish to co, sider the set of all conics and reflect on some of the ·geometric properties ( I" this set. We will need a basic concept from algebra, the so-called resultal of a system of two equations. It is the device which tells us when th two equations have a common solution. Suppose now that we consider two homogeneous polyn )mial equations m
'
L r;(x, y)zi = 0,
F(x, y, z) =
i=O
M
l.l4)
n
L s;(x, y)zi = 0,
G(x, y, z) =
i=O
where r; and si are homogeneous polynomials of degree (m - i) and 11 - i) in (x, y). By elimination theory (van der Waerden [11], vol. 1, pp. ~· \-85), for a fixed value of (x, y) the resulting polynomial equations in z ave a common solution if and only if
rm
rm-l rm
ro ro
rm-l
(n times) rm det
Sn
Sn-1 Sn
ro
rm-1 so
=0. so
Sn-1
(m times) s,. s,.-1
so
This is a homogeneous equation of degree mn in (x, y). If it V< .. 1ishes identically, then elimination theory tells us that the two polynomials F and G [considered to be elements of the ring of polynomials in with coefficients in the quotient field of the ring of polynomials in (x, y)] 1ave a common factor. Applying Gauss's lemma (van der Waerden [11], .'Ol. 1, pp. 70-72}, they must in fact have a common factor in the ring o homogeneous polynomials in (x, y, z). So if F and G are homogenL .-us of deg~ee 2 and have the same solution set, we can concluc;ie that
F = A.G, Thus the set of" conics," that is, the set of solution sets to equation~ of the form (1.6}, ~re parametrized by a complex projective five-space (tha is, the
Chapter I
Ill
set of all complex lines through the origin in C 6 ) with homogeneous coordiuates (A, B, C, D, E, F).
(1.15)
So now we want to look at the geometry ofti1e set of conics that fulfill some particular condition. For example, the set of conics passing through the point
(1,0,0) ts giy.eu.--by the equation A =0,
and so these conics form a complex projective four-space. What happens if we wish to study those conics pas~mg through two or more points? Each new point imposes another linear condition on (A. B, ... , F). However, we must look at the independence of these conditions. Are some of the conditions redundant? lt is very casy to see that if we choose our points sufficiently generally, then each additional point imposes an independent condition until we nm out of conics. But Jet's show this in a way that will work in the projective plane IFV 2
over many fields IF and fot homogeneous forms of different degrees. We start with the set of monomials
(1.16) where (i
+ j + k) =
,I
some fixed numher N. If we make the substitution
.)' = x' '
z = x•
for an appropriate fi·xed choice of r and s, then it will never be true that i
+ jr + ks = i' + j'r + k's,
where xiyjzl' and xi'yFzk' arc distinct monomials in the set (1.16). Recalling the theory-of -t-he ¥an der Mondc determinant (Lang (5], p. 179), we see that if M = max{i + rj + :;k: i + j + k = N}, then the matrix [(. ,·' 1)i+ri+•k],
mu~t
.
I
-I··.J.
+ k. = N ' I = 0' ... , M '
have maximal rank whenever the numbers
are pairwise distinct. So d10osing
Yt
=
xr,
11
Conics
we find among the (M
+ 1) points
(xl> y1 , z1) e CIP2 , a subset whose cardinality is equal to that of the set (1.16) wit! the property that the square matrix
[(xfy{zf)],
i
+j + k =
N, I e subset,
t 1.17)
has nonzero determ.inant. Applying this reasoning to the set of mono.,lials 2. 2 2
x , xy, y , xz, yz, z ,
we conclude that for the proper choice of six points in CIP 2, there :; no conic passing through them all, and for the proper choice of five 1 1ints there is a unique conic containing them. In fact, since the condition th ,t six points be contained in a conic is simply the polynomial condition th< ( the determinant of the matrix (1.17) vanish, the set of sextuples of points ' hich contain no ~onic is ,<:tn open dense subset of
(CIP2) 6 • Similarly, the set of quintuples of points which determine a unique cc 1ic is an open dense subset of
1.5 The Mystic Hexagon There is some beautiful classical geometry connected with ,hese facts-for instance, the problem of how to construct the conic throu~ 1 five given points, which dates from ancient times. The most elegant sol ;tion, perhaps, is given by Pascal's "mystic hexagon." Namely, suppose v are -given five points A, B, C, D, E in order. Referring to Figure 1.5, we nake the following geometric construction: • Construct the lines AB and DE to determine their point of im -rsection P 1 • • Draw any line through A and let P 2 denote its intersection wi h the line CD. • Then, if P 3 is the point of intersection of the line BC with tl line P 1 P 2 , the point P of intersection of the lines AP2 and EP 3 ' ill lie on the conic ABC DE. That is because intercepts of opposite ec ~es of a hexagon inscribed in a conic lie on a line. To see why this last sentence is true, let Q denote the conic ABCDI:. Now
Chapter I
12
Figure 1.5. The: point P traces out a conic through A, B, C, D, E.
consider the following Jist of three cubics (that is, solution sets to homogeneous polynomials of degree 3 in (x, y, z)]:
Q plus P 1 P 2 , AB plus CD plus EP,
(1.18)
PA plus BC plus DE.
All three cubics contain the eight points (1.19) Now we have seen that "in general" we should expect the set of cubics containing eight points to be a C!fl'h that is, the set of homogeneous polynomials in (x, y, z) of degree 3 which vanish at eight general points is a two-dimensional vector space. It can be checked that if A, B, C, D, E are chosen in such a way that the conic is uniquely determined and not degenerate [that is, no e; = 0 in (1.11)], then the eight points in (1.19) are sufficiently general, so that the equations defining the three cubics in (1.18) must be linearly dependent. (See Grifliths and Harris [1], pp. 671-673.) Thus any common solution to two of them will also be a solution to the third. Since P is a common solution for the last two, it must also lie on the conic Q. Another basic prindple we've brushed up against here is Bezout's rheorem, that if C 1 is a plane curve of degree m [solution set of a homogeneous poly normal of degree m in (x, y, z )] and C 2 is a plane curve of degree n, then either C 1 and C 2 have a component in common or they have no more than »Ill points in wmmon. This is clear from our computation of the resultant of the system of equations (1.14). The resultant is a homogeneous ptll) ni:m1ial of degree mn. in (x, y).
Conics
13
1.6 The Cross Ratio Another topic of interest in connection with the study of conics s the
cross ratio. Let's approach this as it must have_been approached
f3 =
sin(1t - {3),
shows immediately that (sin cc)/A = (sin {3}/B. From these last two identities and Figure 1.7 we obtain that
A/B sin cc/sin f3 sin cc A'/B' =sin cc'/sin {3' =sin cc'' and so w.e get the cross ratio
AC/BC sin cc/sin(/3 + y) = AD/BD sin(cc + {3)/sin y
=.-==
--'-----'~~
(see Figure 1.8). This expression is called
th~
cross rat o of the lOints lead~. us to the fact (which is easily checked by direct computation) that if
(ABCD) and is clearly unchanged if the. line Lis moved. This (xt,Yt),
(x2,Y2),
(xJ,YJ),
(x4,y4)
are four distinct points of CIP 1 and if
(x, y)~(~x +by, ex +dy)
(1.20)
is a linear transformation of C IP 1 onto itself, then the expression
(xtY3- YtY3)/(X2Y3- Y2X3) (XtY4- YtX4)/(X2Y4- Y2X4)
I.
Figure 1.6. The law of the sine.
·~
Chapter I
B
Figure 1.7. Ratio of sines of two angles,
C!
and
C!',
with a common vertex.
IS a (nonzero) invariant under transformations of type (1.20) and is called, of course, the cross ratio
(PI, pl' p3' P4),
where P; = ( ,.; , y.). Also notice that the cross ratio of four points. on a well defined by the fact that the transformation f of IC IP\ given by l;;gul-c 1.9 is lineilr. Since all nondegenerate conics are equivalent, it suffices to check this last statement for the conic given in affine coordinates by
~onjc -1S"·
x2
+ y2
=
1,
"-·
---~ .•-
Figure I.S.
·r h.: ..:ross
ratio.
].j(H!
l.
15
Conics
11'
1
fixed vertices
Figure 1.9. Projecting a conic from different vertices.
with fixed vertices (a, b) and (a, -b) and Cifl> 1 represented by the x
2ct +A.= 1t,
2/3 + /1 = 2y + v =
1t, 1t,
so that
ct
+ f3 + ')I =
1t.
From this it follows that the triangles ACS and DBS are similar, anl, so SA SB•= SC · S[).
Thus the product SA SB is independent of the line through Sand ;> the triangles SAC and SDB in Figure 1.11 are similar; therefore
ct =b. Next suppose that in Figure 1.11 we arrange things so that OD l. BC.
Then
a=
o= e/2.
Chapter I
I (I
Figure 1.10. First figure on rays intersecting a circle.
Now we are ready to show why the mapping defined in Figure 1.9 is a lmear automorphism of UP't, that is, a transformation of type (1.20). We consider Figure 1.12. By what we did before we know that b = !(2a) =a so that (J = y = b + 1:. Thus the triangles ORB and OBS are similar, from whid1 it follows that
liR. as·=
Figul"<" 1.1 t. Scwnd figur.:
OB 2 •
<.lll ray~
intersecting ·a circle.
17
Conics
Figure 1.12. Projecting the circle onto a line from vertices (a, b) and (a, -b)
This formula shows why the mapping fin Figure 1.9 is linear, since n the special case in which the conic is the unit circle the mapping is simp
XHljx.
1.7' Cayley's Way of Doing Geometries of Constant Curvatl. ·e Now let's follow an irresistible side topic related to conics but k in an entirely different direction, that of Riemannian geometry. Riem:. geometry found its origins in efforts to show the independen Euclid's fifth postulate-that, through a point not on a line, there p: unique line which does not intersect the given one. Eventuall) dimensional worlds were produced which verified all Euclid's post except the fifth, and each of these "geometries" was uniform in the that it looked the same no matter at which point you stood in it~ dimensional world. The English mathematician Cayley found a love: to study all these geometries at once, and his main tools were the eros· and the theory of conics. In the next five sections we will touch on th points of Cayley's theory. The beginning point is the solution set equation x~
+ y2 + z2 =
0.
This conic in CIJl> 2 will be called the absolute.
tding mian :e of ses a twodates sense twoway ratio high u the
Chapter I
18
Givcu two pomts on the unit sphere P 1, P 2
E
S = i(x, y, z)
E
IR 3 : x~
+ i + z2 = 1},
the distance bdwcen them on S is the same as the angle 6 between the lines OP 1 and OP 2 measured in radians. Write
Pi = (x;, Y;, z;) and denote the complex projective line
by L. Let
{R 1 , R2 } = L n Q, where Q is the standard cone (1.21). These two points R 1 and R 2 correspond to values A. 1 and ). 2 of). given by the roots of the equation
(x 1
+ /.x 2 ) 2 + (y 1 + .1y 2 ) 2 + (z 1 + .1z 2 ) 2 = 0,
that is, I
+ (cos
t~)2/.
+ ,p =
0,
and by the quadratic formula
A. _cos 6 ± (cos 2 (5 1) 1 il _cos {J -1 - ----------· 2·-·-···- ----·-·1 2.. - ---·A. 2 cos J (cos (i- 1) ; cos fJ
+
± i sin b + i sin b
_
- e
±li 6
·
On the other hand, we compute the cross ratio (PI> P2 , HI, Rz)=A.t/A.2.
Thus we have a nice distance formula on the two-sphere S,
where R 1 and R 2 are the two point:. at which the line through P 1 and P2 crosses the standard conic (1.21), the absolute. Of course, we can project half the sphere S onto the plane \, y E IR,
z = 1,
(1.22)
hy central projection. This correspondence then gives a way of measuring distances between points on the plane (1.22) by measuring the angle between the corresponding points on S. But since the cross ratio is a projective iilVariant, we have a way to compute the distance which involves only the complex plane z = 1. Namely, kt p,, p J. E ~:t;2,
19
Conics
and let r 1 and r 2 be the two points at which the complex line through p and p2 intersects the conic x2
+ y 2 = -1
( .23)
in C 2 • Then the (spherical) distance between p1 and .p2_ is given b the formula
±
1
(i.24)
.log(p 1 ,p2 ,rl>r 2. ). 2I
This is because the cross ratio is preserved under central projection. rom the invariance of the cross-ratio, we also conclude that distan is preserved by any transformation · ·· a 11 x X 1---+
+ a 12 y + a 13
a31x+a32y+a33
'
; t.25)
a21x+a22Y+a23 y~---+ a31x + a32y + a33
of C 2 which is real (i.e., aii E IR) and which preserves the conic (1.2:-.1. Of course, this is simply a translation of the condition that the matrix
[aii] be orthogonal. Now suppose we replace the sphereS by the
K(x2 + y2) + z2
surfaceS~:.
given by
= 1.
Suppose we measure the distance between two points (x 1o Y1o z d and (x 2 , y 2 , z 2) on S~:. by measuring the dis~ance between (x 1, Y1o K- 112 z 1) anl (x 2 , y 2 , K- 112 z 2) on the sphere of radius K- 112 , that is, ifwe denote the p Jints by P 1 and P 2 , the distance is given by .Pz
1 (dx 2 + dy 2 + ~
x- 1 dz 2 ) 112,
~.26)
P1
where the path of integration lies in the intersection of S K with the OP 1 P 2 • Via central projection each of these metrics induces one o plane z = 1. Since the sphere of radius K- 112 gets flatter as K appro zero, it is geometrically clear that this induced metric on the plane approaches the.Euclidean one. As before, we have the formula
!ape the ches =l
Chapter I
20
when: R 1 and 1< 2 arc the pointi-> at which the line through (x 1 , y 1, K- 112 z1 ) and (.\ 2 , Yz, K 1 ' 2 .: 2 ) meets the absolute
x2 + .l'l + zl
= 0.
Now the roots of (xl
+ .hz) 2 + (J' 1
t-·
i,y 2 ) 2 t (K-· 1i 2 z 1 + A.K- 112 z 2 ) 2
=0
arc simply those A. such that
lies on ~he complex cone
Thus for the metric induced on the plane z = 1 by central projection, we have
(1.27) where
r" r2 form "the intersection of the Hne through p 1, p 2 with the conic K(x 2
+ /) + I
= 0.
Since the metric (1.27) is mduced from the standard metric on a sphere of ,.ra.d-itrsK - 112 , it is a metric of constant curvature K.
1.8 Through rhe Looking Glass We have seen that the metric (1..26) when restricted to SK stays bounded as K approaches 0, so we can in fact let K become negative and the formula (1.26) will still give a perfectly good metric on SK (or at least on the "top shee'" of what is now a hyperboloid). By "analytic continuation" we still expc:ct that this will be a metric of constant curvature K and that geodesics are intersect ions of S K with planes through 0. Indeed this is the case. By central projection this induces a metric on the part of the z = 1 plane which is interior to the circle K (.\.-<
+ y2) + I = 0.
Again by analytic continuation the metric'
is
still given by the formula
( 1.27), where now all four points pI> p 2 , ,., , r2 are in the real plane z = I. In fact, we have Figure 1.13. Now any linear transformation of IR 3 which takes
Conics
21
circle of radiu·
.q
I.
Figure 1.13. Hyperbolic geometry in the disk of radius (- K) 1'2.
S K onto itself induces an isometry since, perforce, it preserves th symmetric bilinear form
real 1.28)
To study the group of transformations preserving (1.28) it suffices t( consider the cases
K= 1,
K= -1,
since the only important thing about the symmetric real bilinear (1.28) is its number of positive and negative eigenvalues. We have a1 considered the case K = 1. In the case K = - 1 the group of isomc1 called the Lorentz group because of its importance in the theory of vity. To see how it acts, notice thar it contains all rotations around axis and also that it contains transformations which leave y fixed an,
(1, y, 0)~---+ (xto y, zd, (0, y, 1) ~---+ (x 2 , y, z2 ) such that
zi- xi= z1 z2
-1,
z~- x~
= 1,
-
= 0.
X 1X 2
form l.!ady 1es is datithe z take
22
Chapter I
/
/ /
/ /
asympLotc
Figure 1.14. '' Lorentz transformation moving along a system of hyperbolas.
In the (x, z) plane, we have Figure 1.14. Since the involution which interchanges the x axis and the z axis must interchange the points (xl> zt) and (x 2 ,._z_2_),....lhey. m.u..st be "equidistant" along their respective hyperbolas. From this it is immediate that the group of Lorentz transformations is transitive on the hyperboloid S ~.: and also transitive on the set of directions through any fixed point on S~.:. Thus our metric ·must indeed be of constant (negative) curvature.
1.9 The Polar Curve There's another very useful general concept which .can be applied in this setting, namely the concept of the polar curve associated to a given plane curve. Namely, if a plane curve C in C!P> 2 is given by the homogeneous equation Ill
F(~;;, y, :::)
then the condition that the line
=
L rJX", y)zi = 0,
,_.,0
23
Conics
be tangent to C at (1, 0, 0) is simply that the equati9n
r 0 (x, y) = 0 have a double root at (1, 0), that is, that r 0 (l, 0)
~; (1, 0) = 0.
= 0,
This condition can be restated as
oro (1 0) = 0
ox '
oro oy (1 , 0) == 0.
'
by Euler's formula, which says that
oF ox
oF oy
oF ox·
mF=x- +y- +z -·., (deg r)r = X
or OX
or
+ y oy.
We can rewrite the tangency condition simply as
oF oF ox (1, o, o) = oy (1, o, o) = o. Since any point in CIP 1 can be moved to (1, 0, 0) by a linear transform ttion in such a way that a given line through the point goes to the line z = i, we see in general that a line
ax+by+cz·=O is tangent to the curve
F(x, y, z) = 0 at (x 0 , Yo, z0 ) if and only if (x 0 , Yo, z0 )
E
.C and
oF T• oF ToF)/ :::: A.(a, b, c) (T• uX uy uZ (xo,yo,:roJ
1.29)
for some A. e C. (If the left-hand triple vanishes, we will say that ever; line through (x 0 , y 0 , z0 ) is tangent to C at that point. If the. left-hand side Joes not vanish, the fact that ax 0 + by0 + cz 0 = 0 is insured by Euler's forn ula.] The mapping cc;...
;;oc·
(.~o. Yo. zo)
oF oF)/. - -:;-!>(oF l{ T• T• T uX
uy
uz
(xo, yo, zo)
dO)
24
Chapter I
is ·callcd--tlfe poTar mapping associated with the plane curve C. E&c(C) is called the dual curve of C, which we denote C. If the curve C is a nondegen~ ~rate conic, £j) c is of course a linear isomorphism so C is again a nondegenerate conic. To obtain its equation, notice that if C is given by the matrix ~quat ion
PA'v= 0, then Cis given by a matrix M such that
AMA =A. (Remember that A is symmetric.) So
M= A- 1 and
(L31) Now let C be a nondegeneratc plane curve of any degree. (Nondegenerate means that the three partials of the homogeneous defining equation F do not vanish simultancl.:usly.) If~~\ is locally of maximal rank, the equation G = 0 of the dual curve C is determined locally by the formula
ar oF'
G (D/· ox
-·-------) = ).F(x y c':iy' Cz ·' '
z) ·
DitTcrentiating, we get that if the matrix of second partials 2
2
iJ FjiJx iJ 2 F/oxoy [ 2 8 FjiJx
2
D Fjox oy iJ 2 F/iJy 2
o F!iJ y oz 2
is of maximal rank at (x 0 , y 0 , z 0 )
E
C.
2
i'! F/ox oz] 2 F/i'!yiJz
o
2
(1.32)
2
D F/iJz then the point
r»c(~~- ~~· ~~) depends only on the values of the first and second partials ofF at (x 0 , Yo, z 0 ). Since there exists a conic whose defining equation has the same value (namely, zero) and the same first and second partials at (x 0 , y0 , z0 ) that F du~s. the formula (1.31) tor conics implies the same formula for general plane curves. To be more precise, if the mapping ~c is of maximal rank at a__rl)i!"lrP E Clll 2 , then in a neighborhood of P on C
9c
!J:c =identity map.
Via the linearity of the polar mapping in the case of conics, every fact
25
Conics
about conics has a corresponding "dual fact." To begin, let A b ~ the symmetric matrix associated with the nondegenerate conic C. The tmultaneous equations
~] = 0,
1.33)
E:J-o.
1.34)
[x y z]A [
[x y z]A
for fixed (x 0 , y 0 , z0 ) e C IP 2 , mus.t have two solu ti<:ms: (x 1 , Y1> z 1 )
and
(x 2 , y 2 , z 2 ).
Since we have seen that the equation of the tangent line to C at (x;, ·;, z;) IS
[x, y, 'z,]A
[~] = 0,
the point (x 0 , y 0 , z0 ) must constitute the intersection of the tangent !1 I!S to Cat (x 1, Y1> zt) and (x 2 , y 2 , z2 ) (Figure 1.15). Thus the polar mappi .. g !:i!c ean be thought of as assigning to (x 0 , y 0 , z0 ) e CIP2 the line (1.34) vhich meets C in the two points whose tangents pass through (x 0 , y0 , z0 ). ;'rom this it is clear that we have a nice bijection between CIP2 and I
c(2), the set of pairs of points on C (often called the second symmetric p; _,duct
Figure 1.15. Constructing the polar of a point,
Chapter I
26
c
--
Figure L 16. Stereo graphic projection of a conic onto a projective line.
-or C
wrth itself). Since C is bijective to CIP 1 via stereographic projection (Figure 1.16), we see that there is a nice bijection , ··.1· = ~ 1[. -11 2
(If" rrb 'Vtr
1
}(2)
•
All;this is consistent with the !act we saw earlier: if (x 0 , y 0 , z 0 ) E C, then its polar line is tangent to C at (xu, Yo, zo)· Our polar isomorphism then sends points to lines (and lines to points) in a way which is full of geometric meaning. The "polar" or "dual" statements of point p on C two lines passing through a point are line tangent to C at p two points spanning a line. Thus, for example, Pascal's theorem on th!! "mystic hexagon" which we saw earlier translates to the dual theorem: The lines connecting opposite vertices of a hexagon circumscribed about a conic pass tht;ough a CO.(nmon point.
1.10 Perpendiculars in Hyperbolic Space Let's see an application of the polar mapping to the plane geometries of constant curvature that we studied earlier---the so-called spherical and hyperbolic geometries. These consisted of sets of points with a distance
l7
Conics
function.- The distance function was very special in the sense that etch geometry has the largest possible group of isometries, namely, give any point and direction at that point there is a (unique) isometry which akes them to any other point and direction. By central projection we achi( .-·ed a planar representation of each geometry (or, in the case K > 0, at J, ~tst a planar representation of a big. piece of it). One difficulty with this rep1 -.:sentation is the notion of angle. For example, in the case K < 0 the tsual plane Euclidean notion of angle between two lines in our geometry is no good because, for example, the law of the cosines does not even hold infinitesimally and isometries do not preserve angles. We need a way o see geometrically what angles are in this geometry-::::Pr, failing that, atle;: :>t we ought to be able to decide geometrically when two lines are perpend1 :ular. We begin with a sphere of r~dius K- 1 / 2 centered about the origin i i !R 3 • Consider two geodesics on the sphere through a point P = (x 0 , y , z0 ) (Figure 1.17). Let M and M' be the intersections of these two geou~sics ·.with the geodesic X0 X
+ jl 0 y + Z 0 Z = 0
1.35)
on the sphere. If lJ is-the angle between the two original geodesics, th ·n we have seen that
K-tl2lJ =
2i~tl2 log(M, M', N, /t{'),. '
Figure 1.17. The angle between geodesics in spherical geometry.
Chapter I
28
where N, N' are the points of intersection of the complex line (1.35) with the absolute
(1.36) that is, N and N' are the two points of the conic (1.36) whose tangents pass through P! If we translate all this as we did before to a statement about the K-geometry in the plane
z = 1, we obtain that the K-angle between the lines Land£ through p = (x, y) is given by
Jt log(L, £, J, J'), where J and J' are the two tangent lines to K(x 2
+ i) + 1 =
0
which pass through p. (The cross ratio of four lines passing through a point p is simply the cross rauo of their four points of intersection with another line not containing p.) By analytic continuation we expect the same ·fbrrnula to hold for negative K, which it does. We cannot "see" the lines J and J' in either the case or K positive or K negative (i.e.,. the lines J and J' are not real). However, we can see what it means for two geodesics to be perpendicular. Let's get at pctpendicularity, then, in the case K < 0. We have two lines Land £ (geodesics) intersecting in the point p 0 lying in the interior of the disk bounded by (1.37) (Figure 1.18). We have seen that each line has an associated polar point, call them r and p', with respect to the conic (1.37)-p is simply the point at which the tangents to (1.37) at its intersection with L meet. Land £will be perpendicular if log(L, £, J, J') = ni, that is, if
(L, £, J' J') = - 1. But, if A is the matrix of a conic as in Section 1.9,
pA 'Po= 0
29
Conics
p I
I
I
I I
I I I -·-·
~
\-
I
I
\ I
..,
I
I
"
\
\
\ \ \\ \
\
\
---
- - ~ --- p'
Figure 1.18. Angles in hyperbolic geometry.
means that p E (polar line of p 0 ) which we will call H. Alsop' the polar mapping we have ·
E
H. Jnder
point p --. line L, point p' --. line £, point Po --. line H, point n -.line J, point n' --. line J', where n and n' are the points at which J and J' intersect the conic ).37). But n and n' must lie on H since J and J' pass through p0 • The polar mapping is linear, so it preserves ~ross ratio. Thus
(L, E., J, J') = (p, p', n, n'). Suppose now that p
E
£.Then pA 1p' = 0
so that p'
E
L. In this case
(I.., E., J, J') = (p', p, n, n'),
Chapter I
30
since we can compute the cross ratio of four lines through p0 by computing that of their intersection with the line H. Thus if p E I'. (or equivalently p' E L), then
(p', p, n, n') = (p, p', n, n'). Since p, p', n, n' an£ all distinct, the only way that this is possible is if
(p, p', n, n') = -1. Conversely, since (L, I'., J, J') is an injective function of I'., it is clear that if (L, E., J, J') = -1, then p E I'. (and so also p' E L). Thus ··---····
L and I'. are anharmonic with resp.ect to J and J' [i.e., (L, I'., J, J') = -1] if and only if L and I'. meet perpendicularly in the K-geometry. In totally geometric terms:
The perpendici1lar to L passing through p0 is the line p0 p, where p is the polar pont of L.
1.11 Circles in the K-Geometry PI{OI:ILEM:
Find the set of points of fixed distance from a given point
Po· This problem is solved by finding a curve which goes into itself under the isometries which fix p0 . We let L be the polar line of p 0 , which intersects the conic (1.37) in the (imaginary) points nand n'. If
..-
J(x, y, z) = 0,
J'(x, y, z) = 0
are the equations of the two tangent lines to (1.37) at nand n' respectively, then we have the family of conics t 0 [K(x 2
+ y 2 ) + I]+
t 1 .J(x, y,
z)J'(x, y, z) = 0
parametrized by the homogeneous coordinates (t 0 , t t). We will denote this family by c(lo,lt)>
(1.38)
Just as there is, in general, only a I['Ub 1 of conics passing through four points, there is only a ~:a-]> 1 of conics passing through two points and
31
Conics
having fixed tangent direction at each. So the family (1.38) must c tntain the degenerate conic consisting of the line. L= nn' counted twice. Now the isometries of our geometry are the linear tr; nsformations which leave the conic (1.37) fixed. In particular, we have onereal-parameter family of isometries which leave·p0 (and therefore n nd n') fixed and which act transitively on the set of real lines through pl This group of isometries acts linearly on the familyofconics (1.38) but .n fact leaves three conics [2 L, the conic (1.37), (p 0 n) + (p 0 n')] fixed. Th ref9re this group acts trivially on the set of conics (q8). From these tcts it :;, ),; . ... . .. follows immediately that the family . . . .
••[K(x'
+ y') + 'I + ,, [ [x
y , ], [
tf~]:;~r=~.~~:)
e ~ cuts out a family of" circles" in K-geometry, that is, eaci conic in (1.39) which hits the set ··
t0 , t1
(x, y) real,
K(x 2
+ y2 ) + 1 >,o,
I
meets it in a set of points of fixed distance from p 0 • The "dual" fo mutation of the problem, starting with a line L in K~geometry and it: polar point p 0 outside the disk of points of the geometry, gives rise to the c tnclusion that the family (1.39) consists of loci of points of fixed distanc from · the line L (Figure 1.19). In any geometry, a· curve has assigned to it at each of its pl ints a number, called its geodesic curvature at that point, which depends o tly on the distance function of the geometry (O'Neill [7], pp. 329-330; In a geometry with as many isometries as the K-geometry, K < 0, circles therefore must have the same geodesic curve at each of their points. To s, ~ how this curvature behaves, let's consider Figure 1.20. Consider the "K-· ircle" of center (a, 0) which passes through (0, 0). As aapproaches (- Kt r \this circle approaches a conic whose four points of contact with K(x 2 + y2 ) + 1 = 0 all coalesce at (( -Kt 1 ' 2 , 0). [To see this, exam. te our geometric construction of the family (1.38) or the Figure 1.20.] We s: y that the limiting conic and K(x 2 + y 2 ) + 1 = 0 have contact of orde 4 at ((- Kt 112 , 0 ). The order of contact is so great (;;::: 3) that thl· usual Euclidean curvature of these two curves must be the same at thi~ point. The common value is therefore
( ..... K}112. By symmetry the limiting conic has Euclidean curvature (- K) 112 a1 (0, 0).
32
Chapter I
Figure 1.19. The locus of points of fixed distance: from a lin.: in hyperbolic geometry.
But now our K-metril: and the usual Euclidean metric
can be shown to coincide to second order at (0, 0). In differential geometry it is shown that this implies that the geodesic curvatures with respect to the
/
Figun• 1.20. Limit of dr~k:; through a fixed point in hyperbolic geometry.
I Conics
33
two metrics must coincide at (0, 0). We can therefore conclude tl: 1t in K-geometry there are no circles of geodesic curvature less than (- K) 112 but that all curvature values greater than (- K) 1 ' 2 are attained! It's ~ow time to go on to other things, but before we wind u • our treatment of conics, there is one more topic we ought to touch on:~
1.12 Rational Points on Conics We should say a few words about the numb~r th~ory of conics That is, suppose we are given an equation
Ax 2 +Bxy +Cy 2
+ Dxz + Eyz + Fz 2 = 0
1.40)
where the coefficients A, B, ... , F E Q, the field of rational numbe1 ~k when the equation (1.40) has a solution in ·
We
QIP'2' the set of one-dimensional subs paces of the vector space 0 such solution
3 • If there
s one
(xo, Yo, zo), then we can proceed by stereographic projection (Figure 1.21) to qor. ;truct many others as long as the set of complex solutions, that :is, sol uti. )ns in ICIP'2, is a nice, nondegenerate conic. For suppose Yo =I= .0; then givl 1 any point (x 1, 0, z t) E QIP'2 on the line y = 0, the equa~ion . ·.·.· '
A(x 0
+ txt) 2 + B(x 0 + txt)yo + Cy5 + D(xo + txl~(zo + tz1) + Ey 0 (z 0 + tzt)+·F(z 0 + tzt) 2 =
0
has the solution t = 0 and therefore has a second rational solution. problem of deciding what the solution set of (1.40) in QIP' 2 is like rests with answering the single question: Does (1.40) have any r< solutions? Clearing denominators, we can assume that A, B, C, ... , integers and ask the equivalent question: Doe~ (l.40) have any i; solutions? The beautiful thing is that this last question can be effectively deci( any given equation (1.40). First, the process we described earlier of i
t For
(1.41) :o the really :ional F are
tegral ~d for nding
the reader who wants to delve deeper into Riemannian geometry, there is pen .ipS no more beautiful source than the classic Le,ons sur Ia Geometrie des Espaces de Rieman. of Elie Carlan (Paris: Gauthier~Villars, 1963).
Chapter I
34
line y: 0
Figure 1.21. Slcn:ographic projection again.
a linear transformation of !Rif~ 2 which transforms our equation (1.40) to one of the form (1.42) can actually be accomplisht:d over Q, that is, by a 3 x 3 matrix with rational entries which induces an automorphism of i(Jlll~·2·, as long as we do not require that jt:; I =--" I. So WI.! can assume that our equation has the form (1.42). Since to haye rational solutions to (1.42) we must have real ones, one. of-t-he-e; must be positive. Also we can assume that e 1 ~ e2 Replacing x by ax and y by by, we can further assume that e; is a product of distinct primes each raised to the first power. Let's look at two examples:
I l I l. I I
3x 2 + 2y 2
-
z 2 = 0,
(1.43)
3x 2 + y 2
-
z2 = 0.
(1.44)
In (1.43), suppose we ha(.i a solution (x 0 , y 0 , z0 ) of integers with no common factor. Then if 3 { y 0 (3 does not divide y 0 ), we obtain a contradiction by regarding (1.43) as an equation with coefficients in the field If,\
=
7!./37!.
of integers modulo 3. But if 31 J'o, then 31 z0 so 3 2 1Jx5 so 31 x0 • Since Xo, y 0 , and z 0 havl.! no common factor, (1.43) has no rational solutions. On the other hand, the equation (1.44) admits the solution (1, l, 2). Notice that in (1.44) r. 2 is a square modulo
1; 1 .
35
Conics
This sort of reasoning leads to the conclusion,in general, that if L.42) has an integral solution, then \.' · ·
e2 is a square modulo p for each prime number p dividing e1 • So by the Chinese remainder th, >rem
e2 is a square modulo e1 •
IfJ Je1 1/2 and
This means that there exist integers c, e~ such that $ 2 c = e2 + e1 e'~o e1e! = c2 - e2 = (c + e.!f 2.)(~e~~:l· ·--.·. . ·,,;;:-:w··· So, by algebra, the existence of rational numbers a, b such that
e1 =(a+ be~' 2 )(a- be~' 2 } is equivalent to the existence of rational numbers a', b' such that
In other words,
e1 x 2 + e2y 2 - z2 = 0 has a rational solution if and only if eJ. x 2 + t 2 y 2 - z2 = 0 does. If Ie1 j > 1, then
je'll
=I
c2 ~ e21
~Iii
+ 1 <,jell•
so that (eliminating squares which appear in the factorization of It'1 1 ) we have reduced the problem to one in which je 1 1 + I e2 1 is smaller. ¥. : ¢an repeat th!s argument until le 2 1 fails to be a square modulo Ie1 I o' until we reach the case
led = le2l = 1, which admits a solution if and only if e1 or e2 is positive.
CHAPTER TWO
Cubics 2.1 Inflection Points Let us now take up the study of the solution .set E s; CIP 2 of the eq1 ation F(x, y, z) = 0, (2.1) where F is a homogeneous polynomial of degree 3. Again we will ; .sume that the partial derivatives oF oF oF ox' oy' oz
(2.2)
do not all vanish· simultaneously. Unlike the degree-2 case, every cubic polynomial with real coefficients has at least one real root, so if the coefficients ofF are real, then the solution set to (2.1) in IRIP2 will al\'..lYS be a smooth curve. The question whether there exists a solution in OIP;, ifthe coefficients of (2.1) are rational is extremely difficult, and there is as vet no known procedure for deciding in generaL First, let's look at the polar mapping ~E: CIP2--. CIP2 which we introduced in Chapter One. When the degree of E is thr. -.:, this mapping is no longer an isomorpqism. In fact, it is degenerate at p( ·ints p where o2F iJ2F o2F ox2 (p) ox oy (p) oy oz (P) det
iJ2F ox oy (p)
iJ2F ()y2 (p)
o2F oy oz (P)
o2F OX oz (p)
iJ2F oy oz (p)
o2F oz2 (p)
37
=0.
(2.3)
Ch~pter
38
II
Since this dctcrminantal equation is homogeneous of degree 3, we expect that there will be 3 x 3 = 9 points of E where the mappi~g is, in fact, degenerate. But what about
!Z!EIE? Suppose we are at a point peE at which (2.3) holds. If ~EIE were to be of maximal rank at p, then the kernel of ~E at p could not lie in the tangent space to Eat p. This would imply the existence of[x y z] such that
_.-
[x
'(~F.· (p), ~F (p),
z]
y
ox
ay
oF (p)) # O
?z
but such that
[x
y zj M=(O,O,O),
where M is the matrix in (2.3). But multiplying this last equation on the right by 'p, we obtain a contradiction by Euler's formula. Since the mapping !lilEIE is clearly not everywhere degenerate, we see that it is degenerate at most nine points. Now suppose we have changed coordinates so that one of these poiflts is
Po= (0, 1, 0) and at p0 ,
:rx··. oF ay'
ilF) a~ = (0,0, 1).
e(8
Then the equation of E takes the form
y2 z
.\
+ yQ(x, z) + C(x, z) = 0,
(2.4)
where Q and C are homogeneous forms of degree 2 and degree 3 respectively. Let
Q(x, z) = }(ax 2
+ 2bxz + cz 2 ).
The degeneracy of 9i1 E at p0 is simply the degeneracy of the matrix ----·- -
iJ 2 C uy + 3.~2
ax
ax+ b:
+ bz
by
+ a~az
2z
2y
+ bx + cz
cy
+ -Jz2
iJ2C
2
c7 C
bv +-·· · clx Dz
2y
a2 c
+ bx + cz
at (0, l, 0). Thus p0 is an inflection point if and only if u = 0.
(2.5)
39
Cubics
Next we differentiate
a[determinant of (2.5)]
ox
I
= _
p· .. 0
'
2.
o3 C'.
3. ox .'
But
o3 C
ox3 =F 0,
since otherwise the equation (2.4) would have no x 3 term and th would contain the line z = 0 and so would be singular. Thus the ze, the determinant of (2.5) and E meet transversely at p0 • From this conclude that !i)E is d~generate at nine distinct points of E. We also if we restrict the equation (2.4) to the line
curve >set ol we can
z=O, then it becomes
ex 3
= 0,
so we say that the line and E have contact of order 3 at p0 • This Ia fact i5 equivalent to the degeneracy of £i) E at Po, an equivalency which easil) seen to continue to hold for equations and curves of degree higl :r thar three. The same is not true of the computation which allowed us to con· elude that the Hessian curve defined by the determinant of
[a!~:xJ meets E simply (or transversely) at all its points of intersectiOJ. These points of intersection are called the inflection points of the curve, a' d there are
n[3(n- 2)] of them, where n = degree of the curve.
2.2 Normal Form for a Cubic Now suppose we have a nondegenerate (or, as we shall somet, nes cal it, a nonsingular) curve E of degrees 3, and suppose we have rangec things so that one of the nine inflection points of E is the point p<X)
= (0, 1, 0)
40
Chapter II
and the tangent lini.: to E at that pomt is the line
z = 0, that is, the line "at infinity" with respect to the affine-coordinates (x, y). Let us look now at the family of lines
x By our assumption about Poo
y2 z
= ez, E
e
E
C.
(2.6)
E, the equation of E must have the form
+ yQ(x, z) + C(x, z),
and we can compute those values of e for which the line (2.6) is tangent .to E by computing the discriminant of the equation
-
y2
+ Q(e, 1)y + C(e, 1) = 0,
that is, the determinant of
[
,_1.
Q(e•.
0
2
1)
Q(e, 1)
l)l
C(e, 0 · Q(e, 1)
or, simply, we find the set of solutions to
--Q(e, 1)2 + 4C(e, 1) = 0.
(2.7)
Since we have seen that o3 C/ox 3 =1= 0 and o2 Q/ox 2 = 0, this equation has three roots (counting, for the moment, possible multiple roots). Assume that one of these roots 1corresponds to the point
(0, 0, 1).
o Cjoz
3 Then the line X = 0 is tangent to E at that point so that 3 = 2 ,J2Qjoz = 0. If e is to be a multiple root of (2.7), then we must further have that
But then all the partial derivatives of the defining equation of E vanish at (0, 0, 1), contradicting the nonsingularity of E- Thus the roots of (2.7) are distinct and correspond to three distinct points
in C ~D 2 The picture is shown in Figure 2.1.
41
Cubics • P,.
to P,. to P,.
=0
p
Figure l.J. Projecting a cubic from the inflection point p..,.
Now let us consider three cubic curves: E, (line M) +(lineN)+ {linep00 p), (line L) + 2(line p 1 p~),
where p is the third point of intersection of E with the line through ' 1 and p2 • The three cubic curves in the list have eigh( common points, na ;lely, 3p<X> ¥. 2pl, 2p2 ,·p.
But, as we saw in Chapter One when we were trying to cope with l ,tscal's mystic hexagon, we therefore expect the defining equations of th. three cubics to be linearly dependent, which means that the three cubics 1ave a ninth point in common. This point must be p, so that the line (p"" 1 ; must meet E twice at p, that is, p = p3 and the three points p 1 , p2 , p 3 !1 : on a line. By appropriate linear change of coordinates, which disturbs 1 .>thing we have done so far, we can assume
Pt
= (0, 0, 1),
P2=(1,0, 1),
P3
= (J., 0, 1).
Chapter II
42
Since then Q(e, 1) must vanish at
e = 0, 1, A., we conclude that Q(x, z) =. 0 and so our equation of E comes to have the form
y 2 z -- x(x -- z)(x- A.z) = 0.
(2.8)
2.3 Cubics as Topological Groups When we were studying conics in Chapter One we scarcely mentioned their topology in CIJ-1' 2 • This was because everything was so easy. Namely, via stereographic projection (Figure 2.2) we set up a diffeomorphism between our conic and a projective line L, which is in turn diffeomorphic to the standard two-sphere in IR 3 • When we attempt stereographic projection with an elliptic curve, however, things become a bit more complicated (see Figure 2.1.) If the center of projection Poo is an inflection point, then the equation (2.7) is of degree 3; but, if not, it will be of degree 4. Stereographic
Figure 2.2. ProJr,cqng a l.<>.nic from a pomt /1"' on it.
Cubics
projection then presents E as a two-sheeted covering of C.P 1 "ram fied" <J four points (one of which is at infinity if Poo is an inflection point ). W}l<J "ramified" means is that around the point in question the project ,m rna has the same behavior as does the projection : · ~-·.
(curve y 2
= x)
'~, (x axis), .
:· ~~-'..
(x, y)
x
._:;.._·.
near (0, 0). From this it is easy to make a top
X
IR/71.
X
71..
So E is a topological group, in fact. But of course the group structure on E is much more iu ,irhatel related with its complex geometry. Let's now explore this relationShip. W can get a good idea of what is going on by assuming that A. is reat and b looking at the solution seuo (2.8) in IRP 2 or, simply, in the (x, y)-rlane IR (see Figure 2.5). We define a binary operation onE by defining Pt
+ P2 = (x, .- y),
where p 11 p2 , and (x, y) lie on a line. There is no problem that thf-; binar operation is commutative and that each element has an inverse if we defin the point at infinity, Poo, as the identity element. The interesting veflficatio
Figure 2.3. Cutting a cubic into two pieces.
44
Chapter II
----
0
0
0
0
Figure 2.4. Pasting the cubic back together aga~n.
is that of the associativity of the operation. The main idea-that cubics passing through eight points have a ninth in common--is one that we have already used twice before. To see how it works in this case, look at Figure 2.6, a diagram borrowed from John Tate's beautiful lectures at Haverford College in April 1961 on the number theory of c_ubic curves. To show associativity we must show that the points (p!+ q)r and p(q + r} coincide.
Figure
2.5.
A real cubic in normal form, cut by a line.
45
Cubics
Figure 2.6. The associativity of addition on
acubic.
So we must find two cubic curves (besides E) thatpass through the points "
Poo, p, q, r, qr, pq, (q
+ r),
(p
~ight
+ q).
Then if one of these cubics passes through (p + q)r-and the other th1 1ugh p(q + r), we can conclude that the two coincide. The two cubic cun ..:s in .· question are (line through p and q) + (line through qr and (q +.r)) + (line through r and (p + q)) and
/
(line through q and r) + (lim: through pq and (p+ q)) + (line through p and (9 + r) ).
2.4 The Group of Rational Points on a Cubic So it should be clear that the complex solution set to a nondege1 :rate cubic
F(x, y, z) = 0 forms a group with a geometrically constructed group operation wit
(2.9) any
Chapter II
46
given inflection point as identity element. Now suppose the coefficients in (2.9) are n.:al or rational and that we have a solution p E !RIP2 or p E QIP 2 • Now p may or may not be an inflection point, and no linear change of coordinates will change p into an inflection point if it is not already one. Howev.er,--Lhere..is a..birational transformation of IRIP 2 or OIP 2 (and so CIP 2 ) which transforms E in a one-to-one way into a nondegenerate cubic curve and transforms p into an inflection point of that curve. To see this, we again follow Tate's notes and assume that our given point p is not an inflection point. Choose coordinates so that p = (1, 0, 0),
the tangent line to Eat p is given by
z = 0, and the tangent line to E at tbc third point of intersection- of the line z = 0 with E is the line X=
0.
Then the equation for E can be written x2z
+ xQ(y, :) + C(y, z) = 0,
{2.10)
where Q(y, z) = ay 2 + 2byz
+ cz 2
and C(y, z) = syz 2
+ tz 3
Now we transform IQIP 2 (and simultaneously IRIP 2 and CIP 2 ) into itself by the rule {x, y, z)H (xz, xy, z2). (2.11) Let's stop a moment and visualize this transformation of IQIP2 • First, it has an "inverse," namely
(x, y, z)-+ (xy, y 2 zfx, yz)
II
(x 2 , yz, xz).
That is, the transformation (2.11) restricts to an automorphism of {xyz
(OIP 2
= 0}).
The behavior of the map (2.11) on xyz = 0
{2.12)
47
Cubics
this point gets "blown up" to the line z ; 0
this line gets "blown'down" to the point (0,1,0)
this line gets "blown down"· to the point (0,0,1)
~' the first-order infini-
,tesimal--line at this point gets "blown doWn" to (0;1,0); however, the point ''infinitely near' to.this point in-the direcdori-of·the line x ; gets "bloim lip" tcdhe lin· x = o·
Figure.1.7. Analysis of the birational coordinate change (2.11).
can be pictured as shown in Figure 2.7. To describe what is hapr :ning more exactly, we consider the operation c~lled "blowing up a point· Suppose that we want to blow up the point
x=y=O j, in the Cartesian (x, y) plane. We simply reelace the (x; y) plane " .th · the graph of the map 1Ql2 _ _..:.__ iQllfl>1o
(x, y)
(x, y)
With this in mind, it can be shown that the map (2.11) can be descn >ed in the following steps: STEP 1: Blow up the points (1, 0, 0) and (0, l, 0) in 1Ql/P,2 to oi .ain a new "manifold " X 1 (Figure 2.8 ). STEP 2: Blow up the point p in X 1 to obtain a new manifc d X 2 (Figure 2.9 ). It can then be shown that there is an everywhere-defined "algc 1raic .. mapping
f: X 2 ~ ilJliP2 which coincides with (2.11} on the part of X 2 and which was unit uched
48
Chapter II
/
y • 0 • 0
p
Figure 2.8. Pass from ii.U'2 to X 1 by blowing up two points.
under. the modifications of Steps l and 2. In fact, the net effect off is to contrast-to points the sets indicated by the cross marks in Figure 2.10, in tl~t: t)rder indicated by the number of cross marks. The maps (2.11) and (2.12) are t:allcd Cremona transformations. The image of E under the transformation (2.11) is computed by replacing x by x 2 ,
y by yz,
z by xz in the equation (2.10). We obtain x~z
+ x 2Q(yz, xz) + sx 2 yz 3 + tx 3 z 3 =
0,
---·--Figure 2.9;- Pass from X 1 to X 2 by blowing up one more point.
49
Cubics
Figure 2.10. Pass from X 1 to QJP>1 by contracting three curves.: in the order show' .
..
--------·
.
. .... - --···
- ·····--.-···---.--
;::">'·-.:i-J:i;l·
that is,
x 2z[x 3
+ ay 2 z + 2bxyz + cx 2 z + syz 2 + Jx~ ] = o: 2
One sees easily that the part of E which is not being blown up or do m is transformed nicely into the curve
-ay 2z ~ (2bx + sz)yz = x 3 + cx 2 z + txz 2 .:, All this is more transparent if we write (2.10) in affine (x, y) coordim es:
x 2 + x(ay 2 + 2by +c)+ sy + t = 0. If we throw in an extra component (which will be collapsed to a poin, ), the equation becomes
x 3 + ax 2 y2 + 2bx 2 y + cx 2 + sxy + tx
= 0.
The effect of the transformation (2.11) is to replace th(,:.quantity xy i 1 this la~t equation by the quantity y, so we get
x 3 + ay 2 + 2bxy + cx 2 + sy + tx = Q. Now replacing y by [y- (bx/a + sj2a)], we eliminate the .linear term 1 1 yin last equation; and so by replacing (x, y) by ((: 1 x, c 2 y), we can 1 ;duce our equation to the form · ~his
y 2 = x3 + Ax 2 + Bx + C•.:
(2.13)
If we follow the original point p
= (1, 0, 0)
through all these changes, we see that it gets blown up under (2.11) ~· >that to calculate which point of the image cubic it corresponds to, we mu~ look
50
Chapter II
at the equation of E to second order at p. That is, what happens to the tangent line z = 0 toE at p under the transformation (2.11}? The fact that th1s whole line gets collapsed to the point (0, 1, 0} allows us to conclude that the image of pis (0, l, 0). Since the remaining changes of coordinates leave the point at infinity (0, I, 0) fi~ed, our point p corresponds to the unique point at infinity of (2.13 ). Th tis p does in fact go to an inflection point. If p is an inflection point to begin with, the transformation (2.11) is unnecessary in the normaliLation process. In any case, if our original equation (2.9) has coefficients in Q (or !R), so does the final equation (2.13) and there is a nice bijection between (2.9) and (2.13) given in each direction by rational fu!'lctions with rational coefficients. Thus we can find Q (or IR} points of (2.9) by finding those of (2.13 ). Also remember that for a second-degree equation with rational (real) coefficients, if one solution of the restriction of the equation to a rational (real) line is rational (real), so is the other. In the same way, if two solutions of the restriction of a cubic equation to a rational (real) line are rational-(-real),so is-the third. Thus our geometric group law is well defined for cubics in QIP 2 and ~ffl> 2 A high point in the number theory of cubics is the theorem of Mordell, which says that the set of rational points of a rational cubic curve form a finitely generated abelian group.
2.5 A Thought about Complex Conjugation The solution set of a nondegenerate real cubic .curve is topologically either one or two circles, depending on whether the cubic polynomial in x in (2.13)· has one or three real roots. To see how this solution set lies with respect to the set o~ complex solutions, we regard the latter as a twosheeted. covering of the x line (a complex line) ramified above the three points
Pt> P2, fl3 and at oo, where at least p3 is real (Figure 2.11 ). Suppose now that p 1 and p2 are not real. Then the real solution set to (2.13} projects onto a half-line connecting p 3 to infinity so that removing the real solution set from E simply makes it into a tubt:. The complex conjugation map (x, y)~·· (.\', .Y)
leaves the real solution set pointwise fixed. If p 1 and p2 are real, the removal of the two comporu:nts of the real solution set from E breaks it up into two components which are interchanged under complex conjugation.
51
Cubics
r Pz
Figure 2.11. Branch points of the projection o(areal cubic. .........
This last phenomenon carries over to nondegenerate curves of I 1gher degree n whenever the solution set has the maxirllum number [(n- 1) x (n- 2}/2 + 1] of components. ·· ·
2.6 Some
Meromo~phic
Functions on Cubics
Let's look a bit more af stereographic projection from a point p e E. This projection gives a map E
CIFD1
or, what is the same thing, a meromorphic function on E. Given Pt> pz. E E, let p 3 be the third point of intersection of the line p 1 p2 with E. l hen a sterographic projection with center p 3 can be used to construct a meromorphic function/onE whose only zeros are p 1 and p2 and which has a pole at a pregiven point p0 (see Figure 2.12). Suppose 9 is any meromorphic
Figure 2.12. Stereographic projection from p 3 gives a meromorphic function with z.ero set {Pit Pl}·
52
Chapter II
function whose zeros are PI and p2 and which has a pole at p0 • Then the quotient
fig is zero at only one point (the other pole of g) and has only one simple pole (at the Qther pole of f). In other words, under the map
E
Cl?~>
p
(f(p), g(p)),
0 and oo have unique prcimages, so this map is either an isomorphism (which is a topological impossibility) or a constant. Thus all meromorphic functions of degree 2 on E arise via stereographic projection.
2.7 Cross Ratio Revisited, A Moduli Space for Cubics We have seen that lor any stereographic projection, the branch points, that is, the points on the image ICIP\ owr which the map is ramified, are distinct. Let us assign to p E E the complex number
(11I• P1• l'J• P4), that is, the cross ratio (see S..:ction 1.6) of the four branch points of the stereographic projection with center p. Since the cross ratio depends on the nrder in which the four points are taken, we do not obtain in this way a well-defined function on E. To fix this up, let £
of E with itself which does not hit the diagonal locus of ~:;:~·Now we take the jibered product, that is, we take the set X of all ·pafrs (p, (P~> P2. P3, P4)) such that (PI, P2, P3, p4 ) is one of the cross ratios whicb occurs when projecting from p:
1 T
branched (4!)- I cover
E -----·-> £141
)
Then the cross-nd io map is wdl ddincd from X to C. This function is holomorphk, and since the four points are always distinct, it never takes the values 0, J, or oo. So by the maximum principle it is constant. We have
CubiC!>
53
just seen that these sets of branch points are just the branch-point sc s of degree-2 meromorphic functions onE, therefore We ,~nclude: If E and E' are cubic curves which are isomorphic as complex analyt c manifolds, then the set of branch-point cross ratios forE must equal tf·e set forE'. ';",
In fact, if any one cross ratio forE is equal to any one of those forE', hen by elementary algebra ,the others most coincide as well, for if A. is one o the cross-ratios, there are only five oth~r distinct ones;,nainely, 1
I'
1
1 A.-1 A. -A., 1-A.' ~· A.--,1
.?:.:;:.-
Conversely, if E and E' have a common cross ratiQ l; then an easy an: :ytic continuation argument shows that each is isomorphic to the curve y 2 = x(x- l)(x- A.).
Thus we have constructed a moduli space, that is, a topological ~. 1a~e whose points are in one-to-one correspondence with, .the set of iso.norphism classes of cubic curves. This space is
C- {0, 1, oo} equivalence relation where A. "" A.' if A.' is any· of the numbers 1/)., .1, ~ A., 1/(1 - A.), (A. - i)/A., A./(A.- 1).
2.8 The Abelian Differential on a Cubic There is another rather deep connection between the analysis, geon and number theory of cubic curves. Suppose that we take our cur given in the normal form
F(x, y) = y 2
-
x(x- l}(x- A.}= 0,
A.
Differentiating implicitly, we get (
~~ dx + ~~ dy )IE = o.
Now at the points (0, 0}, (1, 0}, and (A., 0} we have
oF =I= 0
ox
'
e Q, IR, or C.
~try,
as
( ..14}
Chapter II
S4
which means (by the implicit function theorem) that y can be used as a local coordinate for E near those points. That is, iff is a holomorphic function on a ndrhborhood of one of these points in ICIP> 2 , then fIE can be written locally as a power series in y. At other (finite) points of E, ~
x- x(point in question)
can be used as a local coordinate. So -1 iJFjiJx dy = - - ---- dx
2
y
is nonzero at the three points (0, 0), (1, 0), and (A., 0) and therefore so _is dx y
(2.15)
We also wish to check the behavior of the differential (2.15) at infinity,, that is, at the point (0, l, 0) of the ICH1' 2 in which the (x, y) plane sits as an open dense subset. First we "homogenize" (2.14) to get _iz
z)(x-A.z)=O.
x(x
We then get the affine equation for J:: in the (x, z) plane [a neighborhood of (0, 1, 0) in ICIP> 2 ] by setting y = I. We obtair G(x, z)
x(x - ::)(x- A.z) = 0.
=;;
(2.16)
Since iJGjiJz =I= 0 at (0, 0), the implicit function theorem again allows the use of x as a local coordinate for E there. So on E z=
L a.x",
and substitution in (2.16) yields
Now let's examine the differential (2.15). We homogenize it by writing it in the form ~
d(x/z)
yTi-- =
x dz - z dx -- ---y~ \
so on the (x, ::) plane given by (xi::) d::
M.:tl mg y =
)
I the differential (2.15) becomes
dx =~ (2 +positive powers -of x! dx.
(2.17)
Thus this differential is finite and nonzero at the infinite point of E. In the language of modcn1 dift\.:'rcutial gcomdry, the section (2.15) of the holo-
Cubics
morphic cotangent bundle of E has no zeros and no poles. Thus the c' Langent bundle of E is trivial (and so the topological Euler characteris· ic is zero). For more information about this see, f~r example, [10, Section a].
2.9 The Elliptic Integral We pick (0, 1, 0) E E as a basepoint, which we call p 0 , and wed, ,1ote the differential (2.15) by· w. Consider the map
E
C,
J---.L ,p
p
I ,x(p)
w = Joo
'.18)
dx [x(x -l)(x- l)]l 12
This map is not well defined, of cour~e, since we have not made precis our path of integration. Remember that we have realized E as a two.-sb l!ted. cover of the x axis cross-pasted along slits (see Figure 2.13). Also .
w = [x(x- l)(x- l)r 112 dx has two determinations, depending on which value of the square· r, ot is picked. The differential w on E is obtained by choosing one of these <
m, n E 7L, where ,).
n:2=2j• 1 W1 •
I I
0
1
1
uultuu,,
,.,...........y
>.
• \\\\\\\\ \\V
0
1
Fiaure 2.13. Floor plan of a cubic.
Po
~·uttom Sheet
Chapter II
56
Also if we write x
J'
i
2 c
=
u + iv where u and
_
W1 t\ W1
l'
are real variables, then
i . (du + i dv) A (du - i dv) =- 1 2.c jx(x-1)(x--")l
1
= j' -
c jx(x- 1)(x- ).)j
(2.19)
du dv
which, in polar coordinates near 0, is
Making a similar development near 1, ;., and p0 , we see that this double integral over all of C is a finite positive number. On the other hand, if we write i = 1, 2,
where ~i and Y/i are of the form I
fn:al-valued function of (u, v)] du +[real-valued function of (u, v)] dv,
then
(2.20) We next wish to reduce this surface integral to a line integral. If we make a cut in the x axis as shown in Figure 2.14, then on the remainder of that complex line we can, by Green's theorem, find a function f such that (~f'= ~ 1
and
lim
.f(p) = 0.
Again by Green's theorem .
I ¢1
•
,1
1\
Y/1
=I
.r+YJt .0
• ).
+ J I-+Y/t +I .1
,00
• ).
I+Y/t
where fl + is the determination off above the slit and fl. is its determination below the slit, etc.
..-----------------
Po
=
0
Figure 2.14. Cutting the ~uhic into two ;lit complex number lines.
57
Cubits
Now along the slit ..too, along U, ,co
j+
=!- ~ 2.'· JA a=!- +'2
,1
I· ~t~J";+ Re ·o ···. ·
7t1;
along 01, Also along ..too, along U., and along 01, +
= -tit- .
'11
This all means that the integral (2.20) can· 6~.. computed via .:Jreen': theorem: ,
,CO
J ~1
1\
'11 =
.
·~J!,'
,CO
J /+t~i 0~
= ( (Re = t(Re
- J ~-~··t 0
n1)tli + ((~e, ndt~i
1t 1 )[Im 1t 1
+ lin n 2 ].
Repeating the same argument with
dg =tit
and
lim •:f!(P) = 0,
I.
p-+co:r::·
we obtain
J~1 "t/1 ... -(lm n,·) [( ~t + ('~t] =
-1
2
(Im nt)[(Re n 1 )
+ (Re n 2 )].
Adding gives (Re nt)(Im n 2 )
-
(Im n 1 )(Re n 2 ) > 0.
(2.21
Chapter II
58
This mcan1> that the two complex numbers n 1 and n 2 are linearly independent over !H so that
.
'•
I[;
E = ...... ·---------Zn1 + Zn 2 is a compact compkx torus. The relation (2.21) is called the second Rimu11111 relation.
Thus (2.18) gives a well-defined holomorphic immersion
£-.. E.
(2.22)
Since E is compact, this mapping must be a finite covering space mapping. lu fact, it is immediat.: to see that the mapping (2.22) induces an isomorphism ·
ll 1 (E·• l.}::::: H I (E·' Z) or, what is the same, an isomorphism of fundamental groups. So by covering-space the::>ry, the map (2:22) is itself an isomorphism '
E
....--
~
1[:
-------+ ------------bt1 Zn 2 •
+
(2.23)
We shall later encounter the inverse isomorphism to (2.23). In particular, the composition of this inverse with the coordinate function x on E [see (2.1 !{ )J has· special S1gnifil:ance in the development of the subject, since it has a particularly nice explicit expression. It is called the Weierstrass P:Jiwct ion.
2.10 The Picard-Fuchs Equation The same differential
d.\
dx
w = y = [;( x - 1)( x--_____,A.)=]t"""'/2 which gave the mapping (2.18) has another 1 very surprising use. Namely, the numbers ·
(2.24)
,.\
n 2 (A) = 2
j •1
w,
59
Cubics
called the periods of E, depend on A.. It is not hard to see that they fact, holomorphic functions of A.. For instance n 2 (A.) can also be o by integrating w around the path y, as shown .in Figure 2.15, so tl moves slightly, the path of integration does not. change and we ca1 entiate n 2 (A.) by differentiation under the integral sign. Let · ,).
are, in >tained at as .A differ-
/,
n(A.) = 1tt(A.) + n,2(A.) = 2jo w. n(A.) can be obtained by integrating (() overthe path (see Figure 2.1' ). As A. approaches 0, this integral becomes
J: x(~~x~) 112 = residueo x(~~l) 112 = 2ni(-,. i) = 2n.
Thus n(A. )/2n has a power-series expansion around A. = 0. We wish 1·) compute this expansion explicitly. To do this, we need to study the dill rential equation satisfied by the ni(A.), the so-called Picard-Fuchs equation. And so we must compute the derivatives of ni(A.) with respe, < to A.; that is, we must differentiate under the integral sign; we must com1 .tte
:A. [x-112(x -1)-lf2(x _ ..ttl/2]
= fx-112(x _
Itlf2(x _ A.t 2,
02 ------ . . oA.2 [x-lf2(x -1)-112(x- ..ttl/2] = h-1(2(x-' 1)-l/2(x- A.t 2 Now those readers familiar with the theory of.deRham cohomolc ;y will recognize that there must be a relation between the three differenti~ s
ow o2 w
w, a..t, a..t2 ,
y
----~~- ~-="~.~j------~r~.---0
',1... ______ ., ,,'
Figure 2.15. One of many equivalent paths of integration
f~r
n 1 ().).
tiO
Chapter II
-Figure 2.16. One vf many equivalent paths of integration for n(..\).
that is, some linear combination of these (with coefficients functions of ..1.) must be an exact Jiffcrential. So the corresponding linear combination of
must be zero. But let's just compute. If we fix A. and differentiate with respect to the variable x; we get
d xt'l(x _ 1)til(x _ ..1.)1/l
(x- A.)l
11__ x- 112(x __ 1 )ltl(x _ ..1.)- 3/l = 2
+ ~xl'l(x -1)-112(x _ ..1.r312
-----
-
~ xlll(x- l)li2(x- ..tr5/2J .dx
2
= (x - 1)
iJw ai +
iJw X
a"w
OA - 2x(x - 1) OA 2 0(1)
00)
= [(x- A.)+ (A.- 1)] oA. +_[(x- A.)+ ..1.] oA. 02(1)
-2[(x- ?c)+ A.)[(x- ..t) +(A.- 1)) o..1.2
Cubics
61
= 'i-w +(A.- 1) 2 ·
ow . l OW - + -w tA. - - 2[(x- ,.) +).] oA.. 2 · • oA. , ·· .
3 ow 0 2 w) .. ·. + (A. 1) -oA. 2 == ·w ·.. ( 2 oA. X
- -
. ow + (U -1) -o), · -
3
- -2 w
ow ow . o2w -3(..1.- 1) oA. __, 3A. oA..:d2A.(A...,. 1) oA.2 = -
1
2w- (4A.- 2)
Integrating both sides around our
ow . :·. o2 w oA.- 2A.(A.-, 1) oA.2.
cycles~--w~·-,·~~f·:
tile
Picard Fuchs
~quation:
(2.25) Now we will appeal to the elementary theory of ordinary differenti<: equations with regular singular points.t The indicia! polynomial of (2.2: ) is
q(r) = r(r -1) + r= r2 •. This means that the vector erated by crl(A.)'
spac~
of solutions o£.(2.25) near A.= 0 ts gen-
holomorphic and nonvanishing at 0,
'
(A.cr 2 (A.) +(log A.)cr 1 (A.))
where cr 2 is~olomorphic and non vanishing .at 0.
Thus it must be that if we normalize so that then
t See E. Coddington, An Introduction to Ordinary Differential Equations (Englewo· ·d Cliffs,
N.J.: Prentice-Hall, 1961), Chapter Four, for more details.
Chapter II
62
Let's solve explicitly for the power-series expansion
of a 1 • First,
a'1 (A.)=
L (n + l}an+tA.", n
a'{(A.)=
L
(n+2}(n+l)an+zA.".
n
So from (2.2:5) we obtain
L [A.(A.- l)(n + 2)(n + l)an+Z + (U ,-,J)(n + l)an+l + !an]A." = 0.
n:2 0
.
Rewriting we get
L [(n + !) an 2
(n + 1) 2 an+ t)A." = 0.
n~O
Smcc a0
= J and we now know that an+ I --
2 [ll + Q/2)1 1 an, n+
we obtain (2.26) Notice that we have obtained along the way that near A.
=0
rr 1 (A.),..., log A.. To sum up, we have a very specific power series
f (- 1/2 )2 A.", n= 0
ll
which gives us "the" solution to the Picard-Fuchs equation that stays bounded near the singular point A. = 0. It is no accident that the coefficients in this power serit:s are rational, as we shall soon see.
2.11 Rational Points on Cubics Ovt':r IFP So far everything we have done with the differential w and the function rr(A.) lies in the realm of analysis or geometry. But surprisingly enough these computations have number··thcoretic applications. For this we must
Cubics
consider cubic curves over the finite fields IFP = 7l.jp7l.,.
where p is an odd prime number. Given A. e 7l., we reduce A. mod p ar.d let C l denote the solution set of · y 2 = x(x -1)(x- A.)
in IFP x IFP. The question we will ask is,
\_,;,,,. -.·. ''-\T'<: ,,
What is the cardinality of C ;.u· ,,· Or, equivalently,
For how many x e IFP is it true that x(x- 1)(x·.:_:·~) t:~,u=;? ',·:)!·:~':,:-.:
Now since the multiplicative group (IFP- {0}) is cycl_ic.;/'x(x- l) x (x - A.) e IF; if and only if · ··. ·
A.)]
= 1 or x = 0, 1, A.. In all other cases [x(x- 1)(x- ,1.)]
this-
in the very neat formula (number of point~ in
·.
CJ=
L {1 + [x(x -1)(x_:))]-t>i
X" IFp
2
}
'., . ·.
( :.27) modulo p. We next wish to simplify the right-hand side of (2.27). For this we use the (character) formulas M if(p-1){k, if (p- 1)lk.x;
c.28)
These are easily proved by noting that
'Lxk=.v"Lxk for any y =I= 0. Now suppose we write
[x(x- 1)(x- ,1.)]
(.. 29)
as a polynomial in x. Then by (2.28) the only term in (2.29) which will contribute to ·
L [x(x- 1)(x- A.)]
Chapter II i~
thl' tt:rm mvolvin'g xr 1 The coefficient of this term is the coefficient of x
l(x- J)(x- il.)]
I(p~t~/2 ( ~ (p
1)/2 )( -·· 1)k x
-k J r~~/2 {(p -11 )/2 )( -1)'il'x
has as coefficient of x
= ( -l)(p-
1112 (p-1)/2 r~o
((p- 1)/2)2A.' r
= (-l)
r
modulo p. This last congruence needs a little explanation. The integer
1p--l r! 2
··--· ···-·······-··
p·-3
p- (2r- 1)
2
2
represents the same element of IFP as does
a(p
I )(p - 3) •
· [p - (2r - 1)),
(2.30)
whSC9- a is any integer such that ..........
ar! 2' == 1 mod p. But then
(2.30) =a· (-1)( ---3) ·
· [ -(2r- 1)) mod p,
which gives the desired congruence. Now if r ~ (p + 1)/2, then
-- 1/2 ) = (_~ ~)( - 3)_ (- p) . . . ( r r!2'
=0,
so the formula for the cardinality modulo p of C_. is
This is the sa111e formula as the one for the period function n(il), the solution to the Picard-Fuch!> equation which is holomorphic at 0! Of
Cubics
course, this fact is not accidental-the reason for it is a rather dcp one discove~ed only i? recent yea~s by,Y. Manin. T~ ~lose this chapter, we wil try to· give some Idea of Manm's result. ·
2.12 .Manin's Result: The Unity of, Mathematics ~
":_.:,.,
The fundamental ingredient in this discussion will be the algebra-gc 1metri' version of the Lefschetz fixed~point theorem fro111 topology. The tt pologi cal version says .that iff is a differentiable mapping ·
f:M -+M,, where M is a. compact differentiabfe~manif~ld::i$~ch 'th~{ the,. gn.'\)il.of meets the diagonal transversely .in M x M, then the Lefschetz nun .'lef 00
L(f)=
2: ap(f)= L (-lttrace(J•: H~(M;C)-+H"(M; ~ peM
n=O
)],
·
where ifj(p)
+p,
'f ( hf) .,., (d' I) :·hjpositive I . . I . grap meet~. Iagona w,I~:i !negative/ onent. Lion:·1 '·<•.''., .
Now transversality at p means, in local cooroinates, that the mapr .ng (identity ,-f) has maximal rank at p. S<>
a(p) = sign det(I ,- J pv)], where J P(f) is the Jacobian matrix off at p. 'Now if A is a comple: -valuec matrix in triangular form · · A=
[a~,*.]· 0 'a · .
then
n
n
det(I-A]= ,L(-1)' r=O
"·
2:
alt •· .. ·a1,= 2:(-l)'trace(A,..~),
iJ·< .. ·<J,
r~o
where N A denotes the linear endomorphism which A induces on the rtl t See, for example, Section 7, Chapter Four, of E. Spanier's Algeb~aic Topology (I·· :w York
McGraw-Hill, 1966).
Chapter II
66
exterior power of C" Thus we can rewrite the Lefschetz fixed-point formula as .--L ( I ')
trace NJ p({},-- = L... ~ (= L\-'( - 1)' ··· ··-·-----. jdet[1 - 1 p(_f)] I
'· P
l)n trace (f*l H•<M> )•
(2.31)
n=O
The main point to notice is that we have two formulas for L(f), one given by local invariants and the other by olohal ones. Now if M is a compact complex manifold, let r be the global sections functor, and: let (J 0 r .9/0, 0 -(,- f'.cll'o. J ------;. r .9/0, 2 -:.......... (2.32) be the·Dolbeault complex on..M;.Jbat:::i_s.~
.91°· q = sheaf of C"' (0, q}-forms on M. (For more details on this, sec Gunning and Rossi [2].) The cohomology groups of this complex are denoted
llq(.M; {I•) or H 0 ·q(M). If M is a Kahler manifold, then Hq(M; (9)1 is a direct summand of the complex-valued deRham cohomology. Suppose now our mapping F:M~M
is liolomorphic. Thcnfinduces a morphism on the sequence (2.32}, and the formula (2.31} continues to hold when the deRham complex is replaced by the -complex (2.32). In thai case formula (2.31} reads · -trace A'J"(f) "' --z:(-1}'-jdet[-1'-=··:/·(-J.)]I = ~ (-lt trace(f*ln•(M,c;>)•. r,p
p
11-0
where J;(.f) is the restriction of the cotangent space mapping J P(f) to the subspace of type (0, 1). But as before
L (-I)' trace N J;(.f} = det(1
- J;(f)],
r
and if J~(.f) denotes the (1, 0)-part of J P(.f), then del[l - Jp(f)]
=- dct[l -
J~(.f')]
• det(l - J~(f)].
Thus our formula becomes
Now the marvelous fact is that formula (2.33} continues to hold in a purely algebraic context. The reader who is unfamiliar with sheaf cohomology ii1 algebraic geometry over an arbiu·ary algebraically closed field k is
6i
Cubics
urged to read on in any case. The results used will be the formal an tlogue! of· the corresponding theorems over C, and. ,that 'formal analo ~Y wil probably carry the reader through. In ·any case, the beauty of th resul should motivate most interested readers to further study of these tt.pics. Suppose, for inst.ance, that k is the algebraic closure of the fielc. !FP of 1 elements, ,1. e (IFP- {0, 1}), and M is the solution set to
y2z = x(x- z)(x- A.z) in kP 2 (one-dimensional subspaces ofk 3 ). In other ,~ords, M is the~ llution set to
• y 2 =x(x...., l)(x-
in k2 with the point at infinity thrown in. Letfb~ the Frobenius n, tpping
f:
M------+M
(x, y, z)
- - - - + (xP,
yP,zP). -: ~;
Now d(xP)jdx = pxp-t = 0, and if n > 1 =dim11 M for the sheaf(!) of regular algebraic functions on lvf., Thus the formuh (2.33) reads · .,
1- trace(f*lut(M;(!)l) =number of.fixed J?Oints off
(2.34)
But the fixed points off are nothing more than .the number of poin ; of M which are represented by triples (x, y, z) of elements of !FP, since X=
xP
if and only if x ~· IFP.
Since the point at infinity:is,one·fixed point off, the formula (2.34)·b, comes ""'"tracef*lni<M;(!)) = numberof points•in
G...
So we will explain the connection between the formula for the c; rdinality of C;. and the period function 1t(A.) by explicitly computing the 1 ace of j• on H 1(M; (!')). We will have to usc the fact that if q, q1 f! M, then
H 1 (M; (!))
~
(algebraic functions with poles at q and q') (functions with poles at q) + (functions with poles at q'j · This comes out of the exact sheaf sequem;e 0-+ (!) -+ (!)( 00 • q)
+ (!)( 00 • q') -t (!)( 00 • q + 00 .• q')-+ 0,
(2.35)
Chapter II
68
where---· -'f (n • q) =sheaf of algebraic functions on M with q as the only pole and tio worse thau order ( -n) at q,
&(oo · q) = lim &(11 q). n-+ ao
Also, using formal power series, we can define differentials on M, and as in the complex case, the k-vector space of everywhere-regular differentials is one dimel)sional with generator written locally at q
w = dx + ···--~··
L a,(A.)[x -
(2.36)
x(q)]' dx.
·:::c..~-='=•'::!:·...1
'"-····
:---.:-·
Also just as befbre,
1+
L a,(J.)[x- x(q)]' r
satisfies the Picard--Fuchs equation, that is,
( J.(~ - 1) ~ 2 + (2-1,2
I)~+~)( 1 + r~1 a,(A.)[x- x(q)]')
xlf2(x _ l)li2(x _ J.)l/2) · = J.~ series expansion for (x _ J.) 2 d (.
(2.37)
Furthermore, referring again to (2.35), we see that the pairing
H 1(M' (ii)
X
lregular
h
x
w
I
k
ld iflerentials / ~----+
(2.38)
resq(hw)
is nondegcnerate, so, in particular, dimkH 1 (M; &) = 1. This is the algebraic version of the Serre duality theorem. Using these facts we are in a position to compute
Namely, the algebraic Rienw1m-Roch theorem (see Chapter Three) will always assure us of the existence of an algebraic function Ji whose only po!crtre 4 and q' and which has a simple pole at q. We write II= _
-~-- + I<:O L h1[x- x(q)]' .
.\- .x(q)
69
Cubics
Then the map f sends h(x) to
h(xP) = [x-
~(q)JP + ~~o~,[~ "\":y,(q)JP'
as long as q is chosen to lie in CA. So the trace. off* must be the coel icient of · ·
1 [x- x(q)] in the series expansion of h(xP)w. That is, referring to (2.36), we ,hav (trace/*) .=o' a~-=-1(2):- .....
But now By {20"37) ·
:+: · ·--·
a2
a
(2.39)
1)
·. . ·
(A.(A.- 1) OA2.+ (2J- 1) oA. + 4 ap_ 1 (A.)(x -x(q)JP,~l. d ..
= dx{c~A.)~x ::- x(q)JP} =
'1,
so that aP_ 1 (2) satisfies the Pica~d-Fuchs equatipn_L·Also aP_ 1 (2) i univalued around X= 0, so to compute its series expansion we rna~
"" (-1/2)2.2';:• aP_ 1 (2)=cL r=O r .. To evaluate c then we need only compute the number of points of one value of 2 (which we have already done). .Thus tracef*lm(M;(I)J
•/for
= -cardinality of CA' = ( -IJ"- "" = ( -1J
.~J-;12)'·· L : : : 1/2 )2;.r.
(p-1)/2 ( r=O
T
2.13 Some Remarks on Serre Duality The usual form of Serre· duality in the theory of complex manir ,[ds is not the one we used to make this last computation. Instead we us d the nondegeneracy of the pairing
(2.40)
70
Chapter II
where (i M, and
+ k) ~ U+ /) =
m = complex dimension of the complex manifold
.01 = !.heaf of holomorphic I forms on M.
H*(M;
.Qi)
is computed from the exact sequence
qnj 0
do· o)
.__1__. qnj ® .9'/o. 1) ~
which generalizes (2.J2). The pairing (2.40) is given by
where w is a a-closed (i, i) form and t7 is a a-closed (/, k) form. Thus in the case which concerns us, the usual complex version reads
H 1(M; l!.')@H 0 (M; .0 1 ) - - - - - - + C ,
I
(tt. (t))
(2.41)
t] 1\ ()).
M
To see that this is the same (in the complex case) as the pairing (2.38), consider the open covering (} 1
= (i\J- q),
U2 = (M- q')
of M. Then iff is a function with simple pole at q and only other pole q', the corresponding Cech one-cochain with coefficients in (!) is given by
(2.42) Next construct the double complex of Cech groups
f!v,,.,v,i6 >--·
....--0 ----+'C (M; t'[;) __ ..., C (M; d
i~
0 _..., C (M; (I') ___ _,. C (M;
i 0
0 0 • )
(y. h) i o 0
0
ia
flu,,.u,ib 1
1
i
si 0 • 0 )
0 0 ----+C 1 (M; .9/ 0 • 1 )----+ (og. ou)
ib
----+ C (M; .91°· 1 ) ___!__. 0
(au) i
(J (: - - - - - + r(.c/0, 0) _ _:;___. r(,Wo, I)
i
i
i
0
0
0
0
----+
71
Cubics
and follow the cochain (2.42) across to the corresponding represent;.; ive og in r .s¥0 • 1 • To do this, pick . jsmooth at q', 0 l = (1/2)/ away from a neighborhood of q', h Jsmooth at q, 'l = (-1/2)! away from a neighborhood of q,
such that g - h = f ev.erywhere except at the points q and q'. Then
on= oh, ~J:ld _t_h.e~eq!:!iy~lep~~-<>.Lthe p~_irJQg~__{2.3~} ;:tnd (±~llows frt m the
equation.
•..
·::;·~:::::::-~·~·:::::.,., ···•7•;--~~'c:'::::~ •... ·. ,, .,.:e';:~~i}3~~~:~::,.
f fro = f
Jim radius->0
f
(JW
9
=
f
M
(J)
A
dg ·
=
f
(J)
.•.• ;~~"
A O(J.
M
The point is that the (equivuent) form (2.38) of the pairing carries >Ver to the general algebraic case, whereas the form (2.41) of the pairing m Kes no sense except when we a·.~ working over the complex numbers.
i
I
CHAPTER THREE-
Theta Functions
3.1' Back to the Group Law on Cubics In this chapter we will attack the problem of giving a nice structure o the set of isomorphism classes of cubics and also the set of "framed" ' tbics. We rely heavily on the analytic tools of theta functions and modular 'orms. The introduction of t~ese concepts will also prepare us for their u~ in a more general setting in later chapters.t In Section 2.9 we saw that if Eisa nonsingular cubic in Clfll2 , a11 l ify 1 and y 2 form a basis for · and 1tj
=
r w,
• YJ
then there is an isomorphism of analytic sets f:
c
E Z1t1
+ Z1t2
=E
(3.1)
r
w.
p
'po
tAn algebraic version of the material of this chapter, which extends many of the r ,ults to cubics in characteristic p, can be found in "Algebraic Formulas in Arbitrary Chara,· ;ristic," by John Tate, which appears as Appendix 1 in S. Lang's book, Elliptic Functions ( eading, Mass: Addison-Wesley, 1973). 73
74
Chapter Ill
Since both E and E arc groups, we would like this map to be a group homomorphism. To see that this is indeed the case, let Cl?t denote the set Of lineS in cu:D2 alid define
c IP!
------•
E,
L
L f--·->
f(p).
(3.2)
pe(LnE)
This mapping is easily seen ro be holomorphic and everywhere defined. Since CIP! is simply connected, the mappingp.2)!ifts to~ mapping CIP! ._... C, since C is the universal covering space of E. This last map must be a constant by the maximum principle, and since the line
z=O must go to 0
E
E by the P1
way we normalized E, we have that
+ P2 + p3 =
0
in E if and only if
{PI• p2 , p 3 } = L n E for L E CIP!, which implies
Thus (3.1) is a group isomorphism as well. We should insert two remarks at this point. First, suppose we have a complex analytic mapping
.
c
c
I------+-----:----=' · 1t1 71. + 1t2 71. n'1 71. + n21l.
(3.3)
Then the function dfldz is doubly periodic and so is constant. This means that if we couple f with an appropriate translation and lift to a transformation C -+C, the resulting map is simply
for some constant·c. Thus: Up to trans/arion, f is automatically a yroup homomorphism.
75
Theta Functions
Second, it is natur.aJ: to, al>lc
wheth~r·all
complex manifolds of the for.
1
··:
..,------:-
(1ttZ
+ n2.z)",
the so-called elliptic curves,.come from cubic curves by the process Wl have outlined. We shall soon see that this is indeed the case. However, dore doing this, we wish to settle anothe.r question which we touched .1n in Chapters One and Two.
3.2 You <;~~'t farametrize a Smoo!h.S~~j~~~J~,;~r,~,i~~lly
_____________
·: :·:~:;:~·s::r;·:: ~~- ·-
'-=·.o:?:'l~-~---= -~~~-
·· :.:-.. .:: i~.. -~~:i~t-~t-t;-~~y~.:._:____ _
We were able to. "parametrize" c.onics by using the inverse of s .:reographic projection ~o give a mapping C-+ (conic),
(3.4)
and now we use the "inverse" of the multi\:alued map
E
C
p
r(J) 0
to give a parametrization C-+E ..
Later we shall give the.mappif,lg. (3.5):morezexplicitly, but for now let"..; just compare it with (3.4),. The mapping (3.4) is of course algebraic (given: by rational func ions) whereas the mapping (3.5) is not :In fact, there is no quotie 1t of polynomials ·
such that
u2 =IV- l)(f ~,tt)
(3.6)
except whenfand g are constaht'functions. For if (3.6) were true, th, n p~qi = q~pt(Pt - P2)(Pt- tlp2).
Then since w,e can (and will)· assume that p 1 and p2 are relatively >rime elements of the polynomial ring C[z] and 'that qi and q 2 are also rei; dvely .prime, we have that
76
Chapter Ill
Thus adJusting q 1 by an appropriate constant gives
qf = 1'1 (PI
- P2)(Pl - AP2).
(3.7)
Also since p~ = (const)q~, p 2 is a perfect square. So now consider the family of polynomials
J3y (3.7) this family has four-distinct entries given by (t 1 , t 2 ) = ( I , 0 ), (1, - 1), (1, - A), (0, 1),
-Whtch are perfect-squares. But now iL
ad - bd = square, crf- dd =square for some relatively prime polynomials r 1 and r 2 , we have a 112 rl -- b 112 r2 =square, alflrl
-t-
[Jl:2,.2
=square,
c 1' 2 r 1 -
dli r 2 =square,
cl12rl t-
rJii2,.2
2
=square,
so that the family of polynomials
becomes a square for four distinct values of (t 1 , t 2 ) in CIF\. Repeating the argument and no~ing that the degree of the polynomials in question is halved at each step, we arrive at a contradiction. ~tice that the preceding argument depended on the fact that
--
.
A =f 0, 1
lf, for example, A= 0, then we do have a rational parametrization of the si11guhtr curve E. This is achieved by means of stereographic projection with center the singular point (0, 0), for every line through (0, 0) meets E in only one more point, just as happened in the case of smooth conics (see Figure 3.1). If the·Jine is given by y =ax, and we parametrize the line by x, then the third point of intersection is given by
7.7
Theta Functions
that is,
x=a 2 +1 So our third point of intersection is
sim~ly
I.
(a 2 + 1, a 3 + a). Now if a = ± i, this point is (0, 0 ). The corresponding' liries are t 1e two lines with contact of order 3 with E at (0, 0). (See Figure 3.2.) T, gether they make up the tangent cone of E at (0, 0 ), and their ~quation is o! ,tained by simply eliminating from the equation ,· y 2 = x 2 (x- l) all terms which are not of lowest degree, to get y2
,. ''
= -x2.
' '·
' 'l 'l
' u
u
/.
/
/
/
/
Figure 3.2. The singular point on the cubic has two tangent liries.
78
Chr.pter III
So stereographic projection from (0, 0) is given by E
(x, y)
t-----t
CfP\, (y, x),
(3.8)
which is a bijecti0n except that it "splits" the point (0, 0), that is: T~oint
(0, 0) with the "infinitely near point" (i.e., tangent direction)
,__,_---a= i goes to (i, 1). ·The point (0, 0) with the" infinitely near point" (i.e., tangent direction) ex = - i goes to (- i, 1).
tg;
We say that 1 is th~ ~i~·~T~g~larizati;;;,~;c£. The map built from the inverse of (3.8) becomes an everywhere-defined algebraic map which is onto E and one-to-one except that (0, 0) has two preimages:
CIP't • E, (x, Y) ,____. ((xjy) 2 + 1, (x/y)((x/y) 2 + 1)).
3.3 Meromorphic Functions on Elliptic Curves Next let us examine the field of meromorphic functions on a nonsingular cubic curve. We want to build functions on E or, what is the same, doubly periodic functions on· C with periods n 1 and n 2 (linearly independent over !R). Replacing w = dx/y by cw for appropriate nonzero constant c, we can assume that
(3.9) Then the second Riemann relation, which we saw in Section 2.9, says. that
(3.10) From now on we will write t instead ofn 2 • Then ifjis periodic with period 1 and is holomorphic, it has a Fourier expansion 0(1
f(u) =
L
a"e2"i"'',
n=- oo
If
f(u
+ t) =f(u),
we would have the contradiction that ali e2 !finr = a,. .
(3.11)
79
Theta Functions
The closest we can come to this is to::demal)d· a second peri,Jdicity property, which results in .·a relation between a~o'.and a11 +1 ~o repl;.ce the relation (3.11). The. set of such functions wilr'then form (at most) ~t onedimensional vector space. We accomplish thi~ l:>y demanding that
f(u. + t) = This translates to the identity
e-2ni(u+"'if(~)::, .\·: i;·::"\f.J~;,!
i_Q other_ words, So if we begin with ao
= 1.
and pick a= (t/2), we get all= exp{21tictl(k-
-n)-r} = exp{1tin t}, 2
and so ao
f(u)
= L:
exp{7ti(n 2 t
n=- ao
+ 2"'~)} .. {·
"
By (3.10), if ulies in a compact subset of C,t~~ri for
jexp{7ti(n 2t
(3.12)
1·-;..
·. -~·-···
In I sufficient!;
large,
+ 2nu)}j = exp{ -n(n1t Im t + 2 Im, u)} ~ exp{ -n(n/2)7t m t}.
So the series (3.12) converges absolutely and uniformly on compa, sets, and the holomorphic function (3.12) is denoted by
8[8](u; t)
t
sub-
(3.13)
or, for the moment, more simply by
O(u). Notice that O(u) is an even function. So from the point of view of Fourier series we have found the ;losest thing to a holomorphic function on
E=
C
z + Zt
that we can. Suppose we start from another point of view. Recall I
:JW
we
Chapter III
110
constructed all the meromorphic functions on the Riemann s_phere CIP\. We can summarize the process in the following steps: ,\
l. Find a homogcnt:ous form (a section of a certain line_ bundle) on Cl? 1 with exactly one simple zero, call it X. 2. Operate on this form with the group of automorphisms of Cl? 1, to get the entire collection of linear forms
aX+ bY,
a, bE C. -3:-Btrild-all meromorphic functions on Cl? 1 by taking quotients of products
r:: (a_!_~_±_b_!_!:) • · ·· • (a, X+ b, Y).
-..,---·-:--,-.--·· -----.-.--·.···
.
(c 1 X
+ d 1 Y) ·
· (c,X + d, Y)
We will do the analogous process on E, and our function O(u) will take the place of the homogeneous form X. We begin by considering the domain inC, shown in Figure 3.3. This is called a fundamental domain for E since under the mapping (3.5), this domain exactly covers E once except that opposite edges of Figure 3.3 are identified on E. To see how many zeros 0 has in Figure 3.3; we compute the boundary integral
1 •
I
-. j
·······-------2m
'i'(Figure3.J)
1 d log 0 = ---; 2m
,>+t
j
•
(-2m) du = 1,
'a+r+l
since O(u
(l
(J
+
+ 1) = O(u),
O(u
+ r) = e-ni(r+2ulO(u).
(3.14)
T
o
+ 1
Figure .3.3. Fundamental domain of elliptic curve with periods 1 and t.
81
Theta Functions
To locate this single zero, we compute
1 .
-. I 21t1 •~Figure
u d log(} 3.3)
1 [ ,«+t+ 1 = -
;
21tl
j '«+ 1
1
,«+t
d log(}+ J
= 2~i {( -ni)[(-r +
«+t+ 1
[-r d log(}+ u( ~2~i). 414 + -r( -2ni) d1 .. · ·"
J
J
.
2a +2) + (2 x integer)]+ [integet: x (2ni-r)]
-t 1) + 2ni-r} m,nel.. (1/2 + -r/2) +m + m, + (ni)(2cx + 2-r
~
This places the zero. of (} in the middle of the fun<:tamental domai. 1, as ·· shown in Figure 3.4. Now suppose that P~>
.. ·•Pr and
q~> ... ,q,
are points of E such that
that is, as points of C,
L pi - L qi = m + m>~ with m, n e l.. We then form the function
f(u) =
I Iie~J--=- (t + -r/2)). njo(u .- qj- (t + -r/2)) ~
~-
zero·of e
Figure 3.4. The zero ofB(u) in the fundamental domain.
o
112
Chapter Ill
Clcarly.f(u
+ 1) =f(u), f(u
but also by (3.14)
+ t)/ f(u) = exp{2ni(l:p1 - l:q1)} = exp{2nint"}.
This means that
is doubly periodic, that is,
g(u
+ 1) = y(u),
g(u
+ •) =
g(u ).
Thus g gives a well-defined meromorphic function on E with zero set Pt, • · ·• Pr
and polar set
Conversely, if g(u) is a meromorphic function on E, with zero set PI • • · ., Pr
.and polar set
--
then the same computation as (3.15) gives
1 .
--: j
u d log g = m + nr
21tt · i'(Figure 3.3)
for some m, n
E
71.. We have therefore proved
Abel's theorem for elliptic curPes: There exists a meromorphic function with zero set {p 1 , ... , p,} and polar set {q 1 , ... , q,} if and only if
L Pi- L £!; E (71. + 7l.r). 3.4 Mcromorphic Functions on Plane Cubics The meromorphic functions on CIP 1 can all be written in the form
p(x, y)
q(x:y}, when.~
p and q arc homogeneous of thc'same degree. Meromorphic func-
83
Theta Functions
tions on a nonsinglliar cubic curve E s;; CI? 2 certainly include 1 ' J ' tions to E of quotients
th~
P(f, y, z) q(x, y, z)'
re tric-
( 1.16)
where p and q are again homogeneous of the same degree. By a theor\ :n of Chow (Gunning and Rossi[2], p. 170), all meromorphic functions on Cl?2 have the .form (3.16), and using the Kodaira vani~hing theorem (~i irzebruch [4], p. 140), it can be shown that any merom()rphic function or. E is the restricti?n of a. function (3.16) to E. Ho~e~e_r/we .show tlu jill meromorph1c functions on E come by restnct10n of functl()ns (3.11 ) bn Cl? 2 in a more elementary._way. Recall that the gr<:>UR~trycture onE h also given geometricallyin-CIP2 (see Chapter Two). The equ'ation '
will
""\
P1
+ P2 =
ql
+ q2
in E
means that the line L through p 1 and p 2 on E meets E again in the arne point that the line M through q 1 ahd q2 does (FigureJ.~). If l(x, y, z) ~ 0 is the equation of the line L and m(x, y, z) = 0 is the equation of the liL M, then the function ·
\ _ l(x, y, z) ( ) mx,y,z
g (x, y, z7 -
restricts to a meromorphic function in E with zero set {pi,
p2} and poL r set
~~
Figure 3.5. The lines Land M have a common point on the cubic..
Chapter Ill
84
-{q:~ (J2}.
Suppose now that we know that for r < r 0 , any function with zero set pI> ... , Pr and polar set q t> ... , qr can be written in the form
f(x, y, z) ··---·-·h(x, )!, z)' where f and hare homogeneous polynomials of the same degree in (x, y, z). If
+
Pt
+Pro= 4t
+
+ qro
in E,
then pic.k p' and q· in E such that P1
+
+- Pro-1 = ql + ··· +.~!o-2 + q'
and
J5' +Pro= qro-- J
+ qro·
Adding these two equations we get
p'=q'. Also by the induction hypothesis, we have
fllht with zero set {p 1 ,
... ,
Pr 0 .. 1 } and polar set {q 1 ,
... ,
qr0 _ 2 , q'}, and
.1~/h2
with zero set {p', Pro} and polar set {qr0 -
1,
qr 0 }. Then
kf~lht h2
has zero set {Pto ... , Pro} and polar set {qJ> ... , qr 0 }. Thus every meromorphic function on E has the form (3.16). There is another easy consequence of these theorems. If a meromorphic function f on E has just one siri1ple zero at p and just one 'simple pole .at q, then
p=q so that the function has no .zeros or poles at all. function on E. we ';"rite the formal sum
(f) =
L (orderP f)p,
Iff is a
meromorphic (3.17)
p€f
which we caii the divisor off Given a_ny q and p2 such that
E
Pt + P2 = 2q,
E, thei'e are many choices of p 1
85
Theta Functions
so there are many meromorphic functions whose only poles are at 'i and those of order ~2. However, iff andg are any t'\Vo such, then ther are. constants a and b such that · .j·i · af + bg
,.
has no worse than a first-order pole at q and' s~· is a con~tant fun•. don. Continuing to reason in this way we arrive at the. ~=:)~
Riemann-Roch theorem for elliptfc curves;. If l:j~~ r1 ~ 2 and ea h 0, then the vector space of meromorphic fun9~ions. with a pole )f order no worse than ri at a point qi, j = 1, ... ; stand, no other po . .:s has (complex) .dimension · ······· ... ·-..·:·C'", ··{'.• ri ~
s
L r,. j=l 3.5 The Weierstrass p-Funct,on • Suppose we look at a trigonometric function ,i,n:Jhe following .vay: The sine function is the inverse function of the function
t
1
•x
a(x) ::;=
(1 -
x2)1/2
dx . .};
With this in mind it makes sense to hope that the iny.erse function tc x 1 .;:;~:;;;) b(x) = J 3 2 .xo (x + ax + bx + c)l.'., J..
d:.tix
will also be a reasonably natural function with' nice properties. T :is is indeed the case. When the problem is suitably normalized, its solukm is called the Weierstrass p-function, to which we briefly alluded in .Ch tpter Two. ··;i; .--. We begin the construction of the p-function with our just-p; oved Riemann-Roch theorem. The Riemann-Roch th~orem implies existe1 ce of a meromorphic function on E with 0 its only pole and that of order 2. Let's explicitly construct such a function. First define 00
om(u;
•> = L:
exp{n:i[(n
+ t)2• + 2(n + t)(u + !)J}.
n==-oo
This series converges for the same reason that the one defining O[g: .u; r) does. Also ·
O[D(u + 1 ;llr) = -O[D(u; r).
3.18)
86
Chapter III
The-function-en] -is an odd function and so has a zero at 0; and
om(u + -r; r) =
( -l)exp{ -ni(r
+ 2u)}O(U(u; -r)
(3.19)
so that just as in (3.14)-(3.16), tJU](u; r) has its only zeros at the points of, (Z + r.l) and a simple zero at each of these. Now
om
is doubly" periodic by (3.l8) and (3.19) and has a double pole at 0 since has a simple zero there. _ _____ · · We next itivesfigate pedoc:lidty properttes of O[U(u; -r) with respect to r. First we have dire~tly from the definition of O[l](u; r) that
O[l](u; -r + l) = e";140BJ(u; -r), and the functions
O[l](ru; r) and O[U(u; -1/-r) both have (Z + (-1/r )Z) as their zero set, both are odd, and both are such that if we apply the operator
to them, we get doubly-periodic functions whose Laurent series at zero is of the form
1/u 2 +(constant) + (even positive powers of u). Thus by the Riemann-Roch theorem
If we now develop the Laurent series expansion
-
d2 log O[D(u; r)
1
------;i~2------- = ~2
+ co(-r) + c2(r)u 2 + ... ,
(3.21)
then .;..-(3.21) implies immediately that if 11 > 0, C2 11 (T
+ 1) =
C2 11 (r),
c2,.( ---1/r) = rl
(3.22)
87
Theta Functions
Finally notice that Jim
e-nltf
t ... ,. 00
4
8[1](u; i) = exp{ni(u + j)} + exp{~ni(~
so that lim (3.21) = n2 csc2 (nu). t-+i.
00 '
all
Thus the C2n(r) stay bounded as t goes to i~finity along the p,.sitive imaginary axis. This last fact, coupled with the relations (3.22), says tJ, 1t the functions c2n(r) are modular forms (see Serre .[8], pp~ 77-79). More precisely:
c2n is a modular form of weight (n + 1) [or of weight 2(n'':j:";l) depend ng on the convention followed in assigning weights to modular forms]. Now a very simple residue computation (given in Serre's book [8)) that the vector space of modular forms of weight 1 has dimension the vector spaces for weights 2 and 3 each have dimensicm 1 'and generators together generate the gtaded ring of modular; forms. T~us. a constant · ,. ·
clr) =
c4 (t) =
L {1/(m + m)4 :_~~ n) e ((Z x z):?.{(O, 0)})}, L: 1/(m + m) 6,
says and their up to
l
(3.23)
since the convergent series on the right-hand side, called the EisHstein series, are clearly modular forms of the appropriate weights. We defi 1e the 1· ~~~n~~oo __ d 2 log O[f](u; r) _ ( ) P(u) du2 .: co t so as to eliminate the troublesome constant terin in (3.21 ). To see why this function solves the problem posed at the beginr 1ng of this section, we again begin with the Riemann-Roch theorem. Nan:dy, if we define the functions 1, p, p', p2, pp', pJ,_ (p')l where p' = dp/du, we have seven functions on
c Z + tZ' each of which has no pole other than 0 and with poles of order 0, 2, 3, 4, 5,
6, 6
88
Chapter-HI
at 0. By tlw Riemann- Roch theorem ·these functions all lie in a sixlhmcnsiona I vector space, so there is a relation
= 0. Develop the Laurent series of the left-hand side around u = 0: A
+ Bp + Cp' + Dp 2 + Epp' + Fp 3 + G(p') 2
-2
p(u)p'(u) = - +au+"·, u·5
2 [ p'( u )]
Thus C = D = L
= 6u4
2 - -8c2
u
-
16c.~-
+
= 0 and we can normalize by setting G = -1 to obtain (p') 2 = 4p 3
-
20c 2 p - 28c4 .
(3.24)
From this we see that we have a well-defined mapping IC
/:7;_ + rZ _ _ __.£, (3.25)
u 1 - - - - (p(u), p'(u)) into tht: cubic curve
If we extend to u mapping IC -- -----··-· - - - - IC iP' 2 ,
z + r7L tl
-
(p, p', 1),
then the mapping is evt:rywhere del1ned and sends the point 0 to (0, 1, 0) as
Theta Functions
89
it should. this mapping is everywhere_oLmaximal_rank; since p' at the four points of ramification of the 2 to l9overing
fero
-~nly
c ----+ CIP11 --7L + ;r7L u 1----- (p(u), 1) and has only simple zeros at each, sop" is not zero there. Also them .pping (3.25) is injective since ifis oLmaxim_~:~,Lr~:~,_~}_c_JlP,4_proper,, tqerefore u -:overing space, but (0, T,-orliasonly one pteiniage. Thus:_ ' ' 0
If we now look at the differential dx
y on E and pull it back under the map (3.25}, we obtain the different:tl du
on the domain. So x(p)
J
x(po)
d
~ = u(f-l(p)). Y
So (3.25} is the inverse map to dx
x(qJ
qr:-+
J
x(qol
(4x
3
-
20c 2 x- 28c4 ) 1 ' 2 •
3.6 Theta-Null Values Give Moduli of Elliptic Curves The relation_ (3.20) allows us to _conclude that
OnJ(u; -1/r) = a(r)exp{{J(r}u 2
+ y(r)u}O[l](ur; r}.
The famous Poisson summation formula from the theory of Fouric series will later allow us to_ compute
a(r), {J(r}, y(r) explicitly. This is an interesting computation which we shall do at l1e end of this chapter. - But for now notice that the functions
(O[i](u; r}f and (0[8](u; r})l
Chapter Ill
90
satisly the san1e periodicity rehitions, namely,
.f(u +-l}=f(u), f(u + r) = exp{ -2ni('r + 2u}}.f(u}.
(3.26}
Again the first relation means thatfhas a F9urier series expansion and the second .relation- means that the nth- ·Fourier coefficient determines the (n + 2}nd. So the vector space of flmctions satisfying (3.26} has dimension ai:,mOsfTwo: :Sri1ce,_:(_a[1J(J1;,t))2 -and_ (0[8](1!; r} 'f
.. ••~.:~
.:~:::.:-· .~ ... _;;;::~" •.••, ................. '"·';;'.~..
_;:;:=;;;r_ ..... :........
-.:
:.".=.: ---·~· --.
__,-- BlFP!; - -
-----~:::
. .~ --~--- ~::~- :.:._,;_ ·. -~-.. ~~,- -:· ~~ -. :·
. . -... ,:::,:, . . ,"-
u ~----- (0[8](u; r} 2, OUJ(u; r} 2 ).
- ·- .... .
.(3.27}
Now h(t + t/2) = (0, I) by (3.15} and his simply branched there, as well as at 11(0) = (1, 0). Since ~~-- 1 ((1, 0}) = {0}, we conclude that h is a double cover and is branched over two other points on CP 1 . We want to compute the projective coordinates of these two other__points. We begin by noticing that CIP 1 q
------E,
t----L p,
p
E
h- 1 (q),
lifts io an entire function on CIP\ and is therefore constant. We see that the two other ramification points, p 1 and p 2 E E, at which h is ramified, must have the property tlJat 2pj = 2 . 0 = 2 . (1/2
+ -r/2} = 0.
Thus the values of h at these points are
(fJ(g](l/2; r} 2 , OnJ(l/2; r} 2 ~ (O(g](r/2; t} 2 , om(r/2; r} 2 ). However, for reasons that appear later, it is preferable to have the values (3.27) of the branch locus of h in terms of the theta-null values, that 'is, the four numbers ,
0[8](0; r),
0(~](0;
r), O[b](O; t}, OnJ(O; -r)
(two of whkh we have yet to define). To achiev~.: this, we bc¥in by defining the even function
O[V](u; r) == O[gl(tl + t; r) =
L exp{ni[n 2 t + 2n(u + -!)]},
91
Theta Functions
which satisfies the, transfqrmation laws
O[f](u + 1; r) = O[?](u, ·r), O[?](u +'r; r) = ....::exp{ -ii(~+ 2u)}O[?](u; r), and the even function
O[A](u; r) = O[i](u,""""j; r) = L h. h
. fi
h
£
exp{1ti[{~-;jJ)2 r +2(n + !)u]}, ···,
I
•
\.. -~·--·:.;:
:~,~i~~~;~~~iti'i[)i' ~~-f;; That is, for any 1>, s e {0, 1},
= (-1)"0[~](u; r)1 O[~](u + r; r) =d -1Y exp{ -1ti(r -t;'2u)}O[:](u; r).
O(:](u + 1; ~)
Also by direct computation it is easy t() see .<'
tha~
•, .. •:::·:,;-
•
e(g](r/2; r) = exp{ -1tir/4}0[A](O;'r), em(r/2; r) = exp{ -1ti(r + 2)/4}0[?](0; r). So now we have the other two branch points of the map h in (3.27) t ivenin · terms of theta-nulls. Namely, they are the, points
(0[?]2' O[A]2}
( -em2, e(~]2} where
em=
0[~](0; r).
Thus our curve E is isomorphic4o the curve
y2
= x(x- 1)(x- A.),
where
A.= -O[A] 4 /0[?] 4 • In fact, then, all six cross-ratios
1
A,
I· 1 -A.,
1 A. A.-1 1-A.' A.-1' A
which uniquely determine E are given in terms of theta-null values
Chapter lii
92 1
0
Figure 3.6. The cut on CIJl> 1 uvcr which )• 1 (,!) lies.
Figure 3.7. The cut on CIJl> 1 over which y2 (J.) lies.
3.7 The Moduli Space of''Lev_ei~Two Structures" on Elliptic Curves Suppose that we denote the curve
. _, y2 =.:.s(,x -
1)(x ~}) •
.. - -~--~-'.·: .. :,..;.:..:.;; -·
. _f~ ···.
._
or rather its closure in (;~» 2 , by E(A.). In Chapter Two we integrated the differential form dx/y over the basis (3.29) of H 1 (E(A.); l). The cycles (3.29) were given by the inverse images in E(A.) of the slits on Cl? 1 shown in Figures 3.6 and 3.7. As A. moves, so do the one-cycles (3.29). Clearly we can move A. around, then return it to its original position, to achieve the homology transformations
Y1 (A.) }'2(..1.)
Y1 (A.), 1'2(A.) + 2yl(A.)
and
yt(..t) Y2(A.)
--
1---+
'Yt(A.) + 2y2(A.), Y2(A.).
The first transformation is achieved by moving A. around the slit 01 once and back to its starting position, the second by moving A. through that slit and. back to its original position. These two transformations, given by the matrices
f~ ~ 1 and f~ ~ J. generate the group of transformations of the A. plane given by
r 2 = IJlac
bl
d
E
...
Sp 1 (l). fac
integer-valued matrices where
b]-· [10
d =
OJ 1
mod 2 \·•
1
93
Theta Functions
rlr
n
To see this, multiply[~ ~]on the Tight by to~get lal < bl;by [A 2t] to get Ib I < Ia 1. etc., urttil b = 0. On the other. hand,. it is cleu that no matter how we move A. around and· return it to its. original positi m, the homology transformation will be the identity modulo 2 .. We say ti ;at the monodromy group of the family {E(.A.)} is r 2 • We have seen that
E(A),.
E(H~,E(l ,.~( 1 ~.<) '"~(~~tl~t~J) -A)
Thus A. does not ·exactly· measure the isomorphism·class ofthe curvl
;2'"'''~::::':'c..)_:'ifc'o~";::j;;'~;it~-:;,;;;-c'<~:;.:i.y2,5. ,~(~";;"'1)(~:T~~)!..~~-"€/(i~~~~;;;_' ··m·•:-'~"~c-· What then does A. measure? If we are given A., then we an~- given Dl •t only E(.A.) but also a specific choice of the identity element of,the grOL p ,E(.A.) (namely, the point at infinity), as well as a specific ordering'ofthe p~>irlts ol order 2 on E(.A.), namely · . · ·L
(0, 0) (1, 0) (A., 0). Now to get the cross ratio
z3 - z1 • z4 - z1 z3- z2 z4- z2 to have the value ...i, we order our branch points as follows:
z1 = oo,
z2 = 0,
z3
= 1,
z4 = A..
To get 1/A. we use the ordedng Zt
= 00,
z2
= 0,
z2
= 1,
and to get (1 -A.) we use Zt
= 00,
z3 = 0,
_So under the isomorphism E(.A.).a:: E(l/.A.) the first point of order 2 goes to the first, but the interchanged. Similarly, under '
seco~d
and tl ird an
·
E(A.) a:: E(l - A.), the first two points of order 2 are interchanged. Since the isomorph sms t<
Chapter Ill
94
are built up by composition, we conclude that the six distinct cross ratios corresponding to the same elliptic curve correspond to the six distinct .Qrderiligs of the points of order 2, or, what is the same thing, to the six ·distinct choices of ordered bases for H 1 (£().); IF2 ). Let's call a choice of ordered basis of H 1 (£().); IF2 ) a framing. Then we say that). gives us the curve£().) together with a framing of H 1 (£().); IF2 ). Conversely, given a cubic curve E and a framing of H 1 (£; IF2 ), there is only one choice of). such that the framing determined by ). is the given one. But we ·must be a little careful, for there-are-two-cases in whicp the ·six numbers
are not all distinct among themselves. First, we have the case in which the six numbers are
-1, -1, 2, }, }, 2, that is, the case of the curve (3.31) Second, we have the case in which the set of six numbers is
p, p, p, p, p, p, where p is a primitive cube root of -· 1. This set of cross ratios corresponds to the curve
·-- -
l
= x3 - l.
(3.32)
Now the curve (3.31) clearly admits the automorphism
(x,
y)~( -x,
iy),
(3.33)
which does not induce tlw identity map on H 1 (£; IF2 ). This automorphism takes a framiug of H 1 (E; IF2 ) into a distinct one. Siuce the automorphism (3.33) is induced by liftiug a linear fractional transformation on CIP\, the cross ratio must be preserved. Similarly, the curve (3.32) admits the automorphisms
(x,
y)~(ax,
-y),
where a is either primitive cube root of unity. These automorphisms also act non trivially 'on a'll framings and preserve cross ratios. Now suppose that we have any isomorphism
a: £(). 0 ) ~ E().o)
Theta Functions
which takes the identity 0 in, E to 0. We then. have the diagram
E(.'.o)
~(A.o)
---'a-
~ ~.r. 1}
!(.f,l)
CP1
CP1<.
where/is a two-sheeted covering branched over·oo'wit_iif(O).= oo. >y th.e Riemann-Roch theorem, ':i'i',;t®1t(ifi'.~~;)~:;: , c·~'+ d - :, -,. f .,.""- :1 -:•.'··''\' for some constants c, d, and so we can complete !h~J2~~~i~g__d_iag, ~tl!l_tq_ obtain -"' ..,,-~;;!n;_,;~?c-'''7::::'''-"- < ·· ···· · ---
E(.'.o)
---'a-
E(.'.o)
!f
l! p
CP1
CP1
where p is a linear fractional transformation. If a changes the fran- .ng of H 1 (E(.'. 0 ); IF2 } then since fJ preserves cross ratios, the set of six cross ratios must have repeated numbers. So E(.'. 0 ) must be one of the two curve~. (3.31) or (3.32). We can sum up this discussion as follows:
There is a one-to-one correspondence between the set of comp,~:!f an. lytic isomorphism classes of pairs · · (E, r)
[where E is a cubic curve and
r
(3 !4)
is a framing ofH 1(E; IF2 )] and the ;et
c- {0, l}. 3.8
Automo~phisms
of Elliptic (:urves
In fact, we can list all automorphisms of elliptic .curves. Let IX:
E .... E
be an automorphism of an elliptic curve E. Having coupled appropriate translation, we can assume ·
IX
w th an
1X(O) = 0. If IX acts trivially on H 1.(E; IF2 ), then in terms of a fundamental dom: m for
96
Chapter HI
1
·-··-•'>···------
'f.;i~~;~c is.""--F'""'u=~'~(J""'-~""·~""'i'e""~!""~t"§'_a~~~i;i'r0'7ciii(;tit
E (Figure!3.8) the points1/2, r/2, 1/2 + r/2, and 0 must all be fixed in E under the action of a. So a acts on line segments as follows
(0, 1/2)H (0, 1/2) or (1, 1/2), (0, r/2) H (O~r7i} or
(r, r/2).
(Remember that a is linear.) Since a preserves orientation, the only possibilities are
a= ±(identity map). From what we have done we can make a complete list of the complex analytic automorphisms of cubic curves: (a) translations (b) ±(identity map) (c) the curve y2 = x(x 2 - 1), which admits an automorphism whose square is -(identity) (d) the curve y 2 = x 3 - 1, which admits an automorphism whose cube is -(identity) All automorphisms of a given curve are obtained by composing automorphisms of that curve which appears on this list.
3.9 The Moduli Space of Elliptic Curves
---Next s'uppose we let .1t _.,
= 11r
E
~'I ov: m r > 0} ,
97
Theta Functions
the upper half-plane. We then have a natural map .Yf
c-
--...--+
{0, 1},
'r 1 - - - - ( -8[&]4/8[~]4).
( ·.35)
[see (3.28)]. We write
E= t
Then we have
soe~ in (3.3) that
if and_pply if there_
C
7L+-r7L
E, "'E,/C"
"~f ~~~~~~
is a comple~ constant e such that.:.: . ._;(.
<;LtiJ\di
·.
. . . ·~·l:i~\1ji (71. +·~i-h.\C? -~ft~;'it;~~~pi~~~~r~;r
e(z+' 71.-r) =
This will happen. if and only if the equations e(d + c-r) = 1, e(b + a-r) = -r'
(3.36)
can be solved, where, as in Section 3.7,
bl
[ac d
E
Sp1(7l.) ..
This happens if and only if 'r
,
a-r + b c-r + d
=--.
So we define an action of Sp 1 (71.) on .Yl' by the~'rule
[: !]
'T-
£[~t·
(aT+ b)(CT
I. (3.37)
and we conclude that the set of orbits of this action is in one :o-one correspondence with the set of isomorphism;classes of elliptic curws, that
.
~ .I(
= (isomorphism classes of elliptic curves) ;;;:;; Sp 1(7L\.Yl').
(deC)
To suin up, we have natural maps
·
(3.38)
Chapter 111
911
As we havt: set things up, the solid arrows are holomorphic maps. We next want to study the structure ofthe mapping given by the broken arrow.
3.10 And So, by the Way, We Get Picard's Theorem Let
(3.39) be the natural bundle with fibre Er for each r E .Yt'. We can then define an action_ of Sp 1(l) on 15' which is compatible with the action defined by (3.37); namely, we define the isomorphism ',:;
where .;:~-
---
e = 1/(cr + d),
as in (3.36). If A.(r) is as in (3.28) and r 1= r' but
A.( r) = A.(r'), then there is an element A
E
Sp 1 (l) such that A· r = r'
Also we have arranged things so that under the isomorphism
Et --> E(A.(r)) given by normalizing (3.27),
oo,
0 l/2
1, 0, A.(r ).
1/2 + r/2 r/2
Therefore the map (3.40) must preserve points of order 2. From (3.36) -~ t-····------~
r/2
1------
--->
(-cr' + a)/2 (dr'- b)/2
=~ = r'/2
Therefore A= I mod 2,
i.e., A E f'2.
(3.41)
99
Theta Functions
On the other hand, if (3.41) iS satisfied, then the composition
E(.).(-r)) ~ E, ~E,. ~ E(.).(-r~)) preserves the ordering of the points of order2 ·so. that
.).(-r) = .).(-r'). Therefore we have an injection
·..-
;.: r 2 \~ .... c -
{O, 1}~'{ ;. '.
-.-~;·
(3.42)
.... ~ ''·!;:·-
1'o see that the mapping (3.42) is actually a surjec~ion, we . reason ; s f()llows. For any . }!ff~...
· ..
). 0 E
there must exist a ·•
E ~
U,
C = {0,
such that
So E(.).(-r)) ~ £(.). 0 ), from which we conclude that X(-r) is in the set of sjx cross ratios dete. mined by .). 0 • Now the group of order 6, '· ·
Spl(z);r2, acts faithfully on the six orderings' of the points o(order exist
7, so the1 ; must
Spl(l)
A
E
t'
=A· t,
such that if then the isomorphism
E(.).(-r')) ~ E(.).()) preserves the distinguished ordering of the points of order 2. So
Ao = .).(-r'). Next suppose there exists tE~,
such that A.
t = t.
Chapter Ill '
100
Then, referring to (3.36), we see that the isomorphism
u-+ eu
is not ±(identity) unless .\
c = .b = 0,
a=d=
±1.
Thus either A= ±(identity) or the curve E, has a noptrivial automorphism. In the latter case, we refer to Section 3.8 to conclude that t
== ior p.,.. __,::_
In fact, from Section 3.7 we can conclude more. Every nontrivial automorphism--ot-E,...and E" gives a nontrivial permutation of the points of order 2, so that A ¢ f' 2 • Our final conclusion is therefore that f' 2 acts on .Yt' without fixed points so that
2:£--..(C- {0, 1}) is a covering space, in fact, the realization of the half-plane as the universal covering space of the complex line minus two points. An aside: A trivial corollary of this last fact is Picards' theorem that any entire function which omits at least two values is constant. This is because the two omitted values can be assumed to be 0 and 1 and, by the theory of covering spaces,- the map then lifts to an analytic map to the universal covering space ,;re of C - {0, 1}. But .Yt' is analytically equivalent to the unit disk so the function must have been constant. I
3.11 The Complex Structure of Jlt We saw in Section 3.10 that there is a natural map
j: (C- {0, l}) --------+ j f ). - - - - + (isomorphism class of E(A.)].
(3.43)
It is just the natural quotient map f' 2 \.:.If'-+ Sp1 (i:)\Yf induced by the action of the group Sp1(if)\r2. Si11ce the map is" generically" six-to-one, the quotient space of (C- {0, 1})
101
Theta Functions
with respect to the action of this group is a one-dimensional co nplex manifold .A whose topological Euler characteristic. i satisfies, the rel Ltion '. ~·:.~;-.-;;<.·
. •'
..
6x-7=-·v<. Thus x= 1 and so the manifold is simply connected. The points of It are in one-to-one correspondence with the set of isomorphism. classes o. cubic curves, so it is called a moduli space. By the iinifonnization theor• ·m for Riemanp surfaces (Springer, Introduction to Riempnn Surfaces [9], 1 224), .A is either C orthe unit disk. Since (C -:{9,)}) admits no b( unded analytic functions, , .. ;(~>:;;
. . . ~-~ . ~:" "" : ··~i1i~~, . '" .
.:
..
In fact, we can see this without using the uriifoiilliiation theorc n. We check directly that the rational function
4>(A.) =
(A.2 -A.+ 1)3 A_2(A. - 1)2
(3.44)
is invariant under the substitutions
A. -d/A. and
A. -d -A.
and so also under the substitutions which send A. to 1/(1 - A.),)/i A. - 1), and (A. - 1)/A. respectively. Since 4> is six-to-one, it follows that o/1'1 is isomorphic to the image of the mapping
4>: (C- {0, 1})- c: Since
L 11 cxiik(t)xitizk = 0,
i+j+k='3
where. the cxiik(t) are algebraic (analytic) functions ·of s'ome comp ..;x par· ameter t, then by the implicit function theorem ~e hav~ the follow :1g: 1. There is an algebraict (analytic) function.
{J: (t-line) __. CIP'2 .....,
t This function is an .. implicit" and therefore possibly multivalued algebraic;·Jiinclt. u.. •.• ,,
Chapter Ill
102
such that p(t) is always one of the nine inflection points of E, (see Chapter Two). · 2. Using P(t) as center of projection, one sees that t_l~e function t t-t A.(t) = cross ratio of branch points of E1
ts an algebraic (analytic) function, since the branch-point set is' given by the zero set of a discriminant polynomial whose. coefficients are algebraic (analytic) functions oft. 3. The induced map (t-lirie)-+ Jlt _ .is algebraic (anaJ.W¥)..... ---·-·-----·-··· . ______ .
·- ·. .t-
.
···~"/""" - .. ·.::.~;: .
•
•
-·. ·- •· .
.,,
.
It is the last property which makes vH a·.. good;;· ~oduli space, and this
property characterizes family
.J{
uniquely. Notice also that if we have an analytic
c 7L
+ t(t)7L,
where r(t) is an analytic function oft, then the formulas (3.28) and (3.44) imply that the induced mapping of the t line into ..Jt is analytic.
3.12 The }-Invariant of an Elliptic Curve We now return to a topic touched on in Section 3.5 and earlier. If E is a cubic or elliptic, curve, then we chose a symplectic basis for H 1 (E; 7L), namely, a pair of cycles y1 and y2 such that Yt • l'2 =
+ 1.
Given another choice, <5 1 and () 2 , then
J1 = ay1- cy2, ,52= ( -b)yl + dy2, where a, b. c, and d E 7l and
Ot • c5 2 =(ad- be)= The group of such matrices [~ ·S] is called
+ 1.
or It is the autom• •rpl11sm group of
(II 1 (£; l.), .intcrsectiot} pairing).
10~
Theta Functions
Now we obtained the number
-r in forming the.m?del
forE as
... ~ ....
(3.45 where w is the everywhere-holomorphic differential on E. Since w has n1 zeros, it is unique up to a multiplicative constantby.the maximum principlf Thus is completely d~termined. by E it~,lfhJ.odulo, the action
-r
b fy 1 co +a Jy2 W
a-r·+ b
'""'~~~~~~*+Z-r r:,a.~"lrr:J.:tt;,;q,_Jr~,-i>~~~··€:~·'·~:;4·:~;::I:~·,,~~:::~~~~~\;. .•.•. . . ·. ·!2= of Sp 1 (Z). Also each number in the upper haJf-plane occursiii·a (3.46), since given we can take
-r
E=
onriul
C
Z + tZ'
w = du, the standard differential on C, and Y~> y2 the edges of he fun damental domain. Thus there is a one-to-one correspondence betv cJn ~li the set of isomorphism classes of elliptic curves, and the set (3.46 where Jlf is the upper half-plane. In Serre's book [8], pp. 77-79, it is shown that Sp 1 (Z) acts l .roper! discontinuously with fundamental domain shown in Figure 3.9. I. is als1 shown that Sp 1 (Z)/{ ±identity} is generated by the elements
f-~ ~]and [~ ~J. that is, by the transformations
tl-+ -1/t, + 1.
fHt
Using this information we obtain another way of ombeddlna Spl(Z)\J'f =.A in projective space. Recall that we found, in (3;22), function!! Cln:
,Jf'--.C
(3,4/
Chapter Ill
104
p
Figure 3.9. Fundamental domain for the action of Sp 1 (Z) on the upper half-plane.
such that
c 2 ,.( -1/r} = r 2 (n+ 1 lc~ 11 (r),
C2n(r + 1) = C2n(r).
And so, since the two transformations (3.47) generate the action of Sp 1 (Z) on £. C2,.
ar+b) d)2(n+1) () C2n T (ci'+ d = (cr + ,
for ail[~, ~Sp 1 (Z). Since the c 2 ,. are also bounded as r -~ i · ao, we say that c 2 ,. is an automorphic form of weight (n + 1). Clearly the set of automorphic forms generates a graded algebra, and if
J'o, ... ,f,. are all homogeneous elements of this algebra of the same weight, then
CIP',.'
.YI'
r
(f0 (r), ... ,j~(r})
1-
descends to a well-defined mapping
Spt (Z}\Jf' -+ CIP',..
lOS
Theta Functions
Actually, the easy residue computation givenin Serre•s;book [8] 'hows that if · '·' . , ··· < · · ·
(3.48) is the vector space of modular forms of weighrk and f is a nonzero el .:ment of Mk, then · . ordCI(l
f +! ord1 f +! ordP f t [~-~~:_()f.orders off at other poin~~-?(!f/Sp 1 (Z)] = k;, ..
(Use the uniformizing parameter q = e2 "if to ccm)pute ordCI(l .) Thus 2 has its only zero at p and hasastmple zero_there,_c}+~~l:l~7its.<;mly zero a1 i and has a simple zero there, and some linear combinatiqn ' - -- .
(3.49) is the unique modular form (up to multiplicative constant) of weight with a zero at infinity, and that zero is simple. Now let-us form the mapp ng Spt(Z)\.n" - - - C~l·
t
1-----
(c2(t)\a{t)}
1
3.50)
So:
1. If we follow Serre's book [8] and add the point <Xl to Sp 1 (Z)\ 'If' by the rule that <Xl is given by q = 0 in the if-disk and
then we obtain a compact complex manifold of dimension 1. 2. The mapping (3.50) extends to an every~her.e-defined morph• in j: .A-+ CP1
such that the point
(c 2 (<Xl) 3, O)has only one inverse image.
Thus j must be an isomorphism. So we see in another way that
via the mapping t~---+d(t)/6(-r).
This last function of t is called the j-invariant of the elliptic curve associated with t and is the complete invariant of that curve. To compute values for A and Bin (3.49), recall that just after (3.: 2) we
Chapter III
106.
computed that c2 ( oo )/n 2 and c 4 ( oo }/n 2 are given by the second- and fourthorder terms; respectively, of the Laurent-series expansion for csc2 (nu). Thus c 2 (oo) = n 4 /15, c 4 {oo) = 2n 6 /189.
(3.51)
So we._c
20c 2 x - 28c;4.
-
'Since we saw in (3.24}that the elliptic curve-corresponding to the period r is simply the curve C given by
y 2 = 4x 3
-
20c2 x- 28c4 ,
we see that if r--+ i · oo, then two of the four branch poin~~ (of C over the x axis} come together. In fact, it is easy to see directly that if, say, A.--+ 0, then
·'
dx
j1 r~(;;-=-xxx - 1)]1 12 becomes infinite as it should.
3.13 Theta-Nulls as Modular Forms We will end this chapter with a few more illustrative relations between the coefficients of theta functions which again show their connection with the theory of modular .forms. To see to what extent the coefficients of theta functions are modular forms, we must compare
+ 1)
O[~](u; r)
and
O[~](u; r
O[~](u; r}
and
o[:](u; -1/r).
as well as
Let's just begin computing. First, the relation (3.20) allows us to conclude that
O[l](u; -1/r) =
IX 1
(r)exp{fl(r)u 2
+ y(r)u}O[D(ru; r).
Now since O[l](u; r) is an odd function in u, we can forget about the term conclude that
e*'" and
O[l](u; -1/r) =
(X 1
{r)exp{p(r}u 2 }0U](ru; r}.
Now fJ(r) will be determined by the· fact that both sides of this last equa-
Theta Functions
107
tion must have the same multipliers, that is, must .t:>ehavein the saJ, ,e way under the transformation · ·uRU+l, URU+t".
So using (3.18) and (3.19) we obtain that 1 = exP.{.B('r )(2u + 1)}exp{ -1ti[~(2u + J )]} so that ,8(•) = 1ti't". Thus OnJ(u; -1/•) = a 1 {•)exp{1tiru 2}00J(•u; •), (3.52) and we have only to compute a 1 (•).This turns qu~·1~'$~:infact, tL,~ most ;·::· ·. interesting part of the computation. We•begin by making the computations for ~[8](u; •) exactly an~dogous to those which led to (3.52) in the case of O[I](u;,f)~ We,obtain .·:
~
·,.
0[8](u; -1/•) = a 0 (r)exp{7tiTu 2}0[g](•u; r). We next compute a 0 {r). We simplify by setting u;,;;; 0 to obtain
L exp{7ti[n~( -1/r)]} =
a 0 {r) L expJiti(n 2•)}.
Next notice that a 0 (•) is an analytic function of• so that it suf 1ces to compute it for yalues x real> 0. • = ix, So we are reduced to the equati&n
L exp{ -.7tn
2
··•· .;.i.:,
/x} = a 0 (ix) L e~p{~1tn 2x}.
(3.53)
Next let's compute the Fourier transform ofthe function
f(t) = exp{ -7txt 2 } ,00
](s) =
j
e-nxtl,e-2nist
dt
/.
-oo .oo
=
exp{ -.1rs 2 /x}
j
• -00
= x -1/2 e - ,.Zj:c•
... :
exp{- [1t 112 x 1t;(t + isjx)JZ} dt
108
Chapter Ill
It is here that we apply the Poisson summationformula to conclude that
L exp{ ·-nxn 2} = x- L exp{ -nn 112
2
jx}.t
Comparing with (3.53), we obtain that
a 0 (ix) = x 112 so that
0[8](u; -·1/r) = (r/i) 112 exp{niu 2 r}0[8](ru; r) . .Also··....
,··~,:-~-=-----~· · ·. · ·······-··· ···-----·-·----
O[?](u; -1/r)~O(g](u+.}; -1/r)
= (r/i)~1 =
2
··'=--~--.::=:-~<~·:r:-;<·:,.,.:
exp{1iiu 2r }exp{ni(u + !)r}0[8](ru + r/2; r)
(r/i) 1 12 exp{niu 2 r}fJ[b](ru; r).
And also, replacing r by --1/r and u by ru in this last formula gives
B[?J(ru; r) = (i/r) 112 exp{ -niu 2 r}O[b](u; -1/r). Finally, using this last identity as well as the definitions of the theta fuq_ctions, we compute
,... ~-
om(u; -1/r) = O[b](u + !; -1/r) = (r/i) 112 exp{ni(u + ·!Yr}O[?](r(u + !); r) =
So,
(r/i) 112 ( -- i) exp{niu 2 r}O[l](ru; r).
all together, O(:](u; -1/r) = (r/i) 1 ' 2( - i)0" exp{niu 2 r}O[~](ur; r).
We also mus• compute O[;~](u; r the definitions, specifically
0£: ](u; r
+ 1) =
I
(3.54)
+ 1). These come almost directly from
exp{ni[(n + !)2 (r + 1) + 2(n + !)(u + s/2)]}
(3.55)
and O[~](u;
r + 1) = =
L cxp{ni[n (r +.I)+ 2n(u + e/2)]} 2
L exp{ni(n 2 (r) + 2n(u + (1 -
e)/2)]}
(3.56)
=0[ 1 ?,](u;r).
t The original rdcrcnce for this formula. togeth~:r wnh an interesting discussion of the origins
of the material of this' section, appears in Chapta 21, Section 5, of E. T. Whittaker aqd G. N. Waison•sdasstc Madem. A11aly8is, Cambridg<~: Cambridge University Press, 1927.
109
Theta Functi()ns
From these formulas we conclude, ,for.exampl~, tha,t.
0(8](0; -r) 8 + O[A](O; !) 8 +8(?](0; -r) 8 is a modular form of weight2 with value
(3.57)
(3.57)
2 aUnfinity; so by (3.51)
2. 15
= - 1t4-
I.
Cz(-r). .
Similarly,
-----
~o[?]~.+~[8] 4 )(8(A] 4
t
0(8] 4 )(8(?J~t,~JAt)
must be 189
.
········ (iC4(-r). 1t
3.14 A Fundamental Domain for
r2_
So certain polynomials in fourth powers of (even) theta-nulls gt nerate the ring of modular forms. · Now let us restrict ourselves to the subgroup
r2.::; Sp1(Z) of those matrices which are the identity modulo 2. If ~e ignore, c1r the moment, conditions at the boundary, then a functi 0 n.c(-r) is a n-.-:">dular form of weight k with respect to r 2 if · ·.
c(-r + 2) = c(-r), c(2-r ~
1)
Recall that we showed before that
[~ ~] Now
g ~ 1 f- ~ =
r= (2-r
+ l)2k~(-r).
r 2 /{.± identi~y Jl1atrix} has gener :tors . ·,-.;. . ~-.
and
.'
r.,:2J---· .... -·---·--·
f0
-·.
1.
~ lf~ -~] {~ -~ J.
so that it is easy to check that 0[~](0,
-r)4
is a modular form of weight 1 with respect to earlier in the chapter we showed that
A.(-r)
= -om4;om4
r2
whenever be= l. No\\ ·
Chapter Ill
110
and that the mapping
A.:
r 2 \X' --+ c
is in fact injective. lndeed we have a diagram
r 2 .Jf'
CIP> 1
..._.
-
{0, 1, oo}
·.··j :r._r-:~-1 (-~om~. o[?] ) .. J algebraic map_ 4
.tf
"·
(3.58)
CP, -· {oo}
.Now_the domain,Qn .n:Lgivcu !:>y,£ig~n:e 3.10is__ma_de up.ofsix.Jun~ damental domains foftheact1onofSp 1(Z) and so is easily-seeri to be afundamental domain for the action of r 2 • Now if e = 0, we have
o·
L: cxp:ni[(n + oj2) 2 r + 2(n + o/2)(e/2)]} = L: exp{ni(n r + Dnr + or/4 + en)}.
OU](O; r) =
1
So if x > 0, we obtain
limOU](O; ix)= (1-o), X·-+
-1
a:·
0
g.lue viet
Figure 3.10. !\ funuamcu1ai domain for Sj),(ii'). ..
r2
broken up into six fundamental domains for
111
Theta Functions
and so by (3.54) lim x 112 0[:](0; ix)
.x->O+
'
.
= f...,«> lim Ofn(O; ix) = (1 ...: e). . .
~;-'
Finally using (3 ..55) and (3.56) gives
2
·:.-,
" J~:ff~i~~~h·'
lim x 1 12 0[~](0; 1 + ix) = e.
From these formulas we conclude that the coordi~ate. function .
.·
. . - 0[~]4/0[~]4
: ..... ·'::·!.:
.
' )~;·;,;;~~·: '' ·. ·.;;,,
in (3.58) has the following;limit values:
ioo
0,
iO+
oo,
1 + iO+
1.
ioo
1,
id+
00,
1 + iO+
0.
Also the function
has the limit values
Thus
1- (-om4;om4) = (0[8]4;o[~J4); in other words, we have Riemann's theta relation: 0[~]4
+ 0[~]4 = 0[8)4.
( L59)
3.15 Jacobi's Identity j,' Finally all the theta-nulls are related by a very nice identity, c !led Jacobi's identity:
( ;.60)
Chapter Ill
112
To see this, l'alse both sides of equation (3.60) to the. eighth power. The right-hand side is an automorphic form of weight 6 for Sp 1 (Z) with a zero at infinity (a so-called cusp form). This follows immediately from equations (3.54 )--(3.56). Using equation (3.55) gives
oO[U~_; T -t-_'1/. OU
= e"i/4
aem(u; r) OU
u=a
I
u=a'
a nq us~I1:8.:J~~54).gives
. ___ '"';"-.~~~n~J~ --lirJ La= (_i)(7) ~ aeu~~~~; ~)La· 112
so that the left-hand side of (3.60), raised to the eighth power, is also a modular form of weight 6. But now computing directly gives
JV[ 1](
T)
1 u- - -- - - '-
011
I u
=a
= L n(2n + 1)( -l)exp{nir(n + 1)2 },
so putting r = ix and letting x~-+ + oo, we get that the left-hand side of (J.60) also has a zero at infinity. Since there is only one cusp form of weight 6 (up to multiplicative constant) by the formula on the orders of the zeros of a modular form given in (3.48)-(3.49), the equation (3.60) must be true up to a multiplicative constant. To evaluate the constant, compute the coefJkicnt of e"i' 14 on both sides of the equation.
CHAPTER FOUR ''-'''
•
<
· .4.1 Cuhumulugy uf a Cumplex Curve Again in this chapter we will assume a certain vague familiarity wit'1 the cohomology of sheaves and the theory of line bundles. This mate1 .al is readily accessible in the modern literature, and some of it can be gl' ;ssed once one is familiar with Cech cohomology of topological spaces. Ou, goal in this chapter is to examine in detail the Jacobian variety which is tssociated in an intrinsic way with each compact complex manifold of d · nension 1. It is the existence ofthis auxiliary variety that makes the the1 ry of compact manifolds of dimension 1 so much more beautiful and con plete than the theory of complex manifolds of higher dimensions. The h: '.herdimensional theory often still consists of a struggle to resurrect or r~ .Jlace in some special case or other the one-dimensional theory. To begin, let C be a compact complex manifold of dimension I C is then triangulable and thereby becomes a finite simplicial complex or a inite cell complex. It is not hard to show that as a cell complex C c; n be pr~sented in the form shown in Figure 4.1, that is, a bouquet of 2y onespheres a1, f3~o et 2 , ... , {3 9 with common point p to which there is attac iCd a · two-cell C. Thus
Ho(C; :l) = Z·{p},
H 1 (C; Z) =
L (Z·{1X1} + Z·{f31}), J
H 2 (C; Z),8:; Z·{C}. 113
114
Chapter IV
The integer g is called the genus of C. Also it is clear from Figure 4.1 that a1 • fJk
= Kronecker
bik,
a1 ak = 0,
fJ.I
pk =
0.
.Because of these formulas we call {a1 , f3k} a symplectic basis of H 1 ( C; Z).
,.~~::·~~• -~-~P3~.:~,:~~-~1 .wi~.I~A!l~\~~.~~U!l;~~~:51~_~~±-·~,~. c 0 ~ l. -
(G
(9*
wh~l'~(R.,.iS::tla.~b~ar'C~f~Jwk>morphic-functions.on
=..
0
(4.1)
c. (\\'~~c~ we ha:ye _al~~~~y
seeri' ilf--ell-a-pterlwo), .·. arid"'((;l*--is'fhe -.-sliearornow11ere:-zFf£T11olomorpfiiC: functions on C whose first cohomology group H 1 (C; (0*)
.
is exactly the group of holomorphic equivalence classes of holomorphic line bundles on'-'· We will build the Jacobian variety of C from a piece of the cohomology sequence associated with the sheaf sequence (4.1 ), namely, the part
0 ~ H 1 (C; Z) -+1-J 1(C; (1 )
~
H 1 (C; {!)*)
~
H 2 (C; Z) ~ 0.
(4.2)
--·--
·Figure 4.1. Culling· open a compact complex one-dimensional manifold in order to lay it out in the plane. · •
115.
The Jacobian Variely
Then we define the Picard variety of C: . o(C) = Hl(C; [g) P lC . . Hl(C; if) ~ kernei(H 1 (C; rg•)-+ H 2 (C;
if)).,_
(4.3)
Even to write the st:quence (4.3) we must assume many things abou; the cohomology of the sheaf (!) on· C, in particular that{ .
~ . . :{;·:f.'t
_,}f~(t:;;_(!)):::: 0
··'·\:,g;L':t{';~~~;;>;;;i,:,
whenever n > dimcC = 1. This fact, as well as many others which w..: will U!!e without.,p~ol:)f. c~111)~Jound (~it~'-e~o~UJ!!:cG1J,g9!P~-PJ. Tllese .act~-~ center arourid the study of the double complex otsheaves•' -
i
a
i
a
r
a
i
-
.910,2 ____. .911,2 _ _ .f112,2:___.
jO
I
io
·· ... ,<
.Ji10,1 ~ .911,1 ~·.Jil2,1. :'··.
io
io
c ____.
(4.4)
--
i 0-
J
a
a
(!)
where
di· 1 = sheaf of C<X> complex-valued, differential forms of type (i, j) c
1
c,
Ql
= sheaf of holomorphic j-forms on C.
(& = 0°.) The point is that the bigraded complex
a: .Jili.l, a+ o)
is a finite resolution of the constant sheaf C, and therefore its global sec-: ion can be used to compute the 'usual deRham cohomology groups
H*(C; C). Much deeper is the consequence of Hodge theory for Kahler mani: >Ids that the spectral sequence associated with the filtration
F(p) = L.Jili,j l
116
Chapter IV
of the complex (4.4) "degenerates at £ 1 ." ln other words,
H'\C; C)= H0 (C; (9), H 1 (C; C)~ H 1 (C; (!') (£> H 0 (C; 0 1 ). 4.2 Duality
(w,-t~)-1-:-'1=--·=--.. . . __. .,; ;-__,~:.
t
;, ... ,. .o .. : ......
fw . 6...n.
(4,5) .•
·c
is nondegenerate and skew-symmetric beqwse it is the complexification of the cup-product pairing on integral cohomology. If 1d•. (
~J'
(4.6)
dn'. •tkfJ,k=l, ... ,g
1
is the basis of H (C; l..) such that 'J.j
y=
r d~j.
··y
we call this basis the Poincare dual basis of the basis {ai, Now if {w 1, ... , w 8 } is a basis for H 0 (C; 0
1
),
f3k}
of H 1(C; Z).
then we have that
must be also closed, linearly independent (0, 1) forms. More precisely, they represent linearly independent elements of H 1 (C; C), and since they lie in
?-close~j~~-!)-form~
= H 1 (C; &) s;; Jil(C; C),
o-exact (0, I )-fonns
they mu:>t, for dimensional reasons, be a basis of this space. Thus the deRham group splits naturally: H 1 (C; C)= (subspa~.:e spanned by {w 1 ,
... ,
+(subspace spanned by {6.i 1,
= 8 1. o(C) + Ho· 1 (C). del'
w 8 })
... ,
w8 })
117
The Jacobian Variety
From this it is easy to see that the matrix
is of maximal rank so that for appropriate choice o( w 11 effect that
I
= (identity. matrix)
... ,
w 9 , w can
',,;:"~:·~:·';!!;fJ.i'?\ji\,
From now on we will assume that. the w1 are so ch9~~?Xi!~~J!,we ha\_
(4.7); where 0 is a complex-valued g x g matrix of maximal rank/ • The formulas ·
Jc
Wj 1\Wk
= 0
imply that
(4.S)
0= 10
(called the first Riemann relation), and the formulas·-
(; Jw1 /\w1 )>o imply that G = (imaginary part 0) =positive definite symmetric matrix
(called the second Riemann relation; we have already case ofit·in Chapter Two). · Consequently we have the formula
co~puted
(4.9)
a s1 ccial
-
I This shows that the column vectors of the matrix
[I o] are linearly independent over IR, since otherwise the matrix
could not possibly be of rank 2g.
Chapter IV
118
4.3 The Chern Class of a Holomorphic Line Bundle We are now iri a position to begin to understand the structure of the Picard variety which we defined at the beginning of this chapter. From the complex (4.4) we see that the mapping
H 1 (C; .l)--. H 1 (C; l9) is simply the restriction of the natural projection
H 1 (C; C ) - - - + H 1 (C; l9),
L aiwi + b/vi
L b/iJi.
(4.10)
o~=Pfo]e~t'i0ntlre111apptng
H 1 (C; IR)--. H 1 (C; CD) is an isomorphism of real vector spaces, since a differential form is real if and only if it is its own conjugate, that is, if and only if it has the form
--
L aiwi +iiiwi.
Thus we have a natural isomorphism
(4.11)
where the complex structure is induced on this real 2g-dimensional torus by means of the mapping (4.10). The next step in understanding (4.3) is to remind ourselves that if we consider the group
H 1 (C; l9*) as computed via Cech cohomology of sheaves, then it is simply the group of holomorpl;ic equivalence classes of complex holomorphic line bundles on C. Topologically these line bundles are classified by their Chern class, an element of H 2 (C; .l) ~ l. (See, for example, Steenrod [10].) We ought to check that the map (4.12) in the sequence (4.2) is indeed the Chern class mapping. Then we will be able to conclude that Pic 0 (C) is simply the moduli space of complex holomorphic line bundles on C which are tdpologically trivial. So let's analyze the definition of the mapping (4.12). We will use the fact from Gunning's Riemann surfaces book [3] that every line bundle on C is a product of bundles of the type
LP = holomorphic bundle associated to the divisor consisting of one point p E= C
119
The Jacobian Variety
or
L; 1 • So we will prove that, wit~ appropriate choice of sign in (4.1 ~ ), {Lp}H +l.
Let us use a covering
U,J't,J'l of C which near p looks like that shown in Figure -4.2. If z is ari an; lyf,ic coordinate on U with z(p) = 0, then
z .. give a global section
on U,
·or
:J•~.~~l•\L
2
1
1 on V and V ':':., Lp', that is, the pasting function q,~~~:4~~ to 1,ut U x C
and
lj x C
~'
together to ~ake LP is
tPuvJ:
un q
and tPvlv2 = 1. The first step in
c•
lj t-----+
1/z(q)
computi~g the image or
{tPuvp tPuv2• tPv 1v2} E H (C; l!J~) 1
(· .13)
2
in H (C; Z) is to select. a Cech one-cochain with coefficients in l!J w:lich maps to (4.13) under the mapping l!)
l!J*
\\\\\ Figure 4.2. The three sets of a Cech covering which computes the Chern class or LP
Cbapter IV
120
We use (4.14) where the branc'1es of log ¢ 111-, and log ¢vv 2 are chosen to coincide along
z real, negative. We next compute the Cech coboundary of (4.14): 1.
. -·-
i5(4.14)uv.vl = · --: (log ¢vv 1 -log ¢vv 2 ) 2'Ttl ,:
~ on{r;;~~~~:~dF:;Jponent of U n
V1 n V2 •
Clearly, then, this element represents the generator of the Cech cohom~ logy group
To compare Cech and deRham cohomology, we use the diagram of complexes: ~
ia 2
0---- C (C; C)----
ia 1
0---- C (C; C)----
ia 0
0---- C (C; C)----+
ia
jb 2
C 2 (C; ,q/ 0 )
____!!_ C (C; d
ia C 1 (C;
..W
.w
0
)
-~ C (C; d
)
~ C (C; d
1
____!!_
1
)
____!!_
)
~-: C (C; d
· (4.15)
ia 0
i 1
0
i
i
i
i
i
T
i
i
0
0
0
0
We can put a metric on the line bundle t~.,d E
bi finding
)
ia 0
jb
C 0 (C;
1
H 1 (C; (0*)
functions
r,:
u,
--t
IRl +
2
)
..,.L
Ill
The Jacobian Variety
such that (4.16)
.
Then
{eap} E H 1 (C; C9)H{log
ThlS last element
n~presents
~ap}
E
C 1 (C; C9)H
£5{4~i log eap} E C 2 (P,;}) £
C?(C ;_ .rt~
the same class iii H.2 (C:}) as does
-- 1 aIo - ~ -} E C 1 (C· d {2ni g afl · ·-•· '·····
~~;:; ;•. :}';~-'
1 )::;__
:
since their difference is a coboundary in the double complex in (4.15 But > by (4.16)
alog ~afl =a log ra -a log rp so that the same class is represented by
{2~i ao log ra} E r d
2
0
£ C (C; d
2
).
So, for example, in the case of LP we can use
1/jzj2 near the edge of U, ru =
{"smoothed-off" positive function near the center,
rvi = 1.
Thus
fu.aolog ru faualog ru fau(-d log z) = 2ni, =
=
and we see indeed that
is a deRham representative of the Chern class of Lp. From the preceding considerations it is also clear that if a line b<'!1dle has constant transition functions 4>aP·• then its Chern class is zero. Convc ·sely; the commutative sheaf diagram ~ ·
o ----7L--.... c--.... c•---. o II 0 ----- 7L.
In
In
(9
(9* - -....
0
122
Chapter IV
gives a commutative
~ohomology
diagram
H 1 (C; C) - - H 1 (C; C*)
l cr
----+
1
H 1 (C; 0)
H 2 (C; Z) II
H 1 (C; @*)
H 2 (C; Z)
and since the mapping a has been shown to be surj~tive, it is clear that a line bundle that has Chern class zero can be constructed using constant transition functions 1
4.4 Abel's Theorem for Curves Next we recall the Lefschetz duality theorem in algebraic topology that there is a nondegenerate pairing H 1 (M;C)®H 1 (M;C)
()', q,)
C
1-----.J q,
(4.17)
y
for any compact manifold M. This pairing, restricted to integral homology and cohomology is integral and unimodular. Define HJ,o(C)= (Ho·t(C))\
H 0 , 1 (C) = (H 1 • 0 (C))\
(4.18)
where the superscript _L means the annihilator of the subspace in question with respect to the pairing (4.17). Then we have a decomposition H 1 (C; C)= Ht,o(C)
+ Ho,t(C)
into complex subspaces, and as before the projection H 1 (C; IR)-+H 1 ,o(C)
(4.19)
is an isomorphism. Also we can identify ------
H 1. o(C) = HLO(C)*,
the dual complex vector space of HL 0 (C), via the pairing (4.17). We define
H 1 (C; IR) Alb( C)= -----H 1 (C; Z)
(4.20),
and give it a complex structure via the projection mapping (4.19). Now it will turn out that, as complex tori,
Alb(C) ~ Pic 0 (C).
(4.21)
123
The Jacobian Variety
(This "common" variety is called the Jacobian variety.}The isomo' phis~ (4.21) is a disguised form of t'he classical Abefs theorem'(which we ;aw in
Section 3.3 in the case g = 1). Before seeing why,,all tl}i~; is true w need ·')•''' some further constructions. First, if we pick a basepoint p0 e C, we can .m~P
:~- .--jtV~~~;;n,
""""-·""''·····, ....., ...... K:
(4.22)
This map makes sense because if we pick a path y'joining p0 top, ti en .
f' 11' ~~~RJ -1:*~--~.,""~- ;;~"
-
r
~
is well defined so that L is indeed an element of If 1 ( C; R) via the 1 airing (4.17). Now if we use a different path from Po to 'A,we get perhaps :1 new element of H 1 (C; IR), but it differs from the ol(f;one by an elen: :nt of H 1 (C; Z:), so. the mapping" is well defined! To check that it is co.nplex analytic, notice that if w e H 1 • 0 (C), then
I. is an analytic function of p. But also the projectiOJ:l of precisely the functional , :
Ho into H
1.
(C) is
·
:':·:.,.
p
w~-+Jw PO
in H 1 • 0 (C)*. Also it is immediate to check (as we did at the begim ing of Chapter Three for cubic curves) that we have isomorphisms
"•:
H1 (C; Z:) - - - H 1 (Alb(C); Z:~ Z:) H1 (C; Z).
K*: H 1 (Alb(C);
t4.23)
We have chosen bases for integral homology and cohomology on t We shall use the same symbols to denote the corresponding bases on AI. ·(C). Now we want to construct an isomorphism between Alb(C and Pic 0 (C). If we ignore their complex structures, this is an easy tasl For example, the intersection pairing on H 1 (C; Z:) gives the Poincare m;: )ping H 1 (C; Z:) ---+H 1 (C; Z:),
a
t-------t
(a'1-+a ·a').
4.24)
Chapter IV
124
(Recall (4.17) which makes H 1 the dual of H d Since {4.24) is an isomorphism, it clearly induces an isomorphism Alb(C) =
~_!(C~~ ~ Hl(C; ~)
H r(C; C) - Hl(C, Z)
= p· o{C) IC
{4.25)
•
The crucial point to check, of course, is whether this mapping {4.25) is complex analytic. J,o. check analyticity we rewrite the map {4.25). Let
---- ~-
,
,;;:;z:a~j "d111 •
(4.26)
which can be considered a deRham two-form on Alb{C) or a skew-
~~nlnetriG~aJ;:fg:rm,;~n.~#ct(§:c:~)~ Jn its::fo~rner r()le ..we ca,t11JSe itJ<> define the topological contraction 1riapping
· · ······ '·
1
· .·.· ·. · ·
H 1 {('; ~) =H 1 (Alb(C); IR) ~ H (Alb{C); ~) = H {C; ~), 1
(a, <1>).
a
The corresponding mapping at the level of simplicial complexes is the "cap product with <1>" mapping, but here we ring isomorphisms JJ*(Aib(C); ) ~ 1\*H 1 (Alb(C); ~ H*(Alb(C); ) ;:: A*H 1 (Alb{C); ~
(4.27)
where the product in H *(Aib(C); ) is the Pontryagin product, which can be defined in an abelian Lie group G as follows: 1( M 1 is a compact rrmanifold and
ai: Mi--+ G is a continuous map, then the product of the elements aAM1)
E
H,iG)
is the image of the distinguished generator of Hr,1(ll 1 M 1) in Hr,J(G) under the mapping
·--
nJaixJ)•
(xi)
So by (4.27) the map a.-... (a, <1>) can be understood as the contraction mapping which is always defined between the wedge algebra of a vector space and the wedge algebra of its dual. Then we check
(ai,
L d¢i AdiJJ) = = ·-· (a 1 , d17 1) d~i := d~j.
so that the contraction mappmg
I!:>
the same as the mapping (4.24).
125
The Jacobian Variety
Now let us rewrite the differential form in terms of the basis
of H 1 (M; C). By (4.7) and the matrix formula following the definiti< n of G in (4.9) · ·
[:<·]. = '7g
-~J-~ -di~::~,;:t[:;]. •': -~·c··cco''
·
R,'i': · (J)g
'
This equation can be .thought of either in H (C; ~) or in H 1 {Alb(( '); $o we obtain in H 2 {Alb(C); C) the following expression for~: 1
C).
(d~1• ... , d~8]/\ [dq,] d'lg
1 4
=
--[w1
=
--[w1
1 4
We say that is a two-form of type (1, 1) on Alb(C), The fact that <.:- 1 is positive definite and is an integral cycle of type (1, 1) makes Alb(( ) into a Kahler manifold. (See for example, Hirzebruch [4], p. 123.) · · Just as we have decomposed H 1 ( C; C)
and
H 1 ( C; C)
in subspaces H 1 , 0 , etc., according to type, we can also decompo · the vector space Hom(:(H 1 (C; C), H 1 (C; C) II
H 1 (C; C)® H 1 (C; C) H1,o ®H1,o
II
+ H1,o ®Ho, 1 + Ho, 1 ®
H1,0
+ Ho,1 ®Ho.1.
11.6
Chapter IV
Also we can decompose~the:coh0m_Q1Qgy,;;;;Qf.Alb{,k)-into t~.,.using_(A.2J)'"c This turns out to be the- same thing as the Hodgg !/gcomposition on the Kahler manifold Alb(C). Suppose we are given an element
------~-
Then
$ = 4) and so ~:-::-;;:;,-'-'
--
:_~
-:
"(/)'liiiiPfJ-cfttii["A:m~
-'
L:_:=~: ---
with Lci,.] a skew:Hennttian. n1~~_ri~_. _13~L!!~~c"_t:latural isomorphism on .. ---, .. ~ ..-'' All-)-(C-),--·---- ·---'-·-·-,--'-~ ·.:.: . .:::·:..::..·.:·::::::-.~-;.-:-;-::-;-:--·:.
Hl.00HO,l __-_._.. 8 w 1 0 wk
f-----+
1,1
w1"wk,
says that if' determines a unittuc element
, 0 • 1 = cj> 1· 0 and claim that
(cp•· o + q,o. 1)(a ® P) =(¢,a A{J)
(4.28)
for all a, PE H 1 (Alb(C); C). To check that ¢ 1 • 0 +~has the desired property (4.28), we compute
(¢1. 0
'+ ¢
0
• I
=
)(a® P) =
since [c.id is skew-Hermitian. Thus, since¢ is actually an element of
E
H2(Alb(C); l~ (¢ 1 • 0
+ ¢ 0• 1)
that is, corresponds to a homomorphism
(4.29) Again using map sends
(4.2~ ),
it is immediate that if ¢ = «1> as in (4.26), then the last
and so is the mapping (4.24). It is also easy to check that (4.29) is an isomorphism if and only if the form ¢, considered a skew-symmetric bilinear form on.H 1 (Aib(C); il~ is unimodular.
The Jacobian Variety
127 '··· ,. . . ....
. .
·~·····
· · · ·' ·· Fmally we can'' cfleCk~e-~analytfut{ of~ny';r'map~irig:-obtaii•; :d -·by' tensoring with~ a map (4.29) coming{r9m ~Jor~.-~5>1!~(!_,1). L1det:9 the composition · ··· ··· ···· ····· ·· · · · ·· · ··· ··· ···· · ··
H 1 , o-+ H 1 (C; ~)-+ H 1 (C;~)'~·•1Io;t is given _by the element f/> 1 • 0
E
Homc(H 1 , 0 , H 0 : 1
,Jb,~·,is~:m:wrphis~4~given qy_J!le.~_(t;::c_r};:fofiiT~~;-The,:s-o.rnm~m Q.J:?.; ~"t ::Aa ""'"(~'Jo)ls
called the Jacobianvariet.V ore ana-·isdenote
u
••.
•
•
·-···
J(C).
_4.5
Th~
Classical Version of Abel's Theorem
To see how this relates to the classical Abel's theorem, notice tltt we have a diagram ~Alb( C)
c
1~
~Pic 0 (C)
where J-l(P) is the equivalence class of the line bundle given by the d p- p0 and K is as in (4.22). The usual form· of Abel's theorem is simply the assertion that the diagram (4.31) is ·commutative. We proceed to prove this. . L~t z bt: a holomorphic coordinatt;Junction qn a disk U0 in C;
qto q2
E
-U1)
visor then now .!t
I
U0 ,
and let
{Uo}
U
{Ua}a=l, ... ,m
be a finite cover of C such that if ex ::f= 0 then U a avoids a neighborhoo, of a simple path connecting q 1 and q 2 in U 0 (Figure 4.3). Now using Secti( 11 4.3 we see that the. line bundle L 91 ® L;;/ is represented in H 1 (C; CQ) b the cocycle {,p} where
4>oa = r/>ap
1 . {log[z- z(q 2 )] -log[z- .z(qt)]}, · 2n1
-
=0
if
IX
# 0,
P#
0.
L32)
Chapter IV
128
In fact, by the Riemann-Roch theorem [see (4.48)], there exists a meromorphic differential
(4.33) 011_ C---such that the only poles of~' are at q1 and q 2 and those poles are -simple with residues 1 and - l respectively. So we might as well take z so that
I . .P I exp 2m
1
I
'J>()
z(p)- z(qt)
1/1 I = ~( ) _ (---). "p z q2
Next we can, and will, assume that elements of our chosen basis d~j' dY/k
of H 1 (C; &:} are such that they are identically zero in a neighborhood of 0 0 c, and that the homology cycles ~Xj, Ih also avoid 0 0 • Define
'on
Then off the path in Figure 4.3,
(if! -
1/J
1
)
=
d.a
for some C«'.function a. Next let ,p·
f:;(P)
=
I
'PO
be dell ned .in a neighborhood of 0 0
•
Wj
119
The Jacobian Variety
Ji
Now we have vectorequations
~ ~]I,~1) (c;(~~q,); =Tt;,;J:t/N ~ ~:) 0
-
- -:(LPJ,•) ~ (Jc'"' "w·);;;(f,' ""'~l
(1:~4 J
e = -(0, !)-component of t/1 1 • On the'\~th~~ hand, ereprese: tstb_e class (~.32)1n H 1 (q; l'9);-since;ifWeaefi~~ 7 : ~;;;;-~'"'"-;;,,_~;;,; - ·· · -·---· where
___Q" O'o
.=:.~lu.·"· . rx ~~4~ft-g·r~;;::;~:t:n,:b:~t;,;rt7-~=:···~::~ = 0,
then and
£5{ 0' 2}
= - {4>2p}
[see (4.32)]. But now the Poincare dual mapping· (4.24) sends J~~ t > that differential form p such that ·· ql
•
r = ·cI p "'1· '1
'ql
~hus by the formula (4.34), the elements:~ must~o, via (4.24), to
(e +e). Thus the mapping
ct>: Alb( C)--. Pic0(C)
W
takes to the class of the bundle L 91 ®L~l,.and the proof of \bel's · ,;c··. theorem is complete. By this theorem, if in Alb(C), then we must have a rational function f on C whose
(f)= L Pr-
d~visor
L q,.
To produce such a function explicitly let
t/1 =I
have poles only at {p,} u {q,}, residues
"'·
+ 1 at p., -1 at q,, etc., as in (4.33).
Chapter IV
130
Then (4.35) may not be a well-defined function because
· r 'fl
r '"·
(4.36)
~::;~~~~~-~;g~:5zi~~~=-~:~~~%~~r:~~,=~~~g~;.~~{:~:;· ... :::.:.:~:~;~=~;.J!-/~;,:~.~:~,~::.=.~:~i{~~s~~:i,~;!.:~,.f~~~~:~., may nofbe integers. However we can adjust .t/1 by a sum such that for all j,
r "'= 0.
'Pi
So, as before, we put
and conclude that there are integers m 1 ,
m9 , n" .. . , n9 s-uch that
... ,
ml
[I
n]
111g
1
= JrwiAtjl 1
·c
nl llg
. = {(4.7)
r "'1·
'aj
Now replace tjJ by
t/1
2:,
niwi.
This new mcromorphic tj1 has integral periods sc t 'mt the function (4.35) is actually well defined.
The Jacobian Variety
131
4.6 · ]'h~ J~~9bi lnYersi~n;;!~~~~~~·;;;":i.:;·· ... Ntiw we want
to do some more wo.rk with' the:'tn~ppi~g IC:
Actually we should write
Kpo
c'-d(C).
since it depends on a choice of the basL ,JOint
p0 • Anyway J(C) is ant:Abelian complex Lie group, so 'Ye,haye inL uced ·,;;i,i~
· '!":;;;, ..,.:''
:,:':"':';\,,
analytic mappings
..
·.~-,.,~··~~~·-~--.-.··-.·.-i·Z.::~rJ~;~~k~~ffG:iiS::,··:.;~X.~c?tt(~'·;· i''.:----V--
r times
• •.. ·.
~.
-~---~
:."
;: .c.'·~; __ -
,. .-..,.-:-~:,:-····:~-!-,:-:<•!•
Now this mapping--i~,i~dependent ofthe o~dering·of domain, and so we have a•commutative diagram
th~
t~·-,--:~...c~--
r-tuple
........... ~ 1.
the
(4 17)
where c is the rfold symmetric product of c. c
(where x is Pontryagin product) since { '~<(()
f
d~i Ad~k = d~i Ad~k = 0, ·
·c
f
d~i Ad'1k =Kronecker J1kt
r
d,J
• I<( C)
• J<(C)
"d,k '= o,
Thus the homology map (K 9 )*:
H2 9(C< 9>; Z)-.H2e(J(C); l)'
132
Chapter IV
must beaR isomorphism smce
·, So by topology,
must be surJective and its sheet number must be one. By ;JJJ~~c~,~OJ!llY:~PIQP~rtie,~,"qf_c:ompl(!~ ~!lalytic morphisms, this can happen_ :()n:ly ifiC; ~~ gerietically'Trijectfve;911iiHs, if'there are open dense sets of c(g> and J(C) which are isomorphic under ,; 9 • This fact is usually called the -Jacobi inversion theorem. Aften:lil;-foranypoint L s;: J(C), Abel's theorem tells us thai K9
K- 1 9
(L)
is a full linear system of divisors of degree g on C (that is, a maximal set of divisors each of which gives the same line bundle L). So "; 1 (L) is natural isomorphic to the projective space
IP'(H 0 (C;
(11 (L))~
where (1i(L) is the sheaf of sections of L. Thus the Jacobi inversion theorem says that "most" line bundles of degree g have only a one-dimensional space of sections. By the Riemann--Roch theorem (4.48), this is simply the assertion that, in general, g points on a curve are not contained in the zero set of a nontrivial holomorphic difft:rential.
4. 7 Back to Theta Functions
t:k
Now let us bring in the concept of theta function in genus g. If b1 , E {0, l}, j, k = I, ... , g, define
.~·"'
---
and o[~](" ; n)
=-I ,,,t=' .
cw{n:ie(m
+ 6/2)n(m
-1-
<'>/2)
?.ll
where
11 E CH
T'.c fact (4.9) that lm f!> 0
+ 2'(m + ~/2)(u + ~:/2)]},
(4.38)
The Jacobian Variety
implies uniform and absolute convergence of this series in a wa) completely analogous to the one-dimensional case which we saw ip C 1apter Three. As before, direct computation gives
O[~](u t Ei; n) = exp{-nioAO[~](u; n), O[~](u
+ nj; n) = exp{ -ni(2uj + + Wjj)}O[~](u; n),
(4.39)
Cj
where Ei is the jth column of the identity matrixL~~:':ld !)1is. the jth c 1lumn Agam we have the task of tracking down the.. i:ero set of the fUJ ..:tions (4.39). We will do this in a very classical way which goes all the wa.' back to Riemann. In fact, the material contained in the rest of .this c tapter appears in the last article publisl,ted by Riemann:'himsel~.t Before :)eginning, we compute
of n~
em( -u; n) = :L exp{ni['(m + o/2)n(m + o/2) + 2'(m + b/2)( -u + t:/2)]} =
L exp{ni['(- m- b/2)0(- m- b/2) + 2'( -m- b/2)(u - e/2)]}
=
L exp{ni['( -m- b/2)0( -m- b/2) + 21( -m- b/2)(u + e/2)- 21( -m- b/2)e]}
= exp{ni'o · e}o[:]
Thus the functions (4.38) are even or odd according to whether 1c'• e is even or odd. For reasons that will become apparent later, we want o say this in a fancy way. Define
m= L O]fl.j +
e)PJ E Hl(C; IFz),
where IF2. is the field with two elements and rxi, logy basis that we have been· using. Then
p1 is the symplectic i.omo-
.2([:1) = 'b · e modulo 2
·T
(4.40)
is a quadratic form on the IF2 -vector space H 1 (C; IF2 ) whose asso ;iated bilinear form is the intersection pairing. Now the mapping C ~J(C)
lifts to an immersion of the universal covering spaces,
C ~ H 1 • 0 (C)*. t The English translation of the title of Riemann's article is "On the Vanishing o1 fheta· Functions"; the article appeared in Borchardt's Journal fur reine und angewandte MCI ·matik,
vol. 65, in 1865, the year before Riemann died.
Chapter IV
134
In Figure
4.f we
have simply given a picture of a fundamental domain
cr;;.c with respect to the action of the covering group tt 1 (C, p0 ). N~w using the dual basis
we get an explicit isomorphism
Ht·o(C)*;;::
1(:11,
where the projection of H 1 (C; 1') into J-1 1 • 0 (C)* corresponds to the lattice generated by the columns
of the matrix (I, n]. Thus
(4.41) Therefore, by {4.39), the zero se[ of O[~](u; n) gives a well-defined hypersurface in J(C). We will later show that up ~p translation (zero set ofiJ[~](u; !1))= "(s-dC( 9 -n). Before proceeding, we should mention that a more modern version of the theory of theta·functions treats them as sections of positive line bundles on abelian varieties. The formulas (4.39) then appear as "patching data" or arise naturally in the cohomology of certain groups associated with abelian varieties. The reader interested in pursuing this direction is referred tct Mumlerd [6].
4.8 The Hasic Computation Now we have shown in genus 1, where C = J(C), that Sa;w = -·rand so (zero set of O(~](u; r))
Jp,w =
+In= (zero set of O[~:~::J(u; r))
1 and
(4.42)
where [;1] is considered a point of J(C) a~.:cording to the r~le
(4.43)
The Jacobian Variety
135
We wish to show that (4.43) is true in general. This fact, together witi the assertion at the end of Section 4.7, will.be proved-with the help o: the · · following essential computation. Fix a constant vector e e C 9• Then it makes sense to re~ :rict 8[~](u + e; n) to C and look at its zer~ set there. A simple computa ion, completely analogous to the one we did in the case g i:;::. 1. iq Secti01 3.3 gives that this restriction has g zeros (unless O(:](u + e;.O)Ic is,identic Jlly 0). A deeper computation of the same sort is the following: . Notice that in Figure 4.1 the vector-valued function (the. coordi 1ate function on U) satisfies the relations ·
u
(u along a1(u along
Pi
)
= (u along a1)+,Eh ,· . '~ 7"7~~~;"(:Li4f
)
= (u along 1)
1 1
p + n1 •
···
So if 8 denotes any of the functions (4.38), possibly translated by a cons ant vector, restricted to C, we compute for e small;. · u[zeros of O(u(
1
) +e)]- u[zeros of O(u( ))] .
=-.I I u · [d log O(u + e) 21t1 .
d log O(u)]
!Jj
1
.
1
.
+-.I I u · [d log O(u +e)- d log O(u)] 21t1 •Pi
--.I I (u + E1)[d log O(u + e + E1) 21t1 .
d log O(u
+ E1)]
d log O(u
+ nJ)]
1
.
21t1
PJ
1
,.
- -. I J (u + OJ[d log O(u + e + nj) =
(4.39)
I 21tl
- - .
1
EJd log O(u
''!Jj
+ e + E1) - d log O(u + E1)]
.
--.I I nJd log O(u + e + nj)d log O(u + nJ)] 2nl ·~ · :3
( )4 9
-
2 ~i L E l[log O(u + e + E 2~i I 1
(4:3'9)- 2rti
=
(4.39)
1)
1
-e.
I
nJ([log O(u
Ei log
-log O(u
+ £1)]~;~~~:-n,l
+ e + QJ) -log O(u + nJ)]~:::~~:+Ejl
(O(u
(4 ~5)
136
Chapter IV
If e is not small, there is some ambiguity in the choice of the branch of the logarithm in the fourth step in (4.45), but the answer is the same modulo an element of the form
What this means is that if
em£ J(C) represents·the zero set.of O[:,](u; 0), then __
_.,_,.~~·'C~c;;;;..,;c......." _,..,;;;;'':C";~JtJ,,,,:- e
reptesefltslhe zero . sei:ofO[~](ii ~e; 0). Identifying can write
c with K(C) £
J(C), we
(sum of g points of (E>[~] +e) C) - (sum of g points of (e[m · C)= e.
(4.46)
Of course, this formula makes no sense if
C c; (0[7.) +e).
(4.47)
However, the functions OU](u; 0) are given by Fourier series and so cannot vanish identically on C 11 since not all the Fourier coefficients are 0. Thus (4.47) only holds f.o.r-those e lying in some proper analytic subvariety of J(C). In a bit we shall sec the significance of this subvariety. First notice the significance of the formula (4.46}. It says that the map
J(C) c<e>, e t-----+((0[~] +e)· C) is.(up to translation by a constant), the.inverse of the map Kg.
C 111 ) - t J(C).
This is, ofcourse, another way to see the Jacobi inversion theorem.
4.9 Riemann's Theorem Before proceeding, w'~ should at Jeust state the Riemann-Roch Lheorem for curws ~xplicitly. A proof is found in Gunning's book on Riemann surl~tces (.1], p. Ill. The theorem states that /(D)--· i(D)
= deg D +
1 - g,
(4.48)
l37
The Jacobian Variety
where D =
L mkp4, p4 e C~and deg D = L mk anct
I( D)= dimension of vector space of meromorphic functions with at most a pole of order mk at Pk, · '· \!
:~;
and
i(D) =dimension ofmeromorphic differentill,lS \Vitll at least a.zero o order mk at Pk ,~='.t'':"· ·· ·· =dim H 0 (C; (!J(K ®
···.. ··
<8hL;k"'t))
= dim H 1 (C; (!)(@ Lmt)).
Serre duality
Pt
Now we have seen that for
P1
+ ... + P11
in an open dense subset of c,
l(Pt
+ ·· · + P11 ) =
1.
This is just another way of stating ttfe Jacobi inversjon theorem. Let us fix
B(u) = e[g](u; Q). Referring to (4.46), we then define Riemann's constan{Kp0 by the formu
L {u(p): p e [zeros of O(u(
) - (e
+Kp
0
))]},;,.
e
(4.· .9)
in J(C). KPo clearly depends on the choice ofbasepoint p 0 used in embedding IC
1e/
C - - - + ' J(C),,
p
Now pick p 1
,1
1-----:---+ ( (
).
+ ·· · + p9 such that L(p 1 + ·· · + p11 ) = 1 and such that if
u(pi) + ·· · + u(p9 ) = e, then
O(u( ) - (e + Kp 0 ))
(4.. ))
does not vanish identically on C. By formula (4.49) and the fact that .o other divisor
138
Chapter IV
has the property that u(qd'+ · .. + u(q9) == e, the function (4.50) vanishes -exadly at p 1 , ... ,.p11 • We concl4de that
0(-u(p 2 ) - '"-u(p11 )
KP 0 )= 0.
-·
Since p2 + ··· + p11 can be made to vary over an open dense subset of •9~11 - 11 ~!}9:si_~~t!.~_i§~-~'l.t!Y~n_fu!Wt\Qt!, __w.<::J}.f!Y~ - --------. . . '
-.. -
.
. ·- ·· . ...::::;......;_..;.....,.:.:.~___;.:__,~.-: ;::·
(K
+ Kp,,):;; e(g).
we··coUld' have rnade the sMne argumeJ'!t (with a diiTe~e~'t' value of K PO) for any of the ; J::'Qilver!)~ly.J'lo:W, suppose th_aF0(4! +~Po)= 0. The set of e such that o(u( ) - e - KPO) = 0 has codimension :2! 2 in J( C). Otherwise 9(u( }. e - KP 0 ) has zero set
em.
PI• ... , Pg.
with u(p,) + .... + u(p11 ) = e. But now if /(p 1 + + p11 ) = 1, then the Pk are uniquely determined, so one of them must be the ·basepoint p0 since ll(!i{(ior- e- KPo) = O(e + K 1, 0 ) = 0. Jf l(p 1 + + p11 ) > 1, we can always find a divisor Po+ q2
such that u(p 0 )
+ u(q 2 ) +
+
+ qg
+ u(q 9 ) =e. So in any case e E "'<~--IJ(('
Thus we have obtained Riemann's theorem: (.('lg-· I l) ( 1\.(g--1)-
0(0] + K PO ) -- Cl 0'
(4.51)
4.10 Linear Systems of Degree g
+ p11 ) >
Next notice that if /(p 1 + there exist q 2 , ••• , tJ 11 such Lhal u(q)
l, then for any pregiven q e C, \
+ u(q 11 ) = u(p 1 ) + ·· · + u(p 11 ) =_e.
+ u(q 2 ) +
Thus by (4.51)
) - e -- K po)
O(u( must vanish at q. So on C O(u(
)
e -
KPo) = 0 ..
139
The Jacobian Variety
On the other hand, if
)-e-KPo)=O,
B(u(
we let r = the maximal integer such that
O(u( ) - (u(pt) + ··· + u(p,))- e ~ ,1(~ 0 ) .
:;:..:~~~!;;.:;.:
=0
·. · :,.·_ :_ ..·;
I
·. ..
for all values of Pt> .. , p,.ThenJor a generic choice ofph ... , Pr+ 1 the
;;;c~;t.;';
function
·'·;,.,,::.
O(u( )._:;; (u(pt)- : .. -'u(Pr+t)) -{~·Kp0 ) has zeros
So by (4.49)
u(pt) + ... + u(p,:q)
f;~(q 1 )
+ ··· + u(q9 -r-t) = u{pt)
+ · ··+ t) + e . u(p,+ ' ·.. :
so that e e "(g-r-l)(c(g-r-11). Since r;:::. 0 we have shown that
l(p 1
+ .. · + p9 ) = 1 if and only if (0[8] +I u(pk) + Kp0 )
";?. K(C),
in which case the intersection is precisely
Pt
+ ...' + P9 •
Intuitively it is then clear that we can obtain the entire linear serie:- for p 1 + · ·· + p9 in any case by taking lim {K(C) n [0[8] e-+0
+I u(pk) + e + Kp0]}
in all possible ways.
4.11 Riemann's Constant Next let Pic'( C) = space of equivalence classes of line bundles of .degree r (Chern class r) on C.
)40
Chapter IV
The correspondence Pic 0 (C) - - Pic'(C)
{L ® L~0 }
{L} I
(4.53)
is bijective and so makes Picr(C) a complex torus. Now in Pic(g-l>(C) we have--illt-l'insieally-given two subvarieties; (i). 0 ={bE Pic :. ''..
.... :·:·; ··:::.···
= tmage in
~::~,'
We call
···-~---~----:->·
..
>:····:~---·
.
.
ofC 111 - 1 > under"Jhe
K( 8 ~n:
c<11 -
n
0}_ ··-· map
Pic(g-l)(C),
+ Pg- l} ·~--~-£~1@ ''' @·L";~~~-;-~~=='"'"·-;
r = {L E Pic18 ··J)(C): L ® L =. f , r
.
Pic19 -- 11
•'--,--,<;;-w,:,-.m=·--=· (pl·+.;~
(ii)
·-·
the cotangent bundle of C}.
the set of theta characteristics of C. Via the mapping (4.53) /( _ 8
l (C(g-l)) HE),
Also, by the Riemann-Roch theorem
+P8 --!)=l(K-(p!+"'+P9 -d)
/(p1+
whent:ver K is a canonical divisor, that is, a divisor which gives the line bundle f . Thus the mapping u(pl
+ ·· · + p8 ·-.) 1-+ u(K) -
u(pl
+ ·· · + P11 - J)
. . II', t hat ts, . must tak e K(g-l)(c.(g- 1 >) mto ttse (c(ll
K(g _ 1)
t
>) --
ll
(K)
-
(C(g-1)) •
K(g- 1)
B_ut 0[8](u; il) is an e~en function, so K(g--l)(c
+ K,,o
= -Kpo- K(g-l)(c
From these two equations we conclude that u(K)
= - 2Kp0 ,
or in other words (- Kp,J corresponds in Pic<11 -l)(C) to a line bundle whose square is the cotangent or canonical bundle .ft'·, that is, to an element of r. Next kt
141
The Jacobian Variety
One computes directly from (4.39) th~t
f(u + Ei) = exp{nibJlf(u), f(u + 0 1) = exp{ni£5 1 wii}f(u)~ .. Thus the function
g(u) = exp{ -ni~u1 b1},·f(u)' is a well-defined m(;!ro01orphic functic;m QJLJ(c):_Now look at
···------~--
u(~,,t,~Lt:~:::
;;;;;k(---'-":"j~ ?4).
for generic choice of constant e. Either this is a constant function or it 1as divisor
(Pt
+
.f.p9 ) - (qt
+ ··' + q9 ),
where l(q 1 + ··· + q11 ) = 1. Thus the only possibility is that the funct on (4.54) is constant for each .fixed e and therefore g(u) is constant. <1ur conclusion is therefore that
er&J =
e[~]- 1·
e b 2 - n· 2
e
£5
(4 .. ·5)
=em+l·2+n·2. To put things together we make a diagram 4 53 J(C) J(C) ( . ) Pic<9 -1l(C)
u
f---,----+
u - K PO'
e(8] - - - - - - - - - e
(4.. 6}
~
(?)
To see how to fill in the (?), recall that in (4.43) we let {[~}
be identified with the set of points of order 2 in Alb( C). But since
2( -Kp0 +
fm = -2Kp
0
= u(K)
and (-KP 0 ) corresponds to an element of~ in Pic
fm.
{[~]} +-+~.
/.
Chapter IV
142
Also
Wt!
should notice that if..
~[:J(u; n) 0[8](u; n)'
/'( ) =
. u
,:,,
then f(u + Ei) = exp{ni<5i}f(u), f(u + ni) = exp{n:iei}f(u). This says, for examplt;!,Jhat ~he line bundle 011 C.given byJhedivisor. ,, :,
0
0
0
0°
',<-:" ... _. '•,
.. , ___ ••
-OM
0
•
--
-·.-·---
0
o
0
0
-~·-=--••
,c,:
•--:-.,.•••.-:-:---:•
(8(3] + !E.;) - (0(8]) is trivial ~ver the set obtained by removing the simple closed path rxi from C and that the bundle
becomes trivial il the cycle f31 is n:moved. So the line bundle of degree 2 in Pic 0 (C) corresponding tom via the Poincare duality map is simply the one which becomes trivial when a smooth (mod 2) representative of the cycle
L birxi + eif3i is removed.
4.12 Riemann's Now the zero set tra·nslate of
eUJ
Singulariti~
Theorem
of O[~](u; il) is irreducible since it is given by a
"r
t
(c
It also has multiplicity I since we have seen that in general
--
e[:.](u + e; n)
t~
has g distinct simple zeros. lt turns out that the multiplicity of a point on the theta divisor has a very nice geometric interpretation. This is given in the Riemann sjngularities theorem, which states that if L E Pic<11 -1)(C), then multiplicity 1. 0 =dim H 0 (C; (!)(L)).
(4.57)
By then by (4.55) and (4.56): mult 1. 0 = multk O(~](u; il), where
(4.58)
143
The Jacobian Variety
A corollary of (4.55), (4.58), and' the fact that em(u; Q) is ey~ll (odd) il and · . ::~ only if 115 e is even (odd )is that if <'"
L e·I: (~he set of theta characteristics), ·I
then
.. , ... -
,.,.1:(~)-:;:;~irn Hll(C'·(!)(.l,)):(~~~:j•f);;t5':"';~·-'>if::L~=•~--'·- -~
is invariant as the curve. Cis continuously deforille
1ooic a:f.ihe foliowlng:''Suppose7fF'~'"'''":'' s.
Then given qh·· .. , q8 , there is a complementary set lls+l>. '/, q,_ 1 such that g-1
g-1
k=1
k=1
L: u(pt) = L: u(qk) = e.
Then for any p1, . .. , p9_ 1 and q 1 ,
... ,
q9_ 1 satisfying this last equatio .. ,
e(tu(pt)- ttu(qt)-e-KPo)
=: so that
o(- kt u(qk) -,-.,t:t~f~(Pt) 'KPO) r
•
=
0.
. :
.,.:..~)4·:;_~:_,·. :·:
(;.59) Conversely, if (4.59).holds for s but not s + 1, then for generic;: choice )f the function
has zeros
t An algebraic proof of this fact has been given in recent years by David Mumford in " heta Characteristics of an Algebraic Curve," Annales scientifiques de rEcol~t Normale Supe1 :ure,
vol. 4, 1971, pp. 181-192.
·
144
Chapter IV
So by (4.49) s+
g
I
..
. ....
L u(q ;) = e + L u(qi) - L u(pi),
j= I
j= I
j= 1
-
that)!i.,--
...
_--'-----"'"'""~-~------~--·=·=-'-'-''---"'= -------~,.,.;cc-~~--.qcc--.-·';:
·
-
L u(pJ + L
j=s+ 2
j= I
u(q1) = e.
Since the p1 were chosen -arbitrarily, we conclude that
...:_, ___________ . _----'-"~--
J(Pt -~-1-
s_. . .--______--+ l:'s_t lJ•-i:,2--±_:·_· Tqll)?: - - - - - --
=-l'rtus~~~=--=~"'=--~=,-=
l(pl
~~------ ·-"·------=---~-,--,.,~------
+ ··· + P11 -d > s if and only if O(~<_,(c<•>)- K.(c(•>)-
Now if z is a local coordinate at p
E
K Po) = 0, then
. o(u(q)- u(p)- e- KPo) 0 = hm ------· ·· ---- = q .... p z(q) z(p) Repeating the argument at each p
E
"""':"""".-.·-:--.---~~
L u(pk)- KPa} = 0.
(4.60)
C and O(K(C)- K(C)- e-
ofJ L oui --(-e- Kp
0
ouj (p). oz
)-
C we conclude that
t)(}
"' (-e-K Po )w1 =0 4-ou) ) so that
80
-----
a~'i- (-e- Kp 0 ) = 0,
j
= 1, ... ,g.
-)
The argument is then repeated to obtain one direction of the Riemann singularities theorem, namely, if
O(K.(C(•I)-
K 5 (C 151 ) -
e- Kp0 ) = 0,
then all partial derivatives of f) through order s vanish at - e - K Po. The j>roof of the other direction is somewhat more difficult, but the general idea is clear.t Now by the Riemann singularities theorem we ean characterize, for example, the set of singular points of e. which we denote
.t
See J.. L.:willes, "Riemann Surfaces and . I . . the Theta Function," Acta Math. vol. Ill, 1964, .PP· 51-55.
The Jacobian Variety
145
as the set Qf line bundles of degree (g- 1) with at. least. two lin :arly independentsections; To getan intuitiv~Iciea aboqJthe dimension 0 this. set, we can argue very roughly as follows. The tangen~ space to the m. ·dulf space of curves of genus g a~ a point C is given by
H 1 (C; (O(ff)), where .r·is.the cqmplex tangent bundle of C.t By Serre duality, whk 1 we saw at the end oU Chapte~ ;I:wo, the cotangent sp~ce. ,.····-:·. istherefore giver by '
'
Ho(c; ;r<2>), which by the Riemarut~Roclttpeor.eiil_has qimensiop. ···''··· ·'
---·····-·.-•-,-·--·-..-..
•~c':
..c:~,;~:i/;2'(2~ :_ 2)"+ 1:._ g== 3;--;_ ;:f'':;:;; ,);c'''
(Genus 0 and 1 are exceptions to thi,s because curves in these genera iave automomorphism groups of dimension > 0.) On the other hand, in 11ow many ways can we make a (g- 1)-sheeted covering of CP 1 which s of genus g? Assuming that all the branch points are simple, we compute the number N of branch points via the Euler characteristic formula to ge, (g- 1) · 2- N = 2- 2g, N=4g -4.
Now assume that three ·of these branch points are 0; 1, oo on C 1Jl1 1 ; the 1 we are free to move (4g- 7) of them. Since there are only an "oo 3 ' - " of distinct curves of genus g, we e}(pect an 00 t~g-7J-<3,- J> = 00 t9- 4) of coverings to correspond to a fixed generic curve C of genus g. In t ther words, for a genus C, we expect dim E> 59 = (g - 4),
( ;.61)
which indeed turns out to be the case if C is not hyperelliptic.t
1' See the introduction to K. Kodaira and D. Spencer, "On Deformations of Complex A .alytic Structures," Annals of Mathematics, vol. 67, 1958, p. 328ff. ~For a proof of this fact, see A. Andreotti· and A. Mayer, "On Period Rellltiofi$ for A! ·!ian .. Integrals on Algebraic Curves," Ann. Scuola Norm. Sup., Pisa, vol. 21, 1967, p. 209.
S.l. ]'opology of Plane Quartics In this .sect_ion we .will_.briefty. examine some of the more entertar 1ing properties of curves
5.1) defined by equations
F(Xe,
t
1,
X2)
= O.
where Fis a homogeneous polynomial of degree 4. We will assume tiLt C is nonsingular, that i_s, ,. the partial derivatives
oF
j =0, 1, 2,.
have no common zeros in CIP 2 • I. Any such C is a Riemann surface of genus 3. To see ~his, form a liu 1ily
( i.2) of equations of degree 4 by taking G to be a generically chosen polynor 1ial of degree 3. · Corresponding to (5.2) we have a family of curves
c,,
(U)
where; ift = (t 0 , tt), then C, is defined by (5.2). Now a small neighborhd)d of C, in CIP 2 can be made into a disk bundle over C, as long as ( is nonsingular. Intersecting normal disks with nearby C,, one conch les easily that the C, in (5.3) are diffeomorphic, with the exception of tl >se finite number of values oft for which C, is singular.
Chapter V
148
One of the singular valuesofTis, cifcourse, t = (0, 1}.
As t approaches (0, l),_we can visualize the "moving picture" of C, by drawing an.anaJogou~_moYing_picnu:e in IRIP>2 , the real projective plane (Figure 5:1); -----.._____ --}?he.:point ofFigure .5.1 is that a_ nearby C, is topologically the same as ::c1 ;;:~-,-~ex£ei)FihaFsmairne~-gfi-6oflloaas of' the crossing points Pi ii1C• which look like solution sets to -
=2:;~~;;:~I9~·;,;;:,~-~=--:. :;;~-;~-~::_:~~--;;;z,:::~~~.~--~:,i--.";. ,. ;;,::~-~;:-~~·~·-" have to be replaced by sets which look like solution sets to
x 'y =e. for some constant 1.: =I= 0. So, in the complex case, we can reconstruct a topol_9.gical model for the nonsingular C, by starting with C(o. 0 with its .-1-hree singular points p 1, p2 , p 3 , cutting out a neighborhood of each in C
G= 0
··------~.'·..;;.:...
Figure 's.l. Curve w1Lh arrows is ·• moving curve" C,.
14§
Quartics and Quintics
Figure 5.2. Two' two-real-dimensional disks meeting transversely at a point in four- .:aldimensional space. -
{X 2 = 0} which is a two-sphere, cut out neighborhoods of PI> p2 , and f1 1 in each, and fill in tubes. Then we get Figure 5.4. In the same way we see that the genus of an nth-degree curve sh< :lid be given by the formula ·
[genus of (n- l)st degree curve]+ (n- 2), or simply "-
2
I k=.l
Figure 5.3. A copy of S1
X
_
k=
(n-l)(n-2) • 2 .
((0, 1)} embedded in four-real-dimensional space.
150
Cluipter V
Figure 5.4. Why a fourth-degree curve has genus 3.
-----5.2 The Twenty-Eight Bitangents There are two very interesting and related facts about nonsingular quartics. We recall the mapping
(5.4)
K:C--d(C}
described in (4.22). Abel's theorem tells us that K is injective. In fact, the map K is always an embedding since, by the Riemann-Roch theorem, given any p E C there is ahvays A l10lomorphic differential (.() E
H 1 • 0 (C)
such that
w!P #- 0. So we can associate with each p tangent space to J(C) at K(p),
E
A p s;
C the one-dimensional subspace of the T/(C), K(p)'
which is tangent to K(C} at K(p). Now the commutative group structure on J(C) induces a unique natural isomorphism of tangent spaces
K(p)
-----=---->
7~(C), 0
•
Quartia and Quintk:s
151
So we get a well-d~ti.!le
C
I?(TJ(C),
····--:~.:..::..
-~·-·
(5.5)
{iP(AP~!·
p .. -where___ __
..
·
o)::.
···--·
!?(vector space V) == (set of all one-dimensional subspaces of ____ _:_,-c·c-,---,-,- ,~;:..~,;::v,eclor.space •• V) .... ".-}~~· ·:~::~;-~ji:::; ,:,·':"'-r:<-+·
L
te
It is asimple exerciSe to show that the mapping ? is the same a the
canonical mapping _·.. , .. .·. .. _ __ .__ __ ..:: ;-~-- __, ·····-······---~~~~w~~;:~7"~:Z;(it.~9(~):);;;
...
{{w eH 1 • 0 (C): wjp=O}} ..
p
So the mapping ? is injective unless there are distinct points p ''
i(P+ q) = i(p) = (u- 1). So by the
Riemann~Roch
theorem
l(p + q)- (g- 1) = 2 + 1 - g, and so the curve C must be hyperelliptic, that is, there is a mcromor >hie function f on C which gives a double-branched cover f: C -.CI? 1 •
Replacing p + q by the divisor 2p and making the same argument we conclude that unless C is hyperelliptic, ? is in fact an embedding of C ·nto a projective space of dimension (g- 1). The degree of p(C) is equal tl the degree of the canonical qr cotangent bundle to C, which is (2g '- 2). In ·ase g = 3 we conclude that every nonhyperelliptic curve,of genus 3 is (ca1 onically) embedded as a nonsingular quartic in 1?2 • , Conversely, if C is a nonsingular plane quartic, then the vector s ace of homogeneous forms of degree 1 9.n 1?2 lies in the space of sections ol the line bundle on C whose associated aivisor is the intersection of a line in IP 2 with C. Again by the Riemann-Roch theorem 3 :S /(C · L) = i(C · L)
+4+ 1-
3.
so that i(C · L)
z
1.
I.
Chapter V
152.
Btll since any ho1omorphic__dif1erential=has",only- four,-zeros, we conclude that in Tact these inequalities tnust actually be equalities and that the hyperplane sections of C must be the canonical divisors of C, that is, the zero sets of holomorphic differentials. So up to a linear automorphism of
.u~l ,Jhe inclu.SlO.J~~---·-··--·
··-----···-· ··-----·
_c:.:_ ___ ···----··-------- ··----------
is':~iiiiplythe~GaiisS:map'Jf::Th:u.siill~!!
non$iflg~ti'!r
plane quartic is hyperelliptic.
~,~1&~:~cii~ii~Ai~ti'Bi!?EIP.:ibil~!ii9i!iQ'16J~~U:tr
we discussed the dual mapping !i': C -~
IP'!
=(set of lines in P2 ).
ln-thecase of the plane quartic the degree of this map is given by
S9(C) ·(line in P!) = C • (zero set of L ai
::i)
= 4. 3.
Since we saw that £Y; is birational onto its image, we expect the genus of ~(C) to be
11
10/2 =55.
However, g(C) = 3 so the only explanation is that 2iJ(C) is singular. In fact, we can analyze the nature of these singularities ..f'irst, S9 is of maximal rank except at the points at whtch C intersects the Hessian curve
o2 F ~j-x~
82 F
---·----
oX 2 oX 0
deL
=0. c]2F
(J2F
<'!XooX 2
axf
There are then:fore 4 · 6 = 24 such points, and generically each contributes a simpk cusp to 0-Y(C), that. is: ~ singularity which is locally analytically equivalent to
153
Quartics and Quintics
that
Next, suppose that tl:le_r~ are two distinct :QQin~LJL~nd.q, on C ; tch · · ·· · · · ···· ~:,.: · " ' ····· .· ~(p)
= ~(q).
J..·
This means. that the tangent line. Lto .C.aLp. cojnc.ideuvith_tl:lat..to,_C.a,. Since deg C = 4, L must have contact of order el(a_~.tlYc.iw.ilb..:<:.::~t ~acl of the two points so that ~ is of maximal rank at' each. To see that ~(p = ~(q) is an ordinary crossing point of !il(C), we notice thatJoc~l coordin;:tes ~t IP>! at L are given: by y and z (Figure 5.5); So the, local"picture at p '1f a 5.6.. .But in Figure 5.6 small variation of the tangent line is as shown in Figure ·-···· . ~==-=:~.' '---~+~ ~;:h· ., <:_ •~:. ~~ distance(Ji;'·p,;.) :;;g,,:;~:f.;~~~·~~rtif~}3 ' :' '
=
1
·y=.·· ·
p'-+p
distance(p, p')
'
': .
so that in terms of the local coordinates y, z the Jacobian matrix of!}} is
t p
(0, a) for some a -=1= 0. Similarly at q the Jacobian matrix
of~
is
(b, 0). Thus ~(C) has a normal crossing, or ordinary double point, ~t ~(p) =f./ ,J). Now we write the exact sheaf sequence 0-.
where
~.&con
&!'J(C)-.
P)*{gc-. !l--.0,
(.-.6)
an open set U is the same as &con the open set I
.@- 1 ( !).
.
line coordinatized by
y
L
c Figure S.S. Local coordinates for the Grassmann variety of lines in
P2 at L.
Chapter V
)'
line
Fig. 5.6. How the tangent line to C moves near p.
From our computations it is easy to see that f2 is zero except at the double points and cusp points of .s&(C), at each of which points it has as stalk a one-dimensional complex vector space. Now the exactness of the- sheaf sequence(I ( -- !2)(
0 --+
C))
II
-t
(9 - t (!; ~((.) - t
0
((·( -12) shows that the Euler characteristic
x(&fi!(C)) =dim H 0 U»(C); &fi!(C))- dim H 1 (~(C); (Q~(C)) depends only on the degree of ~(C). So x(&fi)(C)) must equal (1
55}= ·-54
since that is what it equals for uonsingular plane curves of degree 12. Returning to (5.6) we therefore conclude that
I -- 3 = x((''c) = x(.S?.',,JIJc·) = x(d) + x((~ift) = [number of double points of E»(C)] +(number of cusps of !/'(C)]+ (1 -55) = [number of double points of .01(C)] - 30. So the number of bitangents of C shotild be 28. But there is· another way to look at the bitangellts to a plane quartic. H L is such a bitangem !me, then
(/.. • 'C)
c-::
2p
+ 2q
Qwtrtics and Quintics
!SS
(where possibly for some special cases p = q). Thus 2p +2q is a canon . ..:al divisor for C so that the line bundle LP ®Lq associated with the div•;or p + q must be a theta characteristic tsee Section 4.11). Furthermore, . his bundle has at least one section, and if
:dim,H 0 (C; cP(Lp ® Lq)),> .1,
rr wouta-nave~6efiypereiTip1ic, wmcl:lWeknow- ftisnoi--T 1us:: LP ® Lq is an odd theta characteristic. Conversely, any odd theta charac a~ istic has a section. If (p + q) is the zero set of that secti_()_J:l !h~l!-
-then
2p +2q ,. . - . . .-., ::,.:·: •.. ;_:;;: ~ . ,.·. . . -~_,-·-:.·:.::. -~-~--i:.:.<;~~~~~\~~.4}~~~~::-~:~:~~-l~~~::~:.L.,~.::-;/;: :::;;;;;~~ =is-a carfonlcai-div]s-oralld so is the intersection of a line L with c; T :us there is a one-to-one correspondence between bitangents and odd th ·ta characteristics. But in (4.57) and the argument following H we saw a way to count he number of odd theta characteristics. It was simply the number of sol uti· •ns !J, e E {0, 1}3 to the equation 1 () •
e = 0 (mod 2).
A direct counting argument shows that this nlllil.beE is
2< 3 - n · (2 3 -1)= 28.
5,3 '-';here Are the Hyperelliptic Curves of Genus 3? Our entire discussion of nonsingular plane quartics in Sections 5.1 '' 1d 5.2 is simply a discussion of nonhyperelliptic Riemann surfaces of genw 3. So, we ask ourselves, where do the hyperelliptic curves ·of genus 3 fit i1 lO this discussion? A clue to the answer is provided by examining the Ga, :ss map (~,_7)
in the case that Cis hyperelliptic (see (5.5) and the following discussio 1). Just as any elliptic curve can be writ~.en in the form
y2
= (cubic polynomial in x)
and has holomorphic differential dxjy, one easily checks that any hyper ~t liptic curve of genus 3 can be written in the form
y 2 = (degree-7 polynomial in x)
Chapter V
156'
and has holomorphic differentials Jx
x Jx
x 2 dx
y
y
y
So the Gauss map (5.7)is simply the map ('··· ..., ... cc·............................. , c·•lfl' 2 .. .. p t---
-------~··----------·:·.:...;___:---.";-~
(1, x(p), x 2(p)),
th~t:is~h-reali~es cas a .d<)tJtile-:t>iinchedcoverofacsmooth conicTni?;. So ~ow suppose we iix a smooth conic A in CIP' 2 given by
..........._. ,, ~~~,;:~,~-"'=~vtx-o=;-x l;'·x;:r~'·o'.: .- ---- --~~'7,..~7' "~
, ••: ..... ,
and take any smooth (nonsingular) quartic C given by
F(X 0 , Xt> X 2 )=0 such that C meets A transversely in eight distinct points. Now form the farnily of quartics {C,} given by IF
+ G2
=
0
(5.8)
for t E L1., the open disk of radius c about 0 in C. We picture this family in Figure 5.7. We next form
{(s, x) EA •• x CIP' 2 : s2 F(x) + G(x)Z
= 0},
which we can picture as in Figure 5.8. Finally, we pull apart or separate Figure 5.8 along the locus where it crosses itself. This last can indeed be done algebraically (by a process called normalization), and the resulting picture is as in Figure 5.9. Now it can be shown that the lamily of curves {C.} with C0 put in at .s = 0 is in fact a smooth family, implying that C0 is in fact diffeomorphic to ~~_. . ,..r-f 0. Onr conclusion is that as t approaches zero, then C, approaches
-------1
E
-··
-··------
---~-=-----_-_- . ._-_-----~J)
f - - - _- _
:
I
ct
1...- ...
Hgure 5.7. A family of quanics lying down doubly over a conic.
Quartics and Quintics
IS7
..
.
~···•·:.·:•!~~!~~:~0!.i!~~t~£,!t)~_is:~~i=::~~-~;~Jr:~L~~~l~~~~~~~~~~~~~···"· ..~ the hyperelliptic curve of genus 3 which is a double cover of the conic ·:..~ 0 branched at the eight points · ' ·::' C, n C 0 • Our goal here is not to be precise but rather to visualize geometrically 1 he behavior of curves of genus 3. In this spirit we can say that the hyperell.p{ tic curves of genus 3 are all concentrated in the various directions ofarr,ving at the double conic G2 by families (5.8). It is an interesting little exercise to see what happens to the bit<, ngents of C, as t-+ 0. The answer, which can be guessed either from geon
28
28
G)=28 bisecants of the set C, n C 0 , the set of branch points of (5 -l)
Co-+Co.
That is, the odd theta characteristics of the hyperelliptic curve (5.9) ' ·e simply the line bundles
Lp®Lq on
C0 where p and q are distinct ramification points of (5.9).
r------ . . . - ---------- - l
:
!
/"
_/
L_----------------- --
!c• leo _.J
Figure 5.9. Construction of C0 , a branched double cover of C 0 •
Chapter V
158
5.4 Quintics We will end this chapter with some observations about plane curves of degree 5. Herr a IWW phenomenon occurs. The "general" curve of genus 6 is not a plane quintic even though nonsingular plane quintics give a nice class of curves of genus 6. We can see this by a constant count. As we ~·re;arked··m''Secflon 4.12, the farriily'Of curves of genus _g has 3g-~_par ameters; whereas--plane qumtics can have no more than ·--· . ·----------
..
-····---····-·
- - . :::::_::;~<•c·~,,;;,21:c:;::·>9c:~cc'ft~"''~•'~~·;:·c-
parameters, since the vector space of homogeneou~- (orms ofdegree Sis ~ti~J'..::i2oJ,;wfirfe:'tne:;thrrrenilrri1'-ofY11e- gr'Qup Gt'p 1P2 and so on plane quintics is nine. One nice property of quintics is that their canonical bundle is obtained by restricting· the bundle &(2) on CIP2 to the curve. Thus the set of canonical divisors associated with a quintic Cis obtained by intersecting C in all possible ways with conics in CIP 2 • A quick way to see this is to notice that by the Riemann-Roch theorem, if A is a conic,
;'c}"which--acts- on
i(C · A) 2 6- 10- 1 + 6 = 1.
This is the same proof we used to show that for a quartic, &(1) restricted to give the canonical bundle. J This means that quintics C have a distinguished theta characteristic, -namely,
From the exact sheaf sequence
0~
(0(
-4) ~ (0(1) ~ &(1) lc ~ 0
and the Kodaira vanishing theorem (see Griffiths and Harris [1], p. 155), it is easy to see that &(J)Ic is an odd theta characteristic; in fact,
The Riemann singularities theorem tells us that this happens exactly when the theta divisor of J(C) has a tripk point. Now plane quintics have another special property; they are closely related lo curves of genus 5. This fact was first noticed by the Russian mathematician A. N. tjurin. We shall omit any discussion of the pathology of special cases of the. connection we are about to make. Our assertions will
159
Quartics and Quintics
refer to the connection between the general plane quintic and the ge wal curve of genus ·5 (notice that both form-1-2-dimensionaHamilies). t · Our first assertion then, true only ''generically,'_' is that a cun \! of genus 5 is the complete intersection of three quadrics
Qo. Qt, Q2 in CIJ:D4 , that is, it is th~ base (orfixeq) locus Q[,aJ~milY<.;; . ;.
+~9~
o:. ,;;;~;ki.J_;_~~~~::_ _;_·; . (
-.}go+ JlQt = ..10) -of homogeneous forms of degree 2 on CIFD 4 • Now ':each'Qj is g1ven 'Y a symmetric 5 x 5 matrix (qjP), and the set of those quadricS (5.10) whicl. are singular__!~--~~ven ~~the fifth-degree equation · -"det(Nj~
+ Mi11 + vq2P) = 0
( '.11)
in the CIFD 2 with homogeneous coordinates A., Jl, v. But the connection between curves of degree 5 and curves of ge1. ,Is 5 is much richer than this. The curve of genus 5, which 'w_e shall call .:...
B,
which sits in CIFD 4 as the base locus of the family (5.10), is. canoni ·ally embedded (same proof as for quartics). So as before;we think of Bas 1ing ~
~·
IFD(T;(B). o)
and of our family of quadrics (5.10) as lying there also. This fami :y is simply the family of all quadrics containing the canonically embe, :ded curve of genus 5. There is another way to produce quadrics containing B in IP'('Jj!Bl. 0), namely, by studying the theta function -
8(u) = 8[g](u; 0) on J(B). Suppose u0
E
J(B) is such that
8(u 0 + Kp 0 ) = 0,
ao (u
OU·
0
+ Kp = 0, 0)
J
where u = (u 1 , .•. , u5 ). We then know that u 0 is a singular point o the theta divisor u(.U 4 >). By the Riemann singularities theorem in Section -U2
t The point of view employed in what follows was introduced in the fundamental pap r by A. Andreotti and A. Mayer, "On Period Relations for Abelian Integrals on AJg,· >raic Curves," Ann. Scuola Norm. Sup., Pisa, vol. 21, 1967, pp. 189-238.
Chapter V
160
and (4.60),
O(u(p)- u(q)- u0 for all p, q
E
-_ -:: . -
Kp 0 )= 0
-
B. Differentiating at p = p 1 we obtain
--~-.-.;~j.{Y(R!.L:=-.1!1~)~-=-'f:o;:::;:~f:o,'pi(Pt) = 0,
w.l_l.er~~~~ tht; jth ~l~m!!tlJ .9JJIJ~J-1a~i~...QfJI1• ~(~), N~~t 4ifferentiate with ,, respect_to qat q = Pt toobfairilhalTor all'ji E B
D2 0 ··· · · · -L--'a~jl3u"· (~,~:Q.~-:::4f;;o)O!Af)_~,.(p) = Since 0 is even, we can replace ( -u 0 that the tangent cone
-
KP 0 ) by (u 0
o. + Kp 0 )
and conclude
2:
iSj,k55
to u(U 41 ) at u0 contains B. Thus each point of the singular locus
e.g s;; u(B<
4
>)
(5.13)
has, as its tangent cone, a quadric of the family (5.10). By (4.61), dim (0,u = 1 In fact, it can be shown that (again "generically") is an irreducible smooth curve. Also, the cone ( 5.12) is singular. This is simply because u0 is not an isolated singular point of u(B< 4 l) as it would be if (5.12) were nondegenerate. In this way we see that there is a mapping
e.g
n:
e.g u0
-----+C,
quadric (5.12),
(5.14)
where Cis the plane quintic defined in (5.11). This map assigns to each u0 the value of (?., Jl, v) which gives the tangent cone to u(.a< 4 >) at that point. Again generically, it can be shown that (5.14) is a smooth unbranched double covering. That it is at least a double covering is clear since u 0 and (- u0 - 2K Po) clearly go to the same point.
. ~CHAP'IER_SIX ._
The Schottky < • ..,
·'":~:·::.;~~~;~~·;:,;:·.--·~::. .:.~ ~.:~AA'" '.,.,,~· .-:
Relati,~p
,' _ _,_. _ _
__ ,
,: -·:~;;~;-~:·~~t_;~~~--~:~~~~~~~~~:~~~~~~·:#;~-i~,:~=~~~~
••
6.1 Prym Varieties In this last chapter we will discuss the most famous classical result 1 1 the direction of answering the question, Which symmetric matrices n with positive definite imaginary parts arise from curves in the manner pres· nted in Chapter Four? This is the so-called Schottky problem, and in apprl aching it, we will closely follow the work of H. Farkas.t At the end of Chapter Five we saw a special case of the folio .ving phenomenon. Suppose ~ n:C-+C
(6.1)
is an unbranched irreducible double covering of a curve C of genus (g !- 1}. p 0 e C, then (6.1) corresponds to a subgroup of) 1dex 2 in the fundamental group
If we fix a basepoint
1t1(C, Po).
Since this subgroup contains the commutator subgroup of n 1 (C, p, \, as well as the subgroup of squares in n 1 (C, p 0 ), we conclude that the ~ ;t of double coverings (6.1) of a fixed C is in one-to-one correspondence .vith the set of subspaces of index 2 in
6.2) Since the intersection pairing is nondegenerate on (6.2), each nonzen element of (6.2) corresponds uniquely to a subspace of index 2, namei its t "On the Schottky Relation and Its Generalization to Arbitrary Genus," Annals of Ma1, "rnatics, vol. 92, 1970, pp. 56-81.
161
1.62
Chapter VI
.perpendic~lar. SJJbsp_a~e cwilb: . re~pect to_ the.in.tyrsection p~iriqg,_Thus. the set of double-'cove-rs-('6:t)·ls ln·a·natural one~to"one ·correspondence with the set of nonzero elements of II 1 (C; IF2 ). If y is a simple closed curve representing a nonzero element of fi 1 (C, a: 2 ), then the covering (6.1) corresponds to y if and only if n- 1(y) disconnects C. Now-tnere::K:-a-iiatural:-iiwolurion·
(6.3) ~~~iiid~so~a±"nltttrrttin&llititiD:n::~-'~-~~=,~: ·_ · -':'"'·-· 1*:
JJ1. 0 (C)~ H 1 • 0 (C).
(6.4)
;WJ.;n"!»_giriv~!~~-~ .t1·__ .f\!~<>?:Ie!-:~~~PI~! a'l1a~J~=:~~"?': ' '"". -· -· · for H 1 (C; l) such that Po (reduced mod 2) corresponds to the covering (6.1) and such that the basis is symplectic in the sense of Section 4.1. Then we can build a symplectiC basis for H 1(C; Z) as shown in Figure 6.1. For our symplectic basis for C,
---
n.. (fJ) = fJo,
n .. (a) = 2a 0 ,
n.. (fJj) = n*(fJ'J) =pi, n .. (aj) = n.. (aj) = ai.
c
11
c
---~ Figure 6:1. A symplccllC basasfor 11 1(C; l.).
163
The Schouky Relation
Then the cycleL
As in (4.7) we choose a basis
H 1• 0
,.. ~... (Ct, the
~:~1/Jt,
·--~-~;::
for (-!)-eigenspace of H 1 • 0 (C) with respect to the in ·olution 1*, sucq that
J
1/1" = (Kronecker b1").
Pj-lfi.
We know that H 1• 0 (C):- has dimension g because C has genus 2g
Ht· o(c)+
;?:
+
and
Ht. o(C).
Next we let y
denote the matrix
The matrix Y takes the place of the matrix n in Section 4.2, an. the differentials 1/J", called Prym differentials, take the place of the w 's of Chapter Four. So just as in Section 4.2, we obtain the _Riemann relat1 JllS 1= 11
and Im Y>O. So we can build a complex torus
(H 1 • 0 (C)- )* P(n) = H t(C;
~)
C'
~ L ~EJ + L ~YJ '·
(6.5)
;alled the Prym variety of n, in a manner analogous to the_ constructi< n of Pic0 (C) = J(C) in Chapter Four. Also, since the Riemann relations we e all :hat were needed to construct theta functions, we have them in this se :ing.
Chapter VI
164
Since we. will want to compare.. these:with.the theta functions on J(C), we will use the letter '1 rat_her than 0 to denote these functions. Thus .
'l[:](v; Y) =
L exp{ni('(m + !J)I(m +to)+ 2'(m + to)(v +!e)]}.
(6.6)
1/.B
We.ought to mention that if .. :.
.... .-.' .. ···~··--;
----------·· -----------------------
"-~"=·"~·~'~-·-.::..._:.c-.:~---
:- - .:"·'---,_,_~--,f:·O-e;;;""~c--·----~~~~
o:js~41~qfu:iibl~oove¥i":iWllf~tfie::nlane guintic constructectin'Section ;.;;:.;,::;;:..:;;.....,.~~,;,:,. ·;. ........ ··:·"" ~ ... "" •"" , ....... --~ v··-·o ..n·-···it,h:.:;;.l;;;,·,;::;':;;;;;" .;;,;_;,,;,_,:._..· ·-~~·>.,;f::,"'·"',;;:,·.-: :.~·.:··~,:-·,:, ::. "..
connection with the curve B of genus '5, then - --
1'1 ) J(B) :t t1
.•·; "'" "'" ::·
··..
""
5.4 in ; ....
:·
--
~:::t~~~:·~~~~;~~~~~f-k--;~;}~-:~t~ri~:"~;·:;·~~~~~~~---~--
-
~--------
Im A> 0, we have seen that we have associated t_he whole apparatus of complex torus I(:B
t-l"£j +I
il.A/
theta functions, etc. We call such a matrix A a period matrix. Counting constants, we see that if g ;:::: 4, not every such A will be the "period matrix" n of a curve as in Chapter Four. In particular, for g = 4 the set of period matrices is a ten-dimensional analytic set, while the set of curves of genus 4 has only
3g- 3 = 9 dimensions. Thus there must be a nontrivial (analytic). relation on the entries of A that will always be satisfied if A comes from a curve. One such relation, called the Schottky relation, is the object of the rest of this chapter.
6.2 Riemann's Theta Relation To obtain the Schottky relation, we need two main ingredients. The first is the Prym variety construction, which we have already seen. The second is a beautiful trick using the character formula for finite groups, which results in the most far-reaching of the many identities among theta t For more information about this topic as we:t as recent advances in the theory of Prym varieties, see A. Beau ville, "Prym Varieties and the Schottky Problem," Invent. Math., vol. 41, 1977, PP- 149-196.
165
The Schottky Relation
functions, Riemann's theta relation. We h~~e already deriveci this r~ cation in the ca8e g = 1 in Sectioril14-bydiffe-tent"means~:;<;.:·:·-· ·: · ··· · · __;:_'-'--to begin, we will rederive Riemann's theta·''relatioii-'iii the case .J ::''1. Let e1, ... , e4
4 denote the four product _on -~~ so· that these vectors_ form an orthonOrtl11iLba,sis,. W n~~t. consider two lattices in ~4 : · ·· ··-·. • ·.· - :~·:.:~·-~_;;~~;:;;;: -_,,:;;·
stand~;~('ba~i~ ~ectors in.;~ ~\\i~';y;~1''fh~~~t~~d~rd _;ca'i~~-,
,_l-1 ="~t;!.Le-~~2 + 1Le3 + 1Le4, ··-'"""""h:=id~:t•i.-;,~~"''';;;;~,;.~,.--'"'"''-~;;._.:.: Li;: [submodule oft· L 1 generated bfthe~vectots'e)"'f e;:·-t~;j; ''k ::;; 4, and the vector i(e 1 + e2 + e3 + e4)]. ·
Now (L 1 11 L2 ) is of index 2 in both L 1 and L 2 , and so the scalar pi Jduct is unimodular on L 2 , that is, the matrix giving the scalar product Ji r any given basis of L 2 has determinant 1. As scalar product spaces, L1
;;;;;
L2.
In fact, an explicit orthonormal basis of L 2 is given by !1
= t(e1 + e2 + e3 + e4),
f 2 = t(e 1 + e2 -
e 3 - e4), .· / 3 = t(e1 --e 2 + e 3 - e4), !4 = t(e1 -:- e2 - e3
(6.7)
+ e4).
Notice that if e and fare interchanged, formula (6.7) continues to he J. So the matrix M which gives the isomorphism of L 1 and L 2 satisfies
M 2 = (identity). Finally we let
(6.9) so that [L:L 2 ] = 2 and the extension is generated by any e1 ,j= 1, .. ,4, and [L: Ld = 2 with this latter extension generated by f(e 1 + ·.. + e. ). Next suppose that the compl~x number A satisfies the Riemann relation Im A> 0, and we write the product of four theta functions
6.10) Here the gjs and hjs can be arbitrary elements of ~-we need not rc ;trict ourselves to g-tuples whose entries are all zeros and ones. The ;arne definition of theta function as in (4.38) works in this slightly more gt 1eral
Chapter VI
166.
setting. Suppose now we multipiyout the functions in (6.lO) .. Wecob.tain.. ..c-.:. . exp{2ni(
F'oiJri~rexpaQ~IQQ~.o(Jhe
theta .
)},
where the expression which goes .in the parentheses is
tn~n j ·+ !ni)2 :4-~+im~+~tyj)(u;·+ tii~) ;: · · ·
j= I
.
, · '··: :" 7 ::-~~:''::,·f::,, _,.~~:·"'9!(»t¥1ii}S1(m'+ffi>·+
'(m··+. fi/)(~ ·+. !h l
where
C':·"'c;cCCf."':"~'',:.~~~t~""i:tc" and
So.wc can rewrite (6.10) as
L
exp{2ni[f'(m + -!g)sl(m +!g) + '(m + -!g)(u +!h)]}
mE !.1
=
~/L1
exp{2ni[!'(m + !g).w'(m + fg) +/(m + fg)(u + fh)]}
1m e L
+
L
exp{2ni(m ei}exp{2ni[f'(m + fg)d(m + fg)
m ei.
+ '(m +
-!g )(u + !II)]}.
(6.11)
where (m e 1 ) refers to the standard scalar product mentioned earlier. Next. we break down each of the two sums in (6.1 1) into two parts .a-c.:orafng to whether the index of the summand lies in the subgroup L 2 of L. We obtain · exp{2ni[1'(m + !g)s1'(m + !g) + '(m + -!g)(u +!h)]}
+ exp{2ni[!'(m + e 1 + !y).w(m + e 1 +!-g) + '(m + e1 + !g)(u +!h)]}
+ exp{2ni( -1o · ed}exp{2nH'(m + !-y).W'(m +!-g) + '(w + !o)(u + V1 + e 1 )]} + exp{2ni( ·-lg eJ}}t:xp{2ni(!'(m + !Y + eJ)d(m + + '(m + ~g + eJ)(u l-- !h + eJ)]l.
!-g + et)
The Schottky Relation
167
Now we use that M is an isometry__on. ~"' to· get the finalJorm of (6. i d): -:. ·';.::.!.!!;~::.: :····· ...-!:::;-~ -- _-:~·:=;=-::.: ..'-:~. ·.-
+ !Mg)(Mu + !Mh)!i + exp{2ni[t'(m + !Mg + (t))}d(','l t !Mg + (t)} + '(m + !Mg + (f))(Mu + !Mh)]} +-exp{ -nlg 1}~xp{_~~t[t'(m_±]Mg )d'(~::.f::·_fMg) . .... ···.
exp{2ni[!'(m + !Mg)d'(m + !Mg)+ 1(m
1
. :2L. Lt
..
.
. :+_'_(p + fMgl~_± !Mh ±Jt)]l____· __;~····c;;:::;;~;;;_g.::~!:.f~:- .•. ::d~~
+
. + exp{ ~nigt}exp{2ni(t'{m +
tM~+ (t))d'{m fMg -f- ~~)Ti ±~("!, +;:!~g~~(!~t(~~o,!,~!'!P:-f~,(})JJ},~;';~~'&;l·;,'. ;.{.: where (t) = (t, f, f, t). Thus ....,.,..... ~;---.~-
fr ortmu
1;
j=1
fr
1 L exp{ -ni[a" · gt)} Oft1:~;:.](uj; A), 2 (::) }=1
A)= -
where u' = Mu, g' = Mg, and h' = Mh and the index (::.) runs ov• r fill elements of 1L x 1L whose entries contain only zeros and ones. A. sr .:dial case of this last formula is the identity
0[8] 4
= t(0[8] 4 + 0(~] + O(A] 4 ), 4
which we already saw at the end of Chapter Three. The proof of the corresponding general formula in dimension g d JTers only notationally from the one-dimensional proof. The matrix l is replaced by a 4g x 4g matrix which has a g x g identity matrix in .:ach place that M had a one. The g/s, h/s. u/s are all now vectors, and in step (6.11) we have 29 sums corresponding to the 29 characters of 7l.fl with v ,)ues in the group { ± 1}. So the final, general form of Riemann's theta relat 1 >n is
Ii OftD(uj; A)= ;(/ L exp{ -ni[a" . gt)} Ii ow:;;:.](uj; A)
j= 1
(::)
( >.12)
J= 1
where u', g', h' are as before and the index (;:.) runs over all elements of 7L9 x 7L 9 whose entries c~mtain only zeros and ones.
6.3 Products of Pairs of Theta Functions In order to get at the Schottky relation we need another prelim; 1ary theta relation which is obtained by a simple variation of the argument '1sed to obtain Riemann's theta relation. Again we start with the case g = 1. fhis
Chapter VI
168
time we use the matrix -
-M =
[1 ---cTJ______ 1
-1
so that
M 2 = 2(identity) . .
,-=---'-"~'-"-·'""-
--he-iilC:t st~~dard)affice
.......
--,~->""''·""---"~-"""'",__,_,__----·---~--.
in 'ifii'iri~-~rer·:
·- - -
- '\~-;;,L:L~t/;,,;,;;,,~-~-·--;i;;f 2 :::;= ~='~b..t;;:"~''""-·· Notice 'that L;--';2-'L, b;)genetates -L: (t{e 1 + e2 ) • t(e 1 + e2 )) =!so that the standard scalar product takes halfintegral values on L2 . Now, as in Section 6.2, we write
a;~a;-ror-~~~fm:~l~;:1(i~-+
/L~~- AI;~
O[~i 1 ](u; A)O[~~](u..!; A) =
L L
exp{2ni[!'(m + to)s~'(m +to)+ '(m + to)(u +!h)]}
mELt
=
+ Jg')2.w'(m + !g') + '(m + !o')(u' + 1h')]}
exp{2ni[!'(m
mEL2
(6.13)
where g' = M- 1g, h' = Mh, and u' = Mu. But now we split the final sum in (6:'i)) into two sums, according to the cosets of L 1 in L 2 , and we obtain
O[f:](u; A)O[~~](u 2 ; A)=
L 0[ a'
91
~ta'](u'1 ; 2A)0(112~;.a'](u2; 2A),
(6.14)
.
where r:t' ranges over 0 and 1. In fact, as in Section 6.2, the same formula (6.14) holds in all dimensions. The only d iiTerence is that rl ranges over all elcments.of 71_11 whose entries consist only of zeros and ones.
6.4 A Proportionality Theorem Relating .J.acobians and Pryms We are now ready to bring together the ideas of the first three sections of this chapter. Let n:C--.C
be as in (6.1). We then have the period matrices
169
Tbe Schottky Relation
;.= . [:;0;]~,~~~\'~ ' , ,_
for J(C) and
-r,, -r,
.
for P(n). It is then easy to <;:alculate the period m;i~rixf<>r,/(C) with
ro {1
f<
spect
ilie :::l:,;~:~ven :!7~~~~t:*:;~,) ·~ =
··.···_ -
W,o. t(w,l + T,t) !-(w,' + T11 ) !(w1 ~ +'~~i) • ., , !(w11 - T .1) ~-wi:o ::'!((l):~:;~;;iif::·- ~'!(a>i,-~"~T;r:.;I(;f1~~~tH~)1~~¥}(~:4F!~f
.
w, 0
·,·.
-!(w, - r,) !(w, 1 + r,l)
!(w, 1 - t,t}
'
_!((1.)11 + 1
Let I denote the (g x g) identity matrix, and proceeding as in 6.3, we let '
0 I
+ 1) x
(2g
+ 1) matrix
-I
~ tJ~I -!I~] 0
MOM=
oJ 0 2Y. [2n
::.](w; 0) =
r
We can then write
·or;:
s, ction
whose inverse is
l
Notice also that
11 )
[~ ~ ~]
M = a (2g
•
S.lS}
(mo. m, n) ell>< ll' x lll
where what fits into the square brackets on the right-hand _side i the expression
[=
0
1(m0 + -!e', '[m + -!e'], '[n+ !e'l]t"l'
n
:
:::]
+ te'
e"/2l
I + [mo + !e', '[m + !e'], '[• + !e'l] [ w + e"/2 [. I e"/2J
170
Cbapter VI
;:.Jwe should actually write
(Instead of fJj;.:
e ·--"-···-··· ······'·'·-"-··--'·.
[f] t"
..
t"
.\
---·-·---------------------··---
~~6~i~~hi~:ii:-~6t~ii~ri;iilj2i~~~~~-b~~'6hfit-:r~·"'"::~~-::;=~=~~=~:::£~:7:-
Nqw we <;a11_d9 the _S(l!TlC ~ric;~ to ~h~ lll:t!_i~~2fil'l_~i_c;es in (6.15) that we have at~eady done'-severafflmes:belore; iiamely, sum over
we can
L 2 = M-·
1
(£~
x 7L9 x 7L9).
which can be rewritten as
L:
L:
me 71. x 11.• x 11.•
a'
e2ni[
(6.16)
).
where what goes in the square brackets is
i '(
m+
:~~:/2]) ~~ 2~] (
[ e'/2
m+
+ (m
+
[
&/2:;•'/2])
l ) ( [ /2] ) 2
/2
e' [ e'/2a~ a'j2
Mz
+
e" e'~ ·
.
1n the summation a' runs as always over those elements of 1L 9 whose entries have only zeros and ones. But the summation (6.16} has a much easier form. We let
Th~n
Lor;.:
1
[z 0 ,
U
=
V
= (z 1 1
Z1
+ Zg+h
••• , Z 9
Zg+h ... , Z 9 -
+ z29],
Z 29 ].
(6.16) is simply ,-~"''](u
+ [,~.];
20)rf[~](v; 2Y}
a·
=
L (-l)<<'+a')·•"H[~:.
..
''t•'](u; 20}1][ 0](v; 2Y}. (6.17)
The Schottky Relation
171
Here we use th~,i~~~,e~i~,;~~i~:~ ofOto emph;~ii~th~ttti~tfi~druncl :old~ coming from the period matrix of the Prym variety. ·· The reason for doing this rather involved computationjs that, iS the expression (6.17) shows, a relation between theta functions associate1 with the period Jllatrix n of the curve C and theta fuf!ctions associat¢ wi· h the period -matrix ,r ass~iated with the prym_Y.~d~ty:_e(~)j~_l;Je.girmi lgJp__ emerge. But if we choose carefully the values ofthe various variabks, can actually obtain:a situation in which the e:xpression (6.17) is e.q<~al to zero. The basic idea for this is as follows. Suppose we take an odd theta
we -
ch~~!i~te~!~!ic _L ~?~,£: The:im- '/7ri(c~;-~tiw-~~t.'~-' :::{~i;;:i;+d1[i;¥,;M~(j,,_,,.,_~ But n*(L) is a theta characteristic for
C and so
dim H 0 (C; &(n*(L)) ~ 1. But if we can arrange in fact that n*(L) is an even theta characterist c, we will obtain dim H 0 (C; &(n*(L))) ~ 2. If, on top of this, 8[::. ::. ::.](w; 0) is the theta function on J(C) correspo 1ding to n*(L) under (4.57), (4.58) and the discussion following them, the1 from (4.60) we can conclude that
:,6.18) for all p, q e C. This cah in fact all be arranged. The point here is to trace through the many identifications of Chapter Four to see that with resp~ct to the symplectic bases for homology chosen in Section 6.1, the isomor1 >hism (4.56) works out as follows: _ ~
(points of order 2 on J(C))
e" /2 ]
I [e"/2
+n
[e' /2]
[;,~;;] e"/2
+ fi
-a!-
L
e'/2 - - {L},
(points of order 2 on J(C)) 1
-
[:::;]---+
t, {n* L}.
6.19)
I
e'/2
Thus L is odd whenever
e'e" + e'e" = 1 (mod 2),
6.20)
Chapter VI
172
. -.. . . --------------------).
and we can have.1t!(L) -~veu.s.imply. by having
e'
(6.21)
= 0.
Even the verification of (6.19) is not too bad. Since its correctness (or "not)...is-,invaria,nt-under-deJo.I1Jlation9f C, it suffices in fact to check it for thecfollowing"-'(degeliel;am};:;sas.~he one where C looks Jike Figure .6.2_. (Compare Figure 6.2 with Figure 6.1.) One .then sees that the left-hand con1pol1ent of Figure 6.2 adds nothing to the verification, and what is invol~~sLis verilying (6.19) directly, by hand, in the case in which C and so
.,~~Q:i\:.lot~k:,.~I!!.Rti.&S.~XY~-~:;.. __
. _ __
_ , _ _ __ _ _ - "·""-'-'"''""""~=,~:::::-::'::'.:::==:'"
··-So from··now·on let us assume that (620) and (6.21) are satisfied. Equation (6.17) then gives us that 0
= ~ (-l)
O[?
r.'t"'](.C du.; 2il )t~(O'J(( dv; 2Y)
(6.22)
whenever e' e1' = l. To get even a better relation we will sum (6.22) over all values of c" such tha·t 1; r:" = l. To sec whjlt comes out we simply must compute
y
_.{1
(--])<'
where we sum ovl!r all fJ such that
t•'l (c"+Pl '
fJ
t, 1
= 0. We obtain
if a'= e', if a'= 0, otherwi~e.
So we obtain the relation
O[~
g1(.( du; 20 )tJ[o'](.( dv; 2r) = 0[?
~1(.( du; 2il )t~[&J(( dv; 2f}
Fi¥ure 6.2 A good deg<:ncration of C lor checking Prym identities.
(6.23)
173
The Scho_ttkr Relation
But (6.23} must be true. for all e' =/= 0 since we can always-find aile" such that e' e" = 1. Our conclUsion thtm is that the vectors .: ''·
du; 2o)) ( 0[~ ~1({ . , .·
p .
······~.~
.... «'e{O,l}l .. ,. ···-.. - --· ....... .
-:-----.:::-·
.and
(~[~l({ dv; 2r)). . . ·.
«' e (0, 1}1'
p
are proportional for all p, q e
q.-
6.5 The Proportionality Theorem of Schottky-J ung The proportionality theorem obtained at the end of the last sect m is not quite what we want because of 'he presence of the period matric :> 20 and 21 rather than 0 and Y. However, we can remedy that with the h lp of the relation (6.14) which we derived in Section 6.2. We put u1 =
Jdu,
v1 =
Jdv,
and we write
O[g :.:J(u1 ;
::.](u 2 ; 0) = " (-1 )<•' +a') • ... 0[01 (6.14)~
0)0[~
•' +«')(u' · 0 1'
0 20)0[ "')(u' · 20) . 1 0 2•
- L:a' (-1)<•'+«'> ·•"o[l
•'t«'](u1; 20)0[l ~)(u': 20) .
whereas
O[f ::.](u1; 0)0[8 ::.](u2; 0) = "(-1)<•'+«')·•"0[0 ·•'+«')(u' · 20)0[0 «,(u' · 20)
(6.14)~
1
0
1•
+ L (-l)<•'+a'>·•"O[f
1 OJ
2,
•'6«'](u1; 20)0[f ~](u2 20).
'
~
The subtlety here is that by (4.39)
0[-f ~](u2; 20)= -O[l· ~](u2; 20),
or;.:
2::.](u'; 20) = ( -1)<'. •"or;:.
~)(u'; 20).
Chapter VI
174.
'Next we add
0[8
::.](u 1; !l)O(? = 2
~:.](u 2 ; n)
+ O(?
L. (-1 )<"'h'> · ""0[~ a.'
~:.](u 1 ;
!l)0(8 ~:.](u2; !l)
-'~•'](u'1 ;
2!1)0[? ~·](u2; 2!1)
·.........·· · ···==:'.~constall!:lf~ . . f-J)<"'.t. ~-·> · .-.n[f'~~~](~r-.:·~:f)n_&~]_(P~t'~IL . ._,.._. __ l 6±4i'(c~pstant) ~ IJU'.:](vi, Y)n[f·](v 2 ; !), -where the. essential ppli~fis that.th~ c:onstant is independent of the choice ~ofi;'Ja:nd 'f.". In·th'is:''Way we'o5ta1i1111erheorem~'CWSchoqky and 'J!lJlg that.. 1he vectors ··
1(0[? ::.])(0[8
n(Jdu)) + (0[8 ~:.n(a[? n(J du)) l(n
and [ (11L<]) ( q(:''.]
(J dv))1(::,)
are proportional, where, for the sake of economy of notation, we suppress 11 2 = v2 = 0. In the next section we shall apply this .theorem to· obtain the Schottky relation.
n, r, .and
6.6 The Schottky Relation We are now ready to derive analytic conditions on the period matrices A which must be satisfied if A is to be the period matrix of a curve. Since it can be shown that the (analytic) set of period matrices of curves of genus g has (3g - 3) dimensions (as long as g > 1), the first place to look for a nontrivial analytic relation on the entries of A is in the case g = 4. Given
the Riemann theta relation and the Schottky-Jung proportionality, the idea of constructing relations is very simple. We start with an identity between theta-null values, for instance, the · identity of Riemann
OU:J4 = O[A]'~ + 0[~]
4
IJ[g]~
4
•
Using ='=:
IJ[c\l'~
+ 1J(~J
and Schottky Jung proportionality, we obtain
O[g
:n
2
o[?
:w '-"' o[g 6] 0IV b] 2
2
+ 0[2
~yory
?Jl,
175
The Schortk>: Relation
which must be identically satisfied since the general 2 x2~:period 1: latrix comes from a curve. Again change O's to 17's and use Schottky-Jul}g 1,) get 0[0·o o 0 o]o[o 0 o]o[o o 1 oo o
0 1
o]o[o o 1
o - 0[0o o0 o1]0[01 o0 o1]0[0 .. o 01 o1]0[01 ·10 o1] o. o]o[o. o o]O[~)• o1 o] 1 1 0. 1 . 0 1 1 1 ' + o[o0 o·0 o]o[o
0 0] 1
,'tfh•~? ~g~I~, .llll,l~t 9~.~~~J~fi~ ~ct~Q!!~aJly:: Fffi~l}x;.·.a.~~n~Jl~~~I"~~.,q·s <>. ~·s m this last Identity and replacmg '7[!.,] with · · · ·· · · · ·
(o[g
.:.]e[~
::.])1'2 .
we obtain a theta. identity in dimension 4 which must at lea t be satisfied for the case-iii whieh the period matrix used in building: the theta functions comes from a curve (of genus 4). Finally, it remains to see that the Schottky relation is not s.mply another relation satisfied by all abelian varieties of dimension 4. Fo this we will outline an argument given by R. Accola. The beginning,poin; is an elliptic curve E. It is easy to convince ourselves that given any set ol dght distinct points
P1• ... , Pa on E there exists a Riemann surface
6.24)
h:C-+E
which is a two-sheeted covering of E branched at the points Pi· As in (5.5), we build the canonical mapping
p:C-+IP4. ~
.
Intersections of hyperplanes in IP 4 are in one-to-one correspondenct with canonical divisors on C. Now the double cover (6.24) has a n ,tural involution on it so that hoI= h.
This is just ".sheet interchange" so clearly induces an involution I*: Hl, 0 (C)-+
1 o 1
= identity map. l'
JW 1
Hl, 0 (C),
so by linear algebra, H 1 • 0 (C) decomposes as the direct sum (, the subs paces H 1· 0(C)+ ={we H 1· 0(C): 1*(w) = w}, H 1 • 0(cr ={wE Hl, 0(C): l*(w) = -w}.
Chapter VI
176
Since the el~ments of H L 0 ( C)+ all comc..Jroril holomorpl)J<: qJ(ferenti~ls 011 E, we know that dili1 H 1 • 0 (cr = lands~dim H 1 • 0(C)~._::;A. · This last says something important aboutthecartohlcal mapping ? in this case. It says, that there is a point q in fP> 4 so that under the projection
rr:UJJ4-.fP>3 centered. at q, (C) . oes.onto:::a:tr:::e:Itip:tic:::cmve:·More·Rr®iselywe· ha.Y~fJL con1tnulative 'dfagra~·· :c:ccocc ,. • •.,, ·. . : ·-·-·· ,,,, ··.< .,
.·••·.·
··..
• ..•. •..
••·•··
',.,_
······-----····-- --··-·
C
?
Jh E
IP4
J .!!. .K
--'~
(6.25}
fTl)
lrJ
In fact, it can be easily checked that ./1) embeds E as a curve of degree 4 in
IP3. In what follows, we will only outline the steps. The details are not hard but will make the story a bit too long. Interested readers can fill in the rest with some patience and diligence, or prefer~ply, by conversing with a knowledgeable geometer The first thing to notice about (6.25) is that while IP 3 has a ninedimensional set of quadrics (second-degree hypersurfaces), they cut out on .1f (E) a seven-dimt>nsional linear system of divisors by the Riemann-Roch theorem (4.9). So .Yt?(E) is contained in a one-dimensional family of quadrics. But the intersection of two quadrics has degree 4 and so does .n"(E), §.0 .#{E) is equal to the intersection of any two quadrics of the family. The way we say this is that ./I'(E) is the base locus of the pencil of quadrics Wc
(6.26) There are four va.ues of (..l 0 , A.t) E IP 1 such that the determinant of the matrix in (626) is equal to zero. For each of these four values of (..l 0 , A.t) we obtain from (6.26) a quadratic cone in IP 3 containing .Yt''(E). One such cone is shown in Figure 6.3. Now each line L on the cone intersects .}f'(E) twice and if Land E. are two lines on the cone, then
(L +
q
n }f'(E}:
is the intersection of a hyperplane in P 3 with ,/f''(E').
•
177
The Schottky Relation
L
Figure 6.3. One of the four quadratic cones containing Jf'(E).
What all this means is that any divisor of the form
P1
+ P2 = L
n Jt'(E)
has the property that
is a canonical divisor on C, that is, the line bundle on C associated w, h th~ divisor
is a theta characteristic (see Section 4.11 ). Furthermore, the linear ;. stem associated with this divisor has projective dimension at least'one si1 ;e we can move the line L around the cone. If the projective dimension ( f this linear system were greater than one, then the curve C would have to be hyperelliptic (which it cannot be; if it were, p(C) would be a ndonal curve). Thus each of the four theta charactedstics obtained in this , :ay is an even theta characteristic. So for each of these four line bundles
Chapter VI
178 .. ,. ;···
·;~;;,·,:.::~·.:. :;;-_·..:...:..:.:.:.
we..... have .rhat... . ~:
Under the correspondence (4.56), the line burtdles·-n:i>-corresporrd to four .Pc:>iflt~ ()f 2~~~~~~------__ ----··· __ -'-----=-=--=-=·--:---'.'::-::__·:-::::.--==-:::-_~.'::::· · .. :.;. ..
"'s~'-'l:lyithrli:I~~~.;~·rirr:~S:mgi:ilii~'s~tliebr¢m='(§t~lktti~r.~ ar~. four even thetafunctions
~:~·~: ..···~-:~· ·-.~~-"-·•·;.--':~. =··.;,; ~.:cy;;~,:.,~_,;.tfS:..··~"~c::;c;()[~11J·(U·' _0)~·-·· •: ::::··--.-.~=','•=~~='"'~=:=..,.-=-:-=",{~~~7.)"':;
such fhai O[~Bl](O;
il) = 0.
We will skip over the verification that if 0[~](0; Q) = 0 and b · 'e = 0 (mod 2), then the tangent cone to O[~](u; il) at zero gives a cone as in Figure 6.3. Since there arc.. only four such cones, the four functions (6.27) are the only even theta functions which vanish at u = 0. There is another poin·t .that requires some work (which we will not do). That is, we can choose the symplectic basis for H 1 (C; Z) so that .S(l l] [ £(1) -
[0 () 0 0 0] 0 0 0 0 0,
.1(2)] [ r.(2) -
[0 0 0 0 0]
J(3)] -
[0 0 0 0 0]
[ ~141].r.(4) -
[0 0 0 0 0]
[ r.(3)
-
1 () () 0 0 '
0
I
0
0 0
0 0 0
I •
I •
Now suppose that we choose the double cover of the curve C in (6.24) for which the Schottky-Jung proportionality
O[g ~]u[? ~] = (const)17[~] 2 fiolds. Then for the associated Prym variety we have the identity
11rs
& & &l =
o=
11[&. & &
n
(6.28)
So; finally, recall that1 the Schottky relation has the form j, k,
n
1=0, I
(17[7
r ? gn"
2
... n
=
j,k,I=O, I
(,,r; ~-? wl/2 +
n
(17[/ ~
? ~nl/2.
(6.29)
j,k,I=O, I
If the relation (6.29) were satisfied for the Prym variety in question, then by Jt>.28) we. co-uld conclude that some .,[o o o 1] •t j k I 0
119
The Schottky Relation
O[~](u;
n)
:yanishes ~JJ(:::::OJ9!_ ~he: (Jacobian oLthe) curve C, that is, ~,)me JJE~](u-:::~n-not-it:l,:tht!:H~_tc(6.27-),J3utwehavealready-mentionedthat-for ·-wst c of the type occurring rn-{24)~no oihe~ everitlietafiinction vapishes a til = 0. Thus we can conclude thatthe r(llatjQIJ (§~79,1~ _n,qt ~ll.!J~p~J<:>r_llll x 4 - - • · ·"' · · period matrices, which is what we set out to do: This is not, of course, the complete story even for 4 x 4 P' riod ·1t1atrices. Since· Schottky's time there has _been .consid~ra))l~ _Pro~!e~ ' on this preblem of characterizing period matrices which- come' from ct rves, but the question is still far from being completely answered.
References We include here only those references cited frequently.
1. GRIFFITHS, P., and HARRIS, J., Principles of Algebraic Geometry. New York: Joh1 Wiley and Sons, 1978. · 2. GUNNING, R. C., and Rossi, H., Analytic Functions of Several Complex Variables. Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1965. .· 3. GuNNING, R. C., Lectures on Riemann Surfaces. Princeton, New Jersey: Princeton i ·niversity Press, 1966. 4. HIRZBBRUCH, F., ·Topological Methods in Algebraic Geometry; New York: S1 ·ing.erVerlag, 1966. · I 5. LANG, S., Linear Algebra. Reading, Massachusetts: Addison-Wesley Publishing Co 1971. 6. MuMFORD, D., Abelian Varieties. Tata Institute of Fundamental Research, B· mbay: Oxford University Press, 1970. 7. O'NEILL, B., Elementary Differential Geometry. New York:Academic Press, 1966. 8. SERRE, J.-P., A Course in Arithmetic. New York: Springer~Verlag, 1973. 9. SPRINGER, G., Introduction to Riemann Surfaces. Reading, Massachusetts: Addison·· Nesley Publishing Co., 1957. · 10. STEENROD, N, The Topology of Fibre Bundles. Princeton, New Jersey: Princeton U 1 vcrsity Press, 1951. 11. VANDER WAERDEN, B. L., Algebra. New York: Fredrick Ungar Publishing Co., 1· 70.
181
Index Abelian varieties, 134, 17 5 Abel's theorem, 82, 122, 127 Absolute, the, 17, 28 Accola, vi, 175 Affme-coordinates, 40 Affine set, 8 Albanese variety, 122 Analytic continuation, 53 Analytic manifold, 53 Andreotti, 145, 159 Anhannonic lines, 30 Associativity of cubic group law, 44 Automorphic fonn, 104; see also Modular fonns Automorphism, 100 of elliptic curve, 94, 95
Cartan, vi Cauchy integral formula, 55 Cayley, 17 Cech cochain, 70 Cech one-cochain, 119 Cell complex, 113 Character formula, 63 Characters, 167 Chern class, 118, 139 Chinese remainder theorem, 35 Circle, 30 Cohomology groups, 134 Complex conjugation, 50 Complex manifold, 66, 69, 70, 105 Component, 12 Cone,6,18,176 Congruence, 64 Conic, 8, 9, 17 through five given points, 11 Contact, 39, 77 Coordinates aff"me, 8 homogeneous, 8 Cotangent bundle, 55, 140 Cotangentspace,66 Covering group, 134 Covering space, 58, 89 Cremona transformation, 48 Cross ratio, 13, 18, 28, 52, 91, 93 Cubic, 12, 37, 73, 82, 89 Cup-product pairing, 116 Curvature, 20 Euclidean, 31 geodesic, 31
Base locus, 176 Basepoint, 55 Beau ville, vi Bezout's theorem, 12 Bitangents, 150 Blowing up a point, 4 7 Branch point, 52, 145; see also Ramification point
Canonical bundle, 140, 158 Canonical divisor, 140, 152, 175 ('_anonical mapping, 151, 175 Cap product, 124 Cardinality, 63, 67 183
Index
184
Fundamental domain, 80, 96, 109, 110,
Curve cubic, 95 . ·~r#!i.)s! ~J. ~-4· 95,96 -CUsp,154 · · · · ·· · Cusp form, 112
-Deformation,lc72~~;'.'
134 iilindamentiil group,161
.
·affinani" coiioiitill'(}gy';5-!r;-6t··
:. deRham com,p~«,l~;§L__ 'Desingula.Pzation, 78 Differential, 54, ~9
Gauss map, 151, 155 Gcnus,114, 149 Geodesic,20,27 ----==c· · G · +.;.:.; · · · · · -·~ - ~~~::s~~t curvature, 17 differential, 54 plant, 15 Riemannian, 17, 33
~~,;;::c~;;t-~t-~. ;~-:=~i::c-.·.•,,~,~~-:·z,~~~="'~~'-'C",==~:!~~::~;~f~. holomorphic, 152 meromorphic, 137 Prym,163 Differential equation, 59 Differential forms, 115, 125 Divisor, 84 Dolbeault complex, 66 Double complex, 70,115, 120 Double points, 154 Dual cuiVe, 24 Dual mapping, 152
Eisenstein series, 87 Ellipse, 1 Elliptic integral, 55 Euclid's f1fth postulate, 17 Euler characteristic, 55, 101, 145, 154 Euler's formula, 23, 38
Family of curves, 14 7 Farkas, vi, 161 Fibered product, 52 Finitely generated abelian group, 50 Fourier coefficients, 136 Fourier expansion, 78 Fourier series, 89, 136, 166 Fourier transform, 107 Framing, 94, 95 Frobenius mapping, 67 Function meromorph.ic, 51, 78, 80, 112, 137 rational, 75 theta, 73, .106, 132,159,164 trigonometric, 85 ·
Group of ratjonal points, 45 Gunning, 115, 118, 132
Hessian curve, 39, 152 Hirzebruch, 125 Hodge theory, 115 Homogeneous equation, 5 Hyperbola, 1 Hyperbolic geometry, 21, 26, 32 Hyperelliptic curve~ i51, 155
lgusa, vi Implicit function theorem, 54 Jnfmity, 40, 50 Inflection points, 3 7 Intersection pairing, 102, 162 Invariant, 14 global, 66 local, 66 Involution, 162, 175 Isometrics, 167 group of, 21 Isomorphism class, 93, 95, 97 Jacobi inversion theorem, 131, 136 Jacobian matrix, 65, 153 Jacobian variety, 113, 123, 127 Jacobi's identity, lll j-lnvariant, 102, 105
K:llller manifold, 66, 115, 125 K-Geometry, 30 .
185
Index Ko4aira,145,158
. ,.. ,
__ _ __ PoinC~U:~_mapping,_12_3,l29,_142 ,;,____ -.,.,~;·Pointe~ -- ~'':"1~'""'f~';jf-'c;~;i~~,.~~'-
----~--7.~~~~~~~+7~----~
_______ ,.:~-~~~ :.·:·:::~·--:\
Lattices, 165, i 70 Laurent series, 86, 88 Law of the sine, 13 Lefschetz duality, 122 Lefschetz fixed-point theorem, 65 Lefschetz number, 65 Lewittes, 144 Ue group, 131 Unear fractional transformation, 95 Unear systems, 138,176 Line bundles, 113, 142 ---·· Line integral, 56 Lorentz transformations, 22
Manin, vi, 65 Mayer, 145, 159 Metric, 120 Euclidean,32 induced,19 Modular forms, 73, 87,105; 106, 109,112 Moduli space, 53, 91, 96, 101, 145 Monodromy group, 92 Mordell's theorem, 50 Multiplicity of a point, 142 Mumford,vi, 134 Mystic hexagon, 11, 26
Nondegenerate plane curve, 24 Nonsingularity, 147 Normalization, 156
Orthonormal basis, 165
Parabola, 1 Parametrization, 75 rational, 76 Pen_cil, 1 76 Period, 59 Period matrix, 164, 168, 174 Perpendicularity, 6, 27 ~card-Fuchsequation,58,64,68 ~card's theorem, 98 ~card variety, 115, 122 Poincare dual, 116
--~.:~mit-¥Jr-.near~-4~:,: :;.:c·i~~-+:,,~
sing~,76,77 · Poisson summati<,>n formula, 89,10!. Polar coordinates, S6 Polar curves, 22 · Polar mapping, 24, 29, 37 Polar of~ point, 25
-;e.J!'Iltrr_a.@ e!2d'!Y,.kl~.1 •. 1H-~.:~_.,,
· · 'Power-series-expam.ion't 62" ·· Projection, 176. '-' Projective line, 18 Projective plane,.;, complex; 8 -- real, 148 Projective set, 8 Projective space, 103, 132, 151 · complex, 7 real, 5 Projectivization, 8 Prym varieties, 161, 171, 178-
<'' ;;::,-
Quadratic form, 133 Quadrics, 17 6 Quartics, 147 Quintics, 158
Ramification, 89 Ramification point, 43 Rational points, 33, 62 Rauch, vi Regular singular points, 61 Residue, 59 Resultant, 9 Riemann,133 Riemann relation, 58, 78, 117, 163 Riemann-Roch theorem, 68, 85, 87, )5, 128,136,151 ' Riemann's constant, 137, 139 Riemann sphere, 80 f Riemann's singularities theorem, 142 159, 178 Riemann's theorem, 136 Riemann's theta relation, 111, 165, J ,;7 Riemann surface, 147, 175 Scalar product, 165
Index
ISO
sciiotfky;-i6r···· _Sch!.lttky:::Jung, ..l.71, ..l18____ _ Schottky relation, 164, 174, 178 Serre, vi, 87, 105,132, 145 Serre duality, 68, 69 Sheaf, 67 of holomorphic functions; 114 Sheaf cohomology, 66;·-n'J"''' -S~L:.W.:;l~xn:UtiJWJt:t~1Ji~;J,~!lSI!ectiaf sequerideF1T5_______ - . / Spencer, 145 Spherical distance, 19 Spl~~riCal''geo~~g~:,~~"''·=·· __ ./: ; St
Tangent cone, 77, 160 Tangent line, 154
----__ _. ----~--~...-----
-----------Tlltlgenhpaces;-·1:50-~-~-
Tate, 44, 46, 73
---- rllet8c1laraCieriitics:I4o~-~43~us~-H7~ 158, 171, 177 Theta functions, 132 Theta-null, 89, 106, 109 ,.Xjurin, 158 ·Topological group, 43•· Torus,43 · Transvei:sality~ 65
__ Unbranch.~:double co\'ering, 160, 161 Unfversafcovering spaces, 133 Upper half-plane, 97, 103
van der Monde detenninant, 10
Wedge algebra, 124 Weierstrass p-function, 58, 85, 87
