2-Divisible groups over Z

LITERATURE CITED io

2. 3. 4. 5. 6.

N . N . Kholshchevnikova, "Limits of indeterminate forms and sequences obtained from a given sequence with the aid of a regular transformation," Matem. Zametki, i, No. 6, 887-897 (1974). R. Cooke, Infinite Matrices and Sequence Spaces, Macmillan, London (1950). H. Schaefer, Topological Vector Spaces, Macmillan, New York (1966). V . D . Mil'man, "Geometrical theory of Banach space," Usp. Matem. Nauk, No. 3, 113174 (1970). I . M . Glazman and Yu. I. Lyubich, Finite-Dimensional Linear Analysis [in Russian], Nauka, Moscow (1969). A. Zygmund, Trigonometric Series, 2nd ed., Chelsea, New York (1952).

2-DIVISIBLE

GROUPS OVER Z

V. A. Abrashkin

UDC 519.4

In this paper we construct nontrivial 2-divisible groups over Z which are isogenous to trivial groups and prove the following: THEOREM. If the height h of a 2-divisible group {G(~)} over Z is at most 4, then {G(V)} is isogenous to a trivial group.

Suppose G = Spec A(G) is a finite group scheme over a commutative ring R, and A(G) is a locally free R-algebra of finite rank. If am is a group R-scheme such that am(R) = Z/mZ and A(am) ~ ~ R , then am is an ~tale scheme over R.

If ~m is its Cartier dual, then A @ m )

=

1

R[x]/(x m -

I) and the group law on ~m is the mapping F: A ( ~ m ) - + A ( ~ m ) @ A

(~m), where F ( x ) =

X@X.

Suppose {G (v) } is a p-divisible group over R; it is called trivial if it is a product of powers of {~p,,} or {ap~}. Tate [i] posed the question of the existence of nontrivial p-divisible groups over Z. First of all, it is natural to investigate isogenies of trivial groups. It is easy to show that if {G(V)} is a trivial p-divisible group over Z, where p > 2, then any group isogenous to it is also trivial. In this paper we show that there exist nontrivial 2-divisible groups over Z, which are isogenous to trivial groups, and prove the following THEOREM. If the height h of a 2-divisible is isogenous to a trivial group. Proof. Since group over group over it follows

group {G (~)} over Z is at most 4, then {G(~)}

We will give the proof of the theorem only in the case h = 2. an extension of an ~tale group by an dtale group is again dtale, and an dtale Z has a trivial Z-scheme structure, it follows that if {G (~)} is a 2-divisible Z, G(I) = ~2 • a~ , then G (~) = ~ • ~ , i.e., {G (~)} is trivial. If G(I) = ~2 • ~, then at once from duality that {G(~)} = { ~ X ~=~}is also trivial.

Assertion i_~. If G is a commutative group scheme over Z of order 4 and period 2, then G ~ • ~ , ~2 • ~'~, P~• a2 , o r Go, w h e r e O--~2-.->-Go--~-a2---~O and A (Go) = A ( 9 ~ ) @ Z [i] ( s e e [2, A s s e r t i o n 7 ] ) .

A s s e < t i o n 2~. Any n o n t r i v i a l 2-divisible a 2-divisible g r o u p {G~) }, w h e r e G ~ ) = GO

group

{G(')} o v e r Z o f h e i g h t

2 is

M. V. Lomonosov Moscow State University. Translated from Matematicheskie 19, No. 5, pp. 717-726, May, 1976. Original article submitted March 4, 1975.

isogenous

to

Zametki, Vol.

This material & protected b y copyright registered in the name o f Plenum Publishing Corporation, 227 West 1 7th Street, N e w York, iV. Y. 10011. N o part o f this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording_or otherwise, w i t h o u t written permission o f the publisher. A copy o f this article is available from the publisher for $ Z 50.

] 429

Proof. Suppose G(u ----92 • a2, G(=) ----~' • a2', .... G(~) = ~=n x a2~, G0~+I)~= ~n+1 x a~u+1. Assume that '{G(0 ~)} is isogenous to {G(~)} with the kernel of the isogeny being a.n C G (n)o We will prove that G (n = Go, it being necessary to show that 2-1 (~n)/asn ~ G o , where 2-I (as~) ~ G(~+I). An extension 0 ~

~z X O~z - + G(~+1) --+ ~ ~s" • ~s"-+ 0 is determined

by the induced extensions

0 --* l~s • ~s - , 2 -1 (I~,0 "-, i ~ -~ 0,

(1)

0 -~ ~s X ~ -~ 2-I (~2n) -* a2~ -* 0.

(2)

We will prove that 2 -I (~s~) -- ~2~+i X a2.

Extension

0 -~ ~s -~ 2-~( ~ ) / ~

(I) is determined by the two extensions

-~ ~ , -~ 0,

(3)

0 --, ~s ~ 2-1 ( ~ , ~ ) / ~ --, 1~,~ - " 0

(4)

corresponding to the two projections of ~2 x a ~ on its factors. It is easy to show that ~ is induced by multiplication by 2 in 2-1 (~s~)/=~ and hence 2-~ (~tsn)/~= = ~ + ~ . We will prove that extension (4) is trivial. Let -2-* (~n)/~ = F, and consider the diagram 0 -~ ~s • =s -, G (n+*) -* ~

ii

•

"

a. ~ 0

J

0 -* ~ X ~s -~ ~n • e~ -~~ - I X =~-i -~ 0 It follows easily from this diagram that the inverse image of extension (4) under the mapping ~z~_~-+ ~,~ is trivial. Consequently, ~z,_~ X ~s ~ F, and we have the commutative diagram 0 --* as --, r~ --, p,, - ~ 0 0 --, a , --, r -~ ~,-~ 0

Therefore, F i = FI~,~-,, and since r ~-2 -~ (~s,)/~,, it follows that the image of r under the mapping 2: F - + F, is equal to ~s~-~ ; hence, the period of F~ is equal to 2. By Assertion i, F 1 -----~ X ~ ; henceF -----~,~ X ~z. Thus, 2-I (~n) -----~z~+~ X ~ , and therefore, G("+i)~= ~n+, X =zn+, if and only if 2-~ (=sn) ~= ~ X ~sn+~. If we carry out with extension (2) the same operations as with extension (i), we obtain that 2-* (~n)~= ~z X =sn+~ if and only if we have a commutative diagra m }{ 0 --, l ~ --, t

T

T 2"-~

r'-~ ~mn -+ 0

!

!

I

where F' = 2-~(=s,)/=2, ~, X ~ z ~ - ~ F , rl---F/~z~_~ , the period of F, is equal to 2, and F~ is a nontrivial extension. Consequently, 2-I (=sn)/~s,= I~i = G 0. We now consider 2-divisible groups {G(~)},G(~) -----Go. Let G~ and Go be group schemes containing a group subscheme H. We introduce the following notation: AmH(G~, G=) = G~ • Go/ A (H), where A: H § Gi • G= is the antidiagonal embedding of H into G~ x Go. Assertion

3.

We have the following exact sequences: a) 0 --- l ~ - . G(n)'" a ~ ---,~.0;

b) 0 --, Aml~n (O(n), }~zn+l )

---, O ( n + l ) --,-, a s ---, 0;

c) 0 --, I*~ "-* G (n+~)-~ G (n)• a~+~-~ 0, a2n

Proof.

This is an easy consequence

of [2, Theorem 3].

It follows that

A (G("))= A (Am~,_i(G ('~-~),~n)) ~ B~n) ~ . . . @ ~n-i

B~n) =

t

~

A (a~n)=

A(a~_ 0

~ ~Z

t @...

@ Z

Using the multiplication by m automorphisms, (m, 2) = 1, of the group G (n) , we see that ~(~0 ... ~ ~,,_~-B('~). Spec B(n) represents a fiber over the generator ~ n , hence ~ n acts

t~ansStively on it. Sequence c) implies that B('> ~ B(n-~) and ~ ~ acts on B(n) over B tn-~). The fact that O - + G ~ - ~ G - + G s - + O i s an exact sequence of finite group schemes ira-

430

plies that D ( A ( G ) ) = D (A ly that D (B(~)/B(~-I))=- 2 2.

(GI))IG'ID(A (G~))IV,I(see

[I, Proposition 2]); hence,

it follows easi-

Let us now turn to the construction and proof of uniqueness of a 2-divisible group {G (~)} such that G(I) ----G O 9 Assume that groups G(m), m < n, have been constructed and satisfy all requirements of the definition of a 2-divisible group. Assume also that the G(m) are uniquely 2m+l .--;~

determined by these requirements and that B(~) -~ Z [ ~/IL We will construct G(n) and show that all of the assumptions concerning the G(m) hold also for G(n). Assume that G (n) has been constructed. first the case where BQ = K, a field.

Let B(n) = B, BQ----B @ Q,_~ =

Ifl I.

Consider

Assertion 4. If K D Q ( ~ ) ~ Q is an extension of degree 2 n, and ~=n acts transitively on K/Q, then K/Q is a Galois extension, and if F = GaI(K/Q), then F can be equal only to one of two groups FI, F2, where F1 is Abelian and F2 is non-Abelian and possible only when n > 3. In each case the action of ~2n is uniquely, determined by the choice of some generator of ~2n(Q). Proof. Suppose K(H) is a normal extension of K and H C Gal (K(H)/Q) corresponds to K. Suppose ~ i s some generator of the group ~2n(Q), {a I..... a~} are points generating Spec K, , is the action of ~2n, and, for any h ~ H , h~ = ~. Since the action of ~2n is defined over Q, for any a ~ Gal (Q(Q) we have ~ * ~ i ~ ~ ( ~ * c@. If h ~ H , then h~ = ~, since K ~ Q (~); consequently, ~ * a I =: h (~, el) , and since ~2n acts transitively, we have h~ i = ai for any i = i, . . ., 2n; hence, h = id, i.e., K = K(H). To the tower K ~ Q (~) ~ Q there corresponds an exact sequence 0 - + Z/2Z-+ F =-->Z/2Z • Z/2'~-2Z-+ 0, where Z/2Z = {~}, Z/2n~'Z ~ {5}, with f$ = ~-i, ~ ~ ~_. Let ~ - ~, ~a = 5. Group F acts on ~=:n by means of the projection ~, and for any ~ ~ 2 n ( Q ) we have ~ = - - ~ , gg-----5~. Since ~zn facts transitively on K/Q, any point of spec K _ ~ Q / Q can be uniquely represented in the form go + ~, where go is a fixed point and ~r ~ ~.zn(Q). The action_of ~=n is defined over Q; hence, it can be described by a cocycle {~y}, where ? ~ F , ~ ~ z n ( Q ) , so that ygo = go + ~y. Obviously, ~y = ~yz if and only if y = y~ It is now easy to see that ord q ~ 2'*-;,ord 9 = 2, and, consequently, ~ and r generate P; if F = ~ , then 2~z ~- 4 ~ --~ 0, and if F = F~, then (2 ~- 2n-~)~ @ ~ = ~ 0, where ~ is a generator of ~an(Q) and uniquely determines the action of ~=n. Recall that B ~ Z [ ~ ] ~Z, B is an order of the field K, and D (B/Z IS])--2 ~. If K ~ Q (~) (~a), a ~ Z [~], then for any ~ ~ D i v Z[~]we have %, (a) -----0- (2) , since K/Q is ramified only in (2). It is known that h = ICIQ(~) I ~ I (2); hence, we may assume that c t ~ Z l ~ ] * LE)iVlA. Proof!. field Q(~).

If K/Q is a Galois extension,

then a ----~, e0, g0~, where e0 ----I @ ~ 2 .

Since h = ICI Q (~)I -- i (2), we may assume that a is a cyclotomic unit of the The proof of the lemma is now obvious.

If a = 5, then Gel (K/Q) = F~; if a = So~, then Gal (K/Q) = F=; and if a = ~o, then Gel ( K / Q ) ~ F 1, I'.: and this field must be excluded from consideration. It is easy to show that in each of the other cases the ring of integers of K has the form Z[~, a]; hence, B = Z[G, a] since D(B/Z[~]) = 2 ~. Thus, we have shown that if G (n) exists and BQ is a field, then there are two possibilities for A(G(n)). Let A be one of these two algebras and let G(n) = Spec A. Assertion 5. The action of ~=n described in Assertion 4 uniquely determines on G(n) @ Q the structure of a group scheme over Q, and {G(m)@ Q} ( I ~ m ~ n)constitutes a segment of a 2-divisible group over Q. Proof. We have over Q the obvious group structure G(n) (Q) =- ~ n (Q) X Z/2~Z, where s:a~n (Q)-+Z/2~'Z is some section of a=n. We must turn G(n)(Q) into a Gal(Q/Q)-module: 2n-1

A (G(~)) @ Q = A (Arn~.~,~_x)G(~-~), ~,~) | Q @ K 1

t 2n-1

A (a~,~)@ Q = A (a~.,~_0 @ Q

@ Q 1

431

Put gl = s(T), where i ~ a~n (Q)is a generator._ choose a generator ~i ~ ~ (Q) and define the action of Gal (Q/Q) on the fiber over i ~ ~2~ (Q)by means of Assertion 4. On the fibers ~ ~ ( ~ (m, 2)----I, define the action of Gel(Q/Q) by g m - s (~), ~ m - - - - ~ ~n (4)" On A m ~ n _ i (G(i), ~ n ) Q Q there is already a group scheme structure over Q, and if s : e~(~_i)(Q)-+ Z/2~-IZ c G (~-i)(Q) is the section of a~n-~ induced by the section s for G(n-i)~ G(n), then this structure~is uniquely determined by the choice of some generator ~ E ~2n-~ (Q)for the f i b e r over i ~ c~2n-~(Q) ~ =~n (Q) 9 Obviously, G(n) (~) is a Gel (~/Q)-module if and only if 2~, = ~i. It is easy to show that, up to isomorphism, this group scheme structure over Q on G(n)~ Q does not depend on the choice of the section s and the generator Pl, and {G(m) ~ Q} (i ~ rn~. n) constitutes a segment of a 2-divisible group over Q. Assertion 6. If B----Z [~/~], then G(n) possesses a group scheme structure over Z, which induces the group structure on G(n) ~ Q d e f i n e d in Assertion 5, but if B = Z [e~0~] , then G(n) cannot be turned into a group scheme. We will prove this assertion a little later. Thus, we have completed the construction and proof of uniqueness of a 2-divisible group {G(0~)},G(00 -- Go, under the assumption, though, that BQ is a field. We intend to remove this restriction. Let BQ----Q ( ~ ) ~ Q (~), ~n-1

AQ = A

(Amain_t(-G(,~-l), ~a.)) ~ Q @ (Q (g) @ Q (~)), 1

G = Spec AQ, and suppose {Gim)) (~ ~ m_~ n ) c o n s t i t u t e s

a segment o f a 2 - d i v i s i b l e

group o v e r

Q, where G~m~-- G(m ) @ Q for m < n and G~ n) = G. Suppose g 0 ~ G ( Q ) is a point lying in the fiber over i ~ a ~ ( Q ) . If n > 3 , then, putting ?go = go + ~v, where ? ~ G a l ( Q / ~ ) , we obtain,_ as in Assertion 4, that 2~= + 4 ~ -~ 0 and ord ~= : 2~-e . Consequently, { ~ } ----~ - i (Q) ~ ~ n (Q) but then the point 2n-~(go) is defined over Q, which contradicts the fact that n(~) . (~)/Q)= . .{~}, a. group scheme structure over Q on g!~)is ~i ----Go ~ Qby9 If n = 2, then Gal (Q determined ~ ~ ~4 (Q), ord ~, = 4 and {G~i), G~Iz)}constitutes a segment of a 2-divisible group over Q. If B is an order in the Q(i)-algebra Q (i) @ Q (i),B ~ Z [i],'vi acts on B/Z[i], and D(B/Z[i]) = 2 ~ then it is easy to show that B = A (~) @ Z [i]. Let A -~ Am~,(G0, ~a) @ B @ B and G = Spec A. G cannot be turned into a group scheme over Z, i.e., G @ Q ~-G~~).

Assertion 7.

To prove Assertions 6 and 7 we require the following two assertions. Assertion 8. If G is a group scheme over Q of period 2n, 0-+ ~ n - + G (n)-+ a~n-+ 0,then G coincides with its dual. %

Proof. Let G be the group scheme dual to G over Q, and G x ~ - ~ ~ n the corres__ponding_ nonsingular pairing over Q, which means that there_is defined_ a pairing G (Q) x ~ (Q)_-+~n (Q), compatible with the action of Ga] ~/Q). If ~ ~ n (Q) ~ G (Q) is a generator, go ~ G (Q) is such that ~go is a generator of ~2n(~), and p, go are analogous points for ~ (Q), then e (~, ~) = i, e (go,~)---- ~', e (~t,go)---- ~ , where ~, ~' are primitive roots of unity of degree 2n. Using the bilinearity of the pairinE e, we can alter p and go, leaving unchanged their projections in ~ # ~), so that e (go,~) = ~-*, e (go, ~0) ----~ , and such ~, ~o are uniquely determined by p, go if ~ 0 ~ ~ (Q)is fixed. Since e (go, $0)---e (og0, ~g0)= ~ , it follows t h a t ~ = ~ for ~ Gel(Q/Q); hence, the mapping ~-+~, g0-+~0 defines an isomorphism of G and G, defined over Q. Thus, if o6 G it is possible to introduce a group scheme structure over Z, where G is the scheme of Assertions 6 and 7, then G = ~. Assertion 9. Suppose G = Spec A(G) is a flat scheme over Z such that G O = G @ Q is a finite commutative group scheme over Q, G O = ~q, and e: GQ X ~Q -+ ~t~n X Q is the corresponding nonsingular pairing. Let {hi} be a basis of the Z-algebra A(G) and {~} its dual basis in A (G) @ Q relative to the nonsingular form Tr: A(G) § Z, where Tr $ = ~(g) for any ~ ~ A (G). Then there exists on G a group scheme structure such that G ~ Q -- GQ and G = ~if and only if a) D (A (G)) = t d e t ( e ( g , h ) ) l ,

g,h~G(Q);

b) NTg,h ~ (g)e (g, h)Ni (h) ~ Z f o r a l l

i , j.

Proof. The pairing e is defined by an element e* (x) = ~N~ ~ e~ ~ A (G) ~ i (G) ~ Q, where A(~te,~) ~: Z[x]/(xe~-j), the e ~ A (G) ~ Q form a basis of A (G) (~ Q, a n d ~ N ~ (f)e~ (h) -- e($,h) for g, h ~ GQ (Q). Obviously, for there to exist on G the desired group scheme structure over Z it is necessary and sufficient that {ei} be a basis of A(G). Since D (A (d)) ~ det (Tr (q~]~)) =

432

det'~(N~ (g)), and since the equality ~

(g)e~ (h) = e (g. h) implies that det (~i (g))'det (si(h)) =

det(e(g, h)), then the fact that D(A(G)) = det(e(g, h)) implies that D(A(G)) = D({ei} ). Consequently, for {~i} to be a basis of A(G) it is necessary and sufficient that e~ ~ A (G). If O g ~

A (G) ~ Q are such that Og (h) =

If/~ae (g, h)qi (h) = a, (g),

~g.h, then

then FN, (g)ei = ya, (g)~l~.

The matrix (~ (gj))' is inverse to the matrix

(qi (gj)) ; hence, ai = ~i (~gD~ (g)aj(g))Ni= ~j (Ym a 11~ (g)e (g, h)Ni (h))~j, which proves the assertion. For the schemes of Assertions 6 and 7, condition a) is satisfied automatically. us verify condition b).

Let

Suppose B = z[V~]. Choose a basis {hi} of the algebra A(G(n)) so that it consists of bases of the fibers of the mapping G(~)-+~2n , i.e., Ni (g) v~0 only if g belongs to the corresponding fiber over aan. Note that the sets {~il the ~i correspond to a fiber over a~_~ C ~=~} form a basis of A (Amv~_~ (G~-~, ~n)) and the corresponding dual basis for these nj is a basis of d (G(n-~))a~n-~Xe~n C ~4 (G(~'0); hence it is necessary to verify b) only in fibers over ~ n \ ~=n-~ , and since all of these are isomorphic, we may assume that ~i, ~j belong to a single fiber. Recall that Gal (Q (~)/Q) = {z} ~< {~} acts transitively on this fiber, and _~ = g o - 2~, Tg = go ~- .tq, where go is some point of the fiber and ,~ is a generator of ~ n (Q). If 8 = ~ , 9f:

Gal (Q (O)/Q) -+ (Z/2n+lZ) *,

f (~o b) = (--j)aSb, then ~ (0) = Of(~') and ~(g0) = i - - f(q~)2 P~ q- go, where

~ Gal (Q (0)/Q) = F. If e(g0, ~) = 0~, then e(~ g0,~g0) ----0~(~(~)-~(+))( % ~ P). A basis of the fiber consists of ~]0 =: l, N~ = 0 . . . . . '~gn-1 -- 0~n-~ and we have No = 2 -n, ~i = 2-n0-~. . . . . %n_i = 2-~ 0(-=n-~); hence ~}~(cpgo)~ 2-n 0-~(~)~(0 ~ l < 2~). Condition b) now assumes the form

2-2n

~-Ja, b~(Z/2n+l Z) * O(k-Oa-(~+j)b

-- , (2-" Y,

0

(2-" Y,

J,

0

Z.

Consequently, when B = Z [ 8 ] we o b t a i n t h e g r o u p s c h e m e G ( n ) . The v e r i f i c a t i o n of the that condition b) i s n o t s a t i s f i e d f o r B = Z [~, ~e,-~] and B = A (~2) @ Z [i1 i s o m i t t e d .

fact

T h u s , we h a v e shown t h a t a n y n o n t r i v i a l 2-divisible group of height 2 defined over Z is isogenous to a 2-divisible g r o u p {6~0~)}, Gg1 ) = Go, w h i c h i s u n i q u e l y d e t e r m i n e d b y t h e r e q u i r e m e n t Gg~) = Go. O b v i o u s l y , {Gg~)}is n o n t r i v i a l , b u t i t c a n b e shown t h a t i t i s i s o g e n o u s to a trivial g r o u p w i t h ~ , . C Go as k e r n e l o f t h e i s o g e n y . The t h e o r e m i s p r o v e d i n t h e c a s e h=2. Remark.

With certain

changes,

the

proof

of the

theorem

can be applied

t o any 2 - d i v i s i -

h-n

ble group {G('~)} over Z such that 0-+

~e-+G(t)-+O ~ 2 - + 0 ; in particular, if the height of 1

I

{G(~)} is equal to 4, then this exact sequence holds for G (:) [2, Theorem 2], and we obtain the proof of our theorem for h ~ 4 . In conclusion, I would like to express my deep gratitude to I. R. Shafarevich for his guidance. LITERATURE CITED i.

2.

J. T. Tate, "p-Divisible groups," in: Proceedings of a Conference on Local Fields, Driebergen (The Netherlands), 1966, T. A. Springer (ed{tor), Springer-Verlag, Berlin-Heidelberg--New York (1967), pp. 158-183. V. A. Abrashkin, "A good reduction of two-dimensional Abelian varieties," Akad. Nauk SSSR, Ser~ Matem., 40, No. 2, 262-272 (1976).

433