Subvarieties of moduli spaces - Springer Link

Inventiones math. 24, 95 - 119 (1974) 9 by Sprmger-Verlag 1974

Subvarieties of Moduli Spaces Frans Oort* (Amsterdam and Aarhus) In this paper we try to decide along algebraic lines whether moduli spaces of abelian varieties or of algebraic curves contain complete subvarieties. In Theorem (1.1) we consider abelian varieties and curves of genus three in characteristic p :~ 0. In Section 2 we use m o n o d r o m y in order to show certain families of abelian varieties up to a purely inseparable isogeny are isotrivial; probably the results (2.1) and (2.2) are special cases of more general facts about/-adic monodromy. In Section 4 we answer a question raised by Martin concerning a possible generalization of the fact that two supersingular elliptic curves over an algebraically closed field (of characteristic p) are isogenous. We abbreviate abelian variety(ies) by AV; we use X t for the dual, and 2 for the formal group of an AV X.

1. Moduli Spaces in Characteristic p which Contain Complete Curves All moduli spaces in this section considered will be moduli spaces in the sense of [21] with a field k as base ring. Suppose k = ~ , the field of complex numbers; the coarse moduli scheme Ag of principally polarized abelian varieties of dimension g contains a projective subscheme of dimension g - 1 ; this easily follows from the existence and the properties of the Satake compactification of Ag as for example Shafarevich remarked (cf. [32], p. 111). As Mumford pointed out to me, along the same lines it follows that if k = r and g > 3 , then the coarse moduli scheme Mg or irreducible, non-singular, complete algebraic curves of genus g contains a complete algebraic curve; this can be seen as follows: let Mg be the coarse moduli scheme of good curves of genus g (stable curves of genus g whose Jacobian variety is an abelian variety), and

j: M's~ Ag the Jacobi mapping; let A s ~/~g c IPN_bethe Satake compactification, and denote by C the closure ofj(Mg) in As; each component of C \j(Mg) has codimension at least two in C: this is seen to be true for components of j(M'g)\j(Mg) by counting moduli, and for components of C\j(M'g) it follows from Ag'-,Ag= U Ah; thus we can intersect j(Mg)=IPu with a h 1 connected by one normal crossing); thus it is not clear the method above can be applied to this or another compactification of M~ (and note that j "contracts" U to a lower dimensional j(U) c Ag). Theorem (l.1). Let k be field, char (k) = p 4: 0. a) The coarse moduli scheme A,. a of abelian varieties of dimension g plus a polarization of degree d 2 has a projective subscheme of dimension at least 8 9 1). b) Suppose k is algebraically closed. The coarse moduli scheme M 3 of complete irreducible non-singular algebraic curves of genus 3 contains a complete algebraic curve. Definition (1.2). Let X be an AV over a field k, char(k)=p; we say the p-rank of X equals f notation: p r ( X ) = f if Ivx(g)l = p f , where K is an algebraically closed field containing k, and pX= Ker(p: X - , X ) ; we say X is ordinary if f = d i m X , and we say X is very special if f = 0 . Let Y be a scheme over the prime field IFp; we denote by F=Fy: Y~-~ Y the morphism obtained by raising all sections of (9r to the p-th power; the induced homomorphism F * = h : HI(y,(gr)--~ HI(Y, C0r) is usually called the Hasse-Witt transformation, Lemma (1.3). Let X be an AV over an algebraically closed field of characteristic p of p-rank f; then the number of elements vE H I (X, (gx) such that h v= v equals pl. Proof By duality of abelian varieties, the p-rank of X equals f if and only if the semi-simple rank of h equals f, which, by [10], p. 488, Satz 10, yields the result (also cf. [22], p. 143, Corollary).

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Lemma (1.4). Let S be an irreducible algebraic k-scheme, and X ~ S an abelian scheme over S; let f be the p-rank of the generic fibre; for any field K, and for any s~S(K), pr(X~) 2 ) ; in the proof of [32], Theorem 5 in case of characteristic p + 0 the trace A* need not be a subvariety of A (cf. [ 16], VIII.3, Corollary2 on p. 216).

Proof of (2.1). Suppose k', C', X' as in the theorem. Because C' is of finite type over k', and X' is of finite type over K (and X' of finite type over C'J, there exist k, C,X such that k~k', C'-=C| X'=X| and such that under the identification T~X ~ T~X' (choose a k-embedding k ' c k 's) the groups Gal (KS/K k s) and Gal(K'5/K ' k 'S) have the same image, and such that k is the perfect closure of a field finitely generated over its prime field. Thus it suffices to prove (2.1) in case k has this property. In order to prove the first conclusion of the theorem, we replace k by a finite extension, again denoted by k, so that all places of C where X has bad reduction become rational over the new field, and such that at least one more point of C is rational over k (note that these properties are satisfied already in the second part of the theorem). We choose N = 12; the group scheme N X = K e r ( x N: X - , X ) is constant over an extension L ~ K , L=k(D); because (l, char(p))=l this extension (or: the covering D--, C, klD)=L) can be chosen such that it is separable and unramified at all places where X has good reduction; we denote the new fields again by k, K, with K = k ( C ) , and we arrive at a situation where ~X is a K-constant group scheme, i.e. Gal (K~/K) acts trivially on nX (KS).

First Step. For every a e Gal (K~/K k s) all eigenvalues of p a ~ Aut (T, X) are equal to one. This we prove as follows. Let M c C ( k ) be the set of points where X has bad reduction. Let Kk~L~K

~,

104

F. O o r t

where L is the union of all abelian extension of K k~, of degree a power of t, unramified at all places (discrete valuations) of K kS/k ~ outside M; clearly Gal(K~/L) is a characteristic subgroup of Gal(KS/Kk ~) (i.e. invariant under every automorphism). Let J-= JacM(C) be the generalized Jacobian variety of C constructed with M as conductor (all multiplicities of points in M equal to one), el. [30], Chap. 5: this is a k-group variety, there is an exact sequence of k-group varieties 0 ---+ ( ~ m ) Iml - t =

H --+ J--~ Jac (C) -+ 0

where Jac(C) is the (ordinary) Jacobian variety of C (and H has this form because all points in M have multiplicity one and M=C(k)). Because J is a k-group scheme, T~J is a Gal(kyk)-module in a natural way. Because Gal(K~/L) is characteristic in Gal(KS/K U), and abelian, 0

--~ Gal (KS/K k ~)

, Gal (KS/K) -

, Gal (k~/k)

,0

Gal (L/K k s) ~- Tt J .... e.. ., Aut ( TtX) the action by Gal(k~/k) via inner conjugation inside GaI(K'/K) on Gal(KS/Kk s) induces an action on Gal(L/Kk'); there exists a natural isomorphism of Gal(kS/k)-modules

Gal (L/K k~)'~ TtJ ; this can be seen as follows: a finite abelian extension of K U is induced by an isogeny of a generalized Jacobian of C (cf. [30], VI.11, Prop. 9); if such an extension is contained in L, we can choose the support of the conductor to be contained in M (cf. [301, VI.12, Lemma 1); in fact in that case, the degree of the extension is a power of l, hence prime to char(k), thus the conductor can be chosen contained in M (cf. [301, p. 128, Example 1: if M c N, support IN) = support (M), then Ker (Jac N(C) -~ Jac M(C)) is a unipotent group scheme, hence has no/-torsion points): the isogenies • l': J ~ J are cofinal in the set of all isogenies over J of degree a power of ! (ifG ~ J is an isogeny with kernel annihilated by • Ii, then its factors multiplication by I; on J), thus the natural action o f / - p o w e r torsion points on J on the corresponding coverings of C induce an isomorphism (use [301, VI.11, Proposition 10) between G a l ( L / K k ~) and TtJ; note that the canonical morphism C \ M - - ~ J is defined over k, because moreover C ( k ) \ M + ~ , and it follows that the action of Gal(kS/k)

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commutes with the isomorphism. Thus we obtain a commutative diagram as indicated above. Because k is the perfect closure of a field finitely generated over its prime field, there exists an element o-e Gal (kyk) such that its action on Tt J

r: Gal(U/k) -* Aut (TiJ) has the property: r (a)~ Aut (T~J | II~;) has no eigenvalues equal to a root of unity and the characteristic polynomial (of any power) of r (g) has integral coefficients (cf. [25], Lemma 3.2). Let a'eGal(KyK) be such that

Gal(KyKI-*Gal(KU/K)'~Gal(kyk),

a'~--~a,

and write p(a')=:SeAut(TtX): let asTid, p a=:A~Aut(TiX); choose Q ~ Q~ containing all eigenvalues of A and of S. Consider a~:=(r(a-i))(a), and A , : = p a , ; because of the hypothesis made, the matrices A~ all commute; because of the way ~ acts on Gal (L/K k') ~- T~J,

A i = S - i A S ',

i~7Z

hence any two of these matrices have the same set of eigenvalues. Consider all infinite sequences U=(...,U_t,U0,Ul,U

of eigenvalues of A; let E: = T~X |

2, ...)

Q, and let

E , : = ('1 Ker(Ai-ui)cE; fix ie2g, and consider K e r ( A , - u ) = E~,, where u is an eigenvalue; clearly

0 E~=~. Eiu u

u

(on Eiu the transformation A~ has eigenvalue u and on ~ E~ the map A, V:~U

has no eigenvalue equal to u); thus it follows (because the Ai commute, and dimlE)< o9) that the set U of sequence u with Eu+O is finite and their sum is a direct sum u~U

u~U

Note that

S(EO= Esl~, where (S(u))i:=ui+l, i.e. it is the shift to the left:

x e E u ~ SS-i S-,ASISx=Su~+~ x;

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F. Oort

thus S(E.)cEs~.), thus S F ~ F ; because S is invertible SF=F, thus S (E.I = Espy); thus S induces a permutation

S: U--,U. As U is finite, there exists an integer n so that S"(Eu)=Eu for all ue U. Let T=S", let z=a", let P be the characteristic polynomial of r(z); note P has coefficients in ;r and P(1)4:0 (because r(a) has no eigenvalues equal to a root of units, hence r(z) does not have 1 as eigenvalue). Let 2 be an eigenvalue of A=p(a); choose u e U with Uo=2 (this is possible because Ker (A - 2) 4: 0, etc.); as T (E,) = E~, for all x e E,,

(T-iATi) x=)~ x; note that (P(r))(a)=0, thus

o(P(z)(a))= 1, v (e(~)(a))(x) =,~P(1) x = 1,

and P(1) being non-zero, this implies that ,~ is a root of unity. Because Gal (K~/K) acts trivially on NX (K ~)= TzX/N (TtX), A -= 1 (rood N) (considered as Z~-linear maps), and 2 being an eigenvalue of A, we conclude ;~-- 1 ( m o d l) (in the ring of integers of 11~(2)), thus 2 is an /-power root of unity. Suppose 24:1; in the ring of integers of ~z(2) the element n = l - 2 divides l, and det(A - 2 1 ) = 0 , thus 7"g2g-]-7"C2g-1 . N . a l + . . . + rc2g-i. Ni.ai+ . . . . O, ai~TZt, which implies (because ~z I N = la): nzg_0 (mod ztZg+l); however, from 2 4 1 it follows that n is a uniformizing element of •t(2) (e.g. cf. [351, 7.4.1), contradiction, and hence 2 = 1, which concludes the proof of the first step.

Remark. In case l > 2 g + 1, one can choose N=l: if Gal(KS/K) acts trivially on tX (KS), again 2 is a/-power root of unity, its degree over Qt

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107

is at most 2g because it is an eigenvalue of AEAut((7Zl)2 g) and [Qt(0:Qt] > l - 1 for any/-power root of unity not equal to one, thus it follows that 2 = 1.

Last Step of the Proof. Consider Y: = Trr/k X, i.e. this is an AV over k, plus a purely inseparable homomorphism

t: Y|

X

(with finite kernel) which has a certain universal property (cf. [16], VIII.3, Theorem 8 on p. 213, and Corollary 2 on p. 216); we want to show t is an isogeny (i. e. t is epimorphic); let Z be an AV over K defined by the exact sequence Y| , Z----+ 0; because t is purely inseparable~ and /4=char(k), we obtain an exact sequence

0-+ T~Y--* T~X---, TzZ--,O,

the action of GalIKS/Kk s) on "FlYis trivial and by the first step we conclude all images of p: Gal (K'/K k s) --+ Aut (T~Z) have eigenvalues all equal to one, and this image is commutative: this implies that if Z 4=0, there exists 04=rE ~ Ker{p(b)- 1), the intersection taken over all bEGal(KS/Kk s) (or all be Tfl); we define Z':=Tr,clkZ,

thus

TtZ'~--+T~Z:

by the Lang-N6ron version of the Mordeli-Weil theorem we conclude (cf. [171, p. 97, Theorem 1) that vETrZ', thus Z'4=0; let

Z": = Z/Im(Z' | K-+ Z): by [16], II.1, Theorem 6, we conclude there exists an isogeny (Y| g ) @ (Z'| KI@Z"--+ X, thus the image of the K/k-trace of X contains the image of (Yq3 Z')| K (universal property of the trace), the K/k-trace of X is t: Y| K--+ X, thus Z ' = 0 ; thus Z = 0 , thus t is epi, i.e. t is a purely inseparable isogeny, which concludes the proof of (2.1).

Remark (2.7). Probably the methods above can also be used to prove the statements (2.3) and (2.2) with C replaced by (the function field of) an AV over k. Remark (2.8). In case genus (C)=0, the result (2.2) was proved by Grothendieck (cf. [83, pp. 74/75, Proposition 4.4). Theorem (2.1) was

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F. Oort

inspired by the result of Grothendieck which says that an AV with sufficiently many CM is isogenous to an AV defined over an algebraic extension of the prime field (cf. [22], Theorem on p. 220, and cf. [26]). Possibly these theorems are special cases of more general results. Remark (2.9). It is not difficult to construct examples of non-trivial families X--* C over a complete elliptic curve over any given base field: take an elliptic curve D, a point de D (k) of order (d)> 1, and an AV Y over k with aeAutk(Y), with order (a)=order (d); let C = D / ( d ) and let X be the quotient of D x Y by the equivalence relation (u, y) ~ (u + d, a (y)).

3. Subvarieties of Moduli Spaces Defined by Kodaira Surfaces 1 In the previous section we showed that a fine moduli scheme of algebraic curves in characteristic zero does not contain a complete elliptic or rational curve (use Corollary (2.3) plus Torelli's theorem which says that the Jacobi-morphism from moduli schemes of nonsingular irreducible curves to the moduli schemes of principally AV is injective); in this section we follow a construction suggested by Parshin of certain surfaces, so that we can prove: Theorem (3.1). In any characteristic there exists a number g such that the coarse moduli scheme Mg of (irreducible and nonsingular) algebraic curves contains a complete rational curve E c Mg. By a Kodaira surface, or an irregular algebraic surface, we mean a complete (nonsingular) algebraic surface M plus a smooth morphism h: M--~D onto a complete nonsingular irreducible algebraic curve D; such surfaces were constructed by Kodaira (cf. [-15], also cf. Kas, [14]). The family h of curves parametrized by D defines a morphism M : D --+ Mg, where g is the genus of each of the fibres of M ~ D; we construct M and D such that D --~ Mg factors through a rational curve. (3.2) Parshin's Construction (cf. E28], pp. 1 168/1169, and 1163-1 167). Suppose given a field k, a smooth, irreducible, complete algebraic curve B over k, two positive integers m, n both prime to char(k); then we arrive at an etale covering D ~ B and a Kodaira surface M-~ D; moreover if genus (B)>2, and m > l , and n > l , then no (etale) covering of D makes the fibration M---,D trivial (equivalently: at least two fibres of M ~ D are non-isomorphic). The construction is performed as follows:

D, P~C,_

-

,JxB

- cartesian 1~• • ie BxB=-

i ~JxB

1 I thank K n u d Lonsted for drawing my attention to the paper by Parshin and for stimulating discussion on this topic.

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consider J:=Jac(B), let B---~J be the canonical embedding of B into its Jacobian variety [e.g. cf. [30], V. 6 and V. 9), and define i so that i(b, bj=(0, b) (i.e. the "variable" point b is used to define the canonical morphism ib: B x {b} -* J x {b}); let C be the pull-back of i(B x B) by multiplication by n on the B-group scheme J • B, and let P= )l(b, b)

D w

be the fibre over (b, b) so that P: B x B ~ C is the section defined by P = {0} x B c C ~ J

x B;

then C is irreducible (cf. [30], VI.11, Proposition 10), and n is an etale covering because char(k) does not divide n; note that the degree of the relative Cartier divisor D c C -~ B x B is n 2 "g.... (B)_I, SO bigger than one if n > l and genus(B)>0; because D is an etale covering of B x B, we can choose an etale covering D ~ B CI~ ,

C x~D= :i

B

~[

B ~pro~2 ' B

P,

l

*-

D

and sections P~: D-~ Y so that

D• (in case all points of nJ are rational over k, we can choose D = B ) ; we write P for the section P x B D : D--, Y, and using D,P,,P~, and the integer m we construct M

if" y 4/I h "\.

D as follows: first, let ( e D be the generic point, with fibre Y~over this point: on this fibre there is a point P(() and a divisor 6 = ~ P~((), thus there results a canonical morphism yr = y~.,, Supp (6) --+ Ja% ( Y0 = J~ into its generalized Jacobian variety with respect to 6, sending P(() onto the zero point o f 0 6 J ~ (and this morphism is defined over k(~), because

I lO

F Oort

P(~) and all Pi(0 are rational over k(O); then we construct

I lartsanl" the etale covering Me~ yr defined by multiplication by m on J~ (again t we know the covering is irreducible, and etale because char(k) does not divide m); the curve M~ is the unique smooth complete irreducible curve containing Mg; the minimal fibring h: M -~ D having M; as generic fibre exists, it is smooth (use the fact that Y--, D is smooth, and D ~ P etale over B x B, and apply [28], p. 1164, Lemma 10), and for every point d of D, the fibre M a can be constructed as follows: 2V6=M ~

, J~,.~

r~ = V -

'

,J~,d}, i ( P ( d ) ) = 0 ,

with Ya~ = Yd \ Supp 6 (d) (this follows because Ma is the unique smooth complete curve containing M ~ c M); note that if deg (6) >1, and m > 1 then the morphism ramifies exactly at all points of ~ (d); this can be seen as follows: all points in 6(d) have multiplicity one, thus Java) is an extension over Jac(Yd) with kernel L = (113,,)aeg~a)-1; thus Lc~ Ker( x m: Jo~a~ J6~a))4=0; because the unramified coverings of Ya correspond bijectively with isogenies over Jac(Ya) (cf. [30], VI.12, Corollary on p. 128) it follows fie ramifies at each of the points Pdd)~ Ya; ~

Bx{b}

Cb ,

-

C*------Y~

~'~B*

Ya'

--

M~

~ -

M

D

let d be a point of D mapping onto a point b of B, if genus(B)>0 and n > 1 (so deg(~)> 1), and m > 1 then the composed covering Md--* B x {b} ramifies at the point (b, b).

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Lemma (3.3). Let k be a field, B and Z complete smooth irreducible algebraic curves over k, and T a k-prescheme. Let q~: Z • 2 1 5 be a family of morphisms of Z to B parametrized by T (i. e, q9 commutes with the projections on T ) ; / f genus (B)> 2, then q~ is locally constant on T. Proof One way of proving the lemma is the following: suppose T = Speck [e], then all cr Z • T ~ B • T deforming a fixed (q~o: Z -* B) = ~P@kiwik are in 1-t-correspondence with F(B, ~e~ (CoB, q~o*(gz)) (here ~ stands for the sheaf of germs of k-CB-derivations), and one can show this set of sections to be trivial in case @e~ (60B,s is a negative line bundle (its degree is 2 - 2 g ) . Another way of proving is the following. Because the functor of rnorphisms from Z to B is representable (Grothendieck, cf. S6m. Bourbaki 13, p. 221-20 (1960/61)), and smooth (B is a curve), it suffices to prove the lemma in case k is algebraically closed and T is connected and reduced (and this is the case which we need in applying the lemma). Let t~ T(k), and consider Z ~- J ,Jac(Z)

jo~ Bc_i

I o, = x+as ,Jac(B);

because of the Albanese property of the Jacobian variety (of B) the morphism ~, results, making commutative the diagram; let a,=~,(O); because of the rigidity lemma ([21], Corollary 6.2) the morphism 2 = ~ , - a , (is a bomomorphism which) does not depend on t; fix s e T(k) and consider bEB, (ib)~-,(ib+a~-a,)=:~,(b) because 2jZ + a t = i B = 2 j Z +as we conclude we obtain a family of automorphisms

y:B•215

(b,t)~-~(7,b,t)

(we identified B and B~-iB); because genus(B)>2 we know B has only finitely many automorphisms thus T being connected (and reduced), the family ), is constant, thus a~=at for all t, thus q~ is constant. Q.E.D. Now suppose, notation as in (3.2): genus(B)>_2, m> 1 and n > l ; let T ~ D be a covering, then M x D T is not trivial; in fact, suppose M x o T~_Z x T; then apply (3.3) to (M--~ Y--~ C-~ B• B ) x o T ,

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F. Oort

thus concluding that for each de D the covering Me=(M xDr)t=Z-+ Yd--~ C~--~ B x [b} with tP, d~-~b does not depend on t; however, we have seen that this covering ramifies at b in the given situation, thus while t runs through T, the point b is not constant, and Md-* B x {b} cannot be a constant morphism; hence the fibring h: M --* D cannot be trivialized by a covering of D, and the construction and the claim in the first sentence of (3.2) are established (the arguments essentially can be found on p. 1169 of [28]).

Proof of(3.1). Choose a curve B with genus (B)> 2, and a (finite) group G c Aut (B) with

q: B~. B/G,-,IP 1

(such an example is easy to construct, e.g. take any curve B' with genus(B')>2; its function field k(B') is a separable extension of k(IP1), and take for B the normalization of B' in some finite Galois extension of k (1P1) containing k (B')). Using the construction explained above, with the help of integers m and n (prime to char(k) and both at least equal to two), we arrive at a Kodaira surface M

B ~ ,r

IW . . . .

D

,,E ~

,Mg;

thus we obtain a morphism M: D - * M~, with g=genus(M/D), and we claim this factors through IPt; because h is not locally trivial in the etale topology, the image M(D)=EcMg is not a point, but a curve, and because of Liiroth's theorem we conclude, IPI--~EcMg, this to be a rational curve; the proof of the factorization follows because: if d, e e D (k), and qfd=qfe then Md~-Me; indeed in that case there exists o-EG with f d= ale: the morphism

a: B x {b}-* Bx {~b} extends to a commutative diagram B x {b}

1 B x {~rb}

-~Jac(B)

I" -, Jac (B)

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thus arriving at a morphism a,:

Cb ~ C~b,ff, Pb=P~b,ff,Db=D~b:

this results in an isomorphism a , : M ~ ~, M ~ and because M~ is the unique smooth curve containing M ~ (and the analogous statement for M ~ and M~), we conclude M ~ - M ~ , and Theorem 13.1) is proved.

4. Supersingular AV In this section all fields considered will be of characteristic p + 0. Consider an elliptic curve E over an algebraically closed field k; the following properties are equivalent: a) E is very special [i.e. E has no points of order p); b) Endk(E) has rank 4 as Z-module (note k is algebraically closed); c) the formal group of E is isomorphic to Gt, ~ (notation of [19], p. 35); d) ~p is a subgroup scheme of E (here ~p denotes the kernel of the Frobenius homomorphism F: 6~ ~ G~). An elliptic curve E over a field l is said to be supersingular if it satisfies these properties over an algebraic closure k of l; note that a supersingular curve is isomorphic over k with an elliptic curve defined over the field IFp2, the quadratic extension of the prime field; note that any two supersingular curves are isogenous over an algebraically closed field (in fact they are already isogenous over an extension of degree 12 of a common field of definition, cf. [34], pp. 537/538). Definition (4.1). An AV X over a field ! is called supersingular if the formal group )( of X is isogenous over an algebraic closure k of I to (G1,1) g, i.e. where E is a supersingular curve. Theorem (4.2). Let X he a supersingular AV over a field l; then X is isogenous to E g over k, X|174 where k is an algebraic closure of l, and E is a supersingular curve. Remark (4.3). This gives a positive answer to a question posed by Manin in the case g = 2 (cf. [19], p. 79). In general the isogeny type of the formal group of an AV does not imply the AV can be decomposed up to isogeny in the same manner; e.g. there exists an abelian surface X whose formal group is isogenous with Us, o | G~. ~ such that X is a simple

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AV; this can be seen as follows: consider the closed subscheme V ~ A 2 corresponding to abelian surfaces with p-rank equal to one (notation as in Section 11; as dim(A2)=3, from (1.6) we conclude d i m ( V ) = 2 ; an abelian surface with p-rank equal to one has no positive dimensional families of finite subgroup schemes, thus if such an abelian surface is isogenous to a product of two elliptic curves, it can be defined over a field of transcendence degree one over the prime field; thus the generic point of V corresponds to a simple AV (this fact was already proved by Honda, cf. [11], p. 93). More generally: for any isogeny type of abelian varieties not equal to a power of G1,1 there exist a simple AV having that isogeny type (cf. [18]). Notation (4.4). Let X be an AV over a field l; by a(X) we denote the dimension a IX): = dim k H O M (ap, X | k), where k is an algebraically closed field containing 1(note that Endk (ap)~ k, a ring isomorphism, thus HOM(~p, G) is a right-k-module for any k-group scheme G). Lemma (4,5). Let X be an AV over a field l such that a ( X ) = d i m X ; then X can be defined over a finite field. Proof Take a polarization on X over I, let its degree be d, choose a large integer n prime to p so that Ag, a,, is a fine moduli scheme, g = d i m X, take a level n-structure on X (make a finite extension of I if necessary), and consider the corresponding point x~Ag, a,,(l). Let V be the closure of x in A,, a.,, and let y E V(k), where k is an algebraic closure of the prime field; we want to show dim V=0, thus it suffices to show the tangent space to V at y to be zero. Note that a ( X ) = d i m ( X ) is equivalent by saying that the p-operation in the Lie-algebra Lie(X) is zero. Let X be the abelian scheme over V c Ag, a,, induced from the universal family over this fine moduli scheme; Lie(X) is a locally free sheaf of p-Lie algebras over V, and clearly the p-operation is identically zero; let t: Spec(k[~])--~ VcA~.a,, be a tangent vector, ~2 =0, to V at y; then the fibre Xt:= X X v t has the property Lie(X,) has p-operation equal to zero; from this we are going to conclude t is a zero vector. Consider Y= Xy the AV over k which is the fibre of X at y e V; let ~ = ~ ( Y ; k[e] ~ k) be the set of infinitesimal deformations of Y (notations of [24], p. 274 and p. 277), this set can be canonically identified with the tangent space of the local moduli space M of Y, i. e. s = M (k [s]), and the isomorphism class Xt can be considered as an element t e ~ . Let Go=Lie(Y) be the Lie algebra of Y,, and write R = k [ e ] , e2=0; consider the set ~ = , , ~ f f ( G o ; R ~ k ) of infinitesimal

Subvarieties of Moduh Spaces

115

deformation of Gc : ~f':= {---classes of(G, q~o)[ G is a p-Lie algebra over R, and ~Oo: G| ~Go}; this is a k-vector space in a natural way (see below, or use [29], Lemma 2.10, exactness of the infinitesimal deformation functor of GO is immediate); an infinitesimal deformation of Y yields the same for Go, thus a natural map ~ - , 9(( results, which is k-linear (because the map which associates with Y' over R its Lie-algebra Lie(Y')~J~ff is functorial). We claim: a) the k-linear map LP ---, ogg"is surjective, and b) because a ( Y l = d i m Y, this map is an isomorphism. To prove this, consider the group scheme N o = Ker(F: Y ~ Y~P)),i.e. No is the group scheme of height one uniquely determined by its p-Lie algebra Go, Consider the scheme N : = N o | R By [4], III.3.5 we have a natural identification f

~ H2ymm( N O , Go)

(symm denotes the symmetric cocycles; they correspond to the commutative group scheme structures on N; a group scheme structure on N lifting the one on NO can be identified with a p-Lie algebra structure on Go | extending the p-Lie structure on Go: cf. [3], II.7.3.5), moreover ~Z~~ H~rmm(Y, Go) (cf. [4], III.3.7), and the map s ~ ~ is induced by i: No ~--+Y. After choice of a k-basis Go = kg, one can make the following identifications, resulting in a commutative diagram: ,.~ ~

Hsaymm(Y,G0)

~

~

n2ymm(No, Go) ~

(Ext(Y,~.)) g

(Ext(No,I/3,))g

(where Ext stands for the group of isomorphism classes of k-group scheme extensions); the exact sequence 0

,N O

,y

r , yIp)__~.0

results into an exact sequence ...--* Ext ( Y,,Ga)---~ Ext (No, 113,)-~ E 2 ( y (p~,~ u ) = 0 9 Inventionesmath,Vol 24

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F. Oort

(the last equality because of [23], Lemma 12.8), thus Claim (a) is proved. Because a(Y)= dim Y= g, we have N o ~ (~p)g, thus dim~ .r = gZ (because Ext(%, ~3,)~ k as k-modules, the action of k is via End(G,, ~3,), cf. [23], Proposition 10.5); moreover d i m k d = g z (because dimkExt(Y,~,)= dim Y, cf. [30], VII.17), thus Claim (b) is proved. The fact that Lie(Xt) has p-operation zero, i.e. Lie(X,)=0~JY, implies via (b) that t = 0 . Thus dim V= 0, the point XEAg, ,t,. is rational over the algebraic closure k of the prime field, and X = Xx; thus the lemma is proved.

Remark(4.6) (Mumford). From the lemma one can deduce that abelian varieties of dimension two can be lifted to characteristic zero. Proof of (4.2). Let X and l be as in the theorem; we assume 1 is algebraically closed; we can replace X within isogeny by an AV over 1 (again denoted by X) such that X'---(G~,I)g; then a(X)=g, because av ~--*Gu l; thus by (4.5) we know X can be defined over a finite field K (and this AV again is denoted by X); let f: X---, X be the geometric Frobenius of X over K, i.e. if K has q = p" elements, raising to the q-th power of the elements of 0x is a K-endomorphism: f = (X

v , XiV)__,...__, X(V-)~ X).

Let P be the characteristic polynomial off. Because P is monic and has integral coefficients (cf. [16], p. 187, Corollary 2; [22], p. 180, Theorem 4), the zeros of P are algebraic integers. Let v be an extension of the p-adic valuation on Q to ~, thus v(p)= 1. Let 21 ..... 22g be the zeros of P in C. By the Riemann-Weil hypothesis we know ]2i] = l / q ,

l