On the equations defining abelian varieties. II - Springer Link

Inventiones math. 3, 75--135 (1967)

On the Equations Defining Abelian Varieties. H D. MUMFORD (Cambridge, Mass.) Contents

w6. w7. w8. w9.

Structure of the Moduli Space . . . . . . . . . . . . . . . . . . . . The 2-Adic Limit . . . . . . . . . . . . . . . . . . . . . . . . . 2-Adic Theta Functions . . . . . . . . . . . . . . . . . . . . . . . The 2-Adic Moduli Space . . . . . . . . . . . . . . . . . . . . . .

76 99 112 123

In the first part of this paper, we have analyzed a single abelian variety X. In particular, if L is an ample invertible sheaf on X, we have analyzed the vector space F(Z, L) and the ring @ r (X,/Y) n

and have shown 1) how to choose canonical bases of these vector spaces, 2) how to express this ring as a quotient of a polynomial ring by an explicit homogeneous ideal involving coefficients which are essentially the "theta-null werte" of X. In this second part, we shall apply the first part to embed both the moduli spaces of abelian varieties, and the inverse limit of these spaces over successively higher levels, as open sets in projective schemes associated to homogeneous coordinate rings defined by explicit homogeneous ideals. We also introduce algebraic theta functions, defined on a 2-adie vector space in terms of which our results on moduli take on a simple form. I want to offer some explanation of why the 2-adics play such a central role in this theory. The situation is this: if you stick to abelian varieties of char. p(p:t:2), then you can build up a theory of theta functions for these over any (restricted) product

l-['Q,

ItS

where S is any set of primes containing 2, but not containing p. In other words, Q2 always has to be there, but you can throw in plenty of other factors if you like. Using only Q2 seemed to have two advantages: (i) you can deal simultaneously with all characteristics except 2, and (ii) the resulting theta-functions are more concise, i.e., are defined on the smallest locally compact group which admits them. I might have written a general theory for some arbitrary set S - clearly this is the accepted French approach - but there seemed no point in not sticking to the simplest and most basic case. The essential features are related to the fact that multiplication by 2 does not preserve H a a r measure. 6

Inventiones math., Vol, 3

76

D. MUMFORD:

w 6. Structure of the Moduli Space To study questions of moduli, we must first have a theory of families of the objects to be classified. Therefore, we must generalize our theory to abelian schemes:

Definition. Let S be a scheme. A n abelian scheme ~ over S is a g r o u p scheme s finitely presented over S such that the projection n: ~ -* S is proper and s m o o t h and the geometric fibres of n are connected. F o r some of the basic facts a b o u t abelian schemes, we refer the reader to [9], Ch. 61 Definition. Let L be an invertible sheaf on S. Then 1. H(L) is the group of sections a: S - * ~ of ~z such that if T~: ~r -* ~r is translation by a, then T*L_~L| for some invertible sheaf M on S. 2. H 0 (L) is the subgroup of those a such that

T~*L.~ L . 3. ~ ( L ) is the group of pairs (a, ~ TS,S then L'=F*L. Then f ~ H f ( L ) , f ~ s ( L ) way.

are functors in an obvious

Proposition 1. Assume that L is relatively ample over S. Then the 2 functors f ~ H r and f ~ , (~f are representable by group schemes H(L), ~(L) flat and of finite presentation over S. H(L) is a closed sub-group scheme of ~ itself, finite over S, and there is a canonical exact sequence: O~Gm, s ~ ~_(L)--* t t ( L ) ~ O . 1 If any doubt should arise as to whether results in [9] are still valid if S is nonnoetherian, it should be dispelled by noticing that any abr scheme over an affine S is obtained by base extension from an abelian scheme over Spec(R), where R is a Z-algebra of finite type, [5], Ch. 4, ~ 8, 11.

On the Equations Defining Abelian Varieties. II

77

Proof. First of all, H(L) is nothing but the kernel of the canonical homomorphism: A(L): ~ _,~r. A(L) is flat and finite and of finite presentation (cf. [9], p. t22), hence H(L) is flat and finite and finitely presented over S. Secondly, the morphism from the functor ff$ to the functor H I is represented (relatively) by Gin-bundles. In fact, if ~: T ~ X x s T is an element of HY(T), and if M = n , ( r : ( C ) | C -1) [here L' is the sheaf on Y" x sTinduced by L and n is the projection from /g'XsTto T], then T* E_~ E | n * M , and multiplication by non-zero sections of M defines the isomorphisms from L' to T*L'. Therefore the relative functor in this case is represented by the line bundle M on T corresponding to M - 1 , minus its O-section.

Q.E.D. Definition. Let ~ =(dl, d2 . . . . , dg) be any set of elementary divisors (di integers, > 1, di+~ [di). Define a functor f~(6) on the category of all schemes S to the category of groups by: f~s(0)=group of triples (co,x, l), cteF(S, 0~), x is a map from the set no (X) of connected components of X to K(6), the discrete group @ Z/diZ. l=(ll ..... l~), where li is a dl-th root of 1 in F(S, 0"). Multiplication is: (e, x, I). (a', x', I') = (e. ~'. l'(x), x + x', 1+ l') where I+

Ii, . . . , g

l'(x),

on the component Y,

= l-[ IT' i=1

if x ( Y ) = ( a l .... , ag). [We add the l's instead of multiplying them to be consistent with our previous notation.] It is easy to check that S ~ ~s(6) is represented by a group scheme _~(6), flat and of finite type over Z, that fits into a canonical exact sequence:

O--,a.

H (6) O

of group schemes, where H ( 6 ) = E@ Z/d,Z] G [-~/ld,] --

i

i

and where Z]d~ Z is taken as a discrete reduced group scheme and/Ja, is the usual group scheme of drth roots of 1. 6*

78

D. MUMFORD:

Definition. A S-structure on a relatively ample invertible sheaf L is an isomorphism over S of the group schemes fg(L) and ff__(6)xS, for some 6, which is the identity on the sub-group schemes G,,,s. When this exists, 6 is called the type of L. In this definition, we have included some types of non-separable invertible sheaves [in fact, all ample invertible sheaves on an abelian variety with pg points of order p have a type in the above sense] because in the present categorical approach it is no trouble. However, this was just for the fun of it and we shall now restrict ourselves to the separable case. Fix 6=(dl, dg). Assume all d~ even. Assume all schemes are schemes over Spec Z [d-1 ], where ...,

g

d= I-I di . i=1

Definition. An invertible sheaf L on YC[S is symmetric if t*L~-L, where t: Y ' ~ Y " is the inverse. It is totally symmetric if there is an isomorphism q~: L - ~ t * L which restricts to the identity on L | O~r2, where ~ f 2 = ~ is the kernel of 26. It is normalized if e*L~-O__s where e: S ~ r is the identity section. Definition. Let 2: ~(L)--%~(6) • S be a 8-structure for a symmetric relatively ample invertible sheaf L on Y'. Let ~k: L--~ t*L be any isomorphism. We define: i) an automorphism 6-1 of the functor f--*fC:(L). Given f : T ~ S , let L' = L | O r be the induced sheaf on Y" • s T. Let a: T--* 5f • s T be a section and

q~: E

"~ , T * ( E )

an isomorphism, so that (a, q~)~f#f(L). Let ~ ' : L ' ~ t * L ' be the isomorphism induced by ~k. Then _, (Ca, ~)) = (~ o a, (TT. ~~') -~ o ,* ~ o 4'3,

i.e., E

r ,t*L'

'*q' , t * ( T * E )

il

~ T,o. O' TT.

,

where t: f • s T ~ ~r • s T is the inverse. ii) The automorphism 6_ t of the functor induces an automorphism

6: fg(L)--+ff__(L) of the scheme. iii) Similarly, the map (a, x, l) ~ (a, - x, - I)


79

where if l=(ll, ..., lg), then -l--=(l~ 1..... I; 1) gives an automorphism D_ 1 of the functor S ~ ~s (3). iv) This induces an automorphism D of the group scheme ~(b). v) Then 2 is symmetric if _D o 2 = 2 o fi_.

Definition. We consider triples of the following type: i) an abelian scheme X over S, ii) a relatively ample, totally symmetric, normalized invertible sheaf Long, iii) a symmetric ,9-structure 2: _~(L) -%_(r

x S for L.

We shall call this triple an abelian scheme with a b-marking.

Definition. For all schemes S, let Jg~(S) denote the set of abelian schemes X over S with b-markings, taken modulo isomorphisms. As S varies, these sets form a functor Jg~ in S. This will be called the moduli functor for abelian schemes with b-marking. The object of this section will be to show that ~g~ is representable, and to represent it by an open subset of a definite projective variety. The next step is to study the representations of ~(b).

Definition. Let V~ be the free Z[d-~]-module of functions from K(b) to Z[d-2]. Then for all schemes S, and invertible sheaves L on S, V~| is the sheaf of functions from K(b) to L. The discrete group ~s(b) acts O__s-linearly on this sheaf, exactly as ~(b) acted on V~ in w 1: e. g., if (a, x, l) e (~s (3),

f~r(u, V~| Assume for simplicity that U is connected, that x(U)=(al . . . . . ag) and that l=(/1, .--, lg). Then (~, x , / ) takes f into f * , where

f * (b, .... , b , ) = a . H l~' . f (a l + b~ ..... a, + b,). Let E~ = V(V~): this is a vector bundle of rank d over Spec Z [d-2], and it is a direct sum of trivial line bundles L a with canonical sections [a]: Recall that, in general, an S-valued point of gn corresponds to a homomorphism from V~ to F(S, O_s). Then [a] corresponds to the homomorphism

f~,f(a) from V(b) to Z[d-1]. All the actions of (gs(b) on Va| can be put together dually in an anti-representation of _(~(b) on Ea, i.e., a representation in which the order of multiplication is reversed. This is clear from

80

D. MUMFORD:

a functorial point of view, but it may be useful to define this anti-representation directly by putting together representations on the various subgroups: the subgroup Gin, with S-valued points (~t, 0, 0), acts by homotheties on E6,

the discrete subgroup ~) Z/d~ Z, with S-valued points (1, a, 0), acts by permuting the sections [a]: thus the point b takes [a] to [a + b], the discrete but twisted subgroup ~9/~d, with S-valued points (1,0, l), acts diagonally: thus the point (11. . . . , lg) takes [a] t o / 7 ft. [a].

Proposition 2. Let

S be a scheme and let p: [~(~) x

S] x sF--,F

be an anti-representation over S of ~(~) x S on a vector bundle F over S of rank d. Assume that the subgroup Gm acts on F in the standard way. Then there is a line bundle L over S and an isomorphism F

,[n~• |

(where | denotes tensor product of vector bundles over S) such that the action of ~(6) on F corresponds to the above action of ~_(6) on E~ tensored with the trivial action on L. Moreover, this isomorphism is unique up to multiplication by an element of F(S, 0~). Proof. This is nearly the same as that of Prop. 3, w 1, except that we must use a basic result on the representations of /~d established in [4]. It is shown there that if a group scheme of the form ~)/~d, is represented, over S, in a vector bundle F, then F is a direct sum of sub-vector bundles f , , aeK(~), where the group acts on Fa by the character (~ #ld,--+ G m g

(1~, ..., 18) ~ l~ 1~'. i=l

Now we realize ~/~d, as the subgroup of ff__(tS)of triples (1, 0, l); and decompose F accordingly. Exactly as in w the action of the Z [ d - 1 ] valued points ~ra = (1, a, 0) of if(f) permutes these ,ca transitively. Therefore all F a are non-empty; since the rank of F is d, each Fa is a line bundle over S. Let L = Fo. We set up the required isomorphism by first identifying Fo=L~(L oxS)| then identifying Fa with (L~ x S ) | L by using the actions of aa and the first identification; and then taking the direct sum. The details work out exactly as in w1. Q.E.D.


81

N o w start with an abelian scheme ~ / S with 3-marking. Let L be the given sheaf on ~ , and let F=F(n,L): this is a vector bundle over S of rank d. Let f : T ~ S be a morphism of schemes. Then a T-valued point of F/S, i.e., a morphism g is the diagram: T

g

~F

~ S~/Z~non|calmorphism is the same thing as a homomorphism y:

f*(n,L)

r ,O r

of Qr-modules. I claim that ~ ( L ) is anti-represented over S on the vector bundle F in a canonical way. In fact, if f : T ~ S is a morphism of schemes, a T-valued point of ff(L)/S is given by a section

o~: T---*~'xsT and an isomorphism

q~: E N , T* E if L' is the induced sheaf on ~r • s T. Then tp induces

rc, E ~" , n.(T* C) f*(n,L)

(Tro T,),(T~*E) n,(T~). T*(E) re,/_~ f * (n, L)

by standard canonical identifications. Let the composite isomorphism on f * (~, L) be called [(p]. Then this acts on a T-valued point y : f * ( n , L) of F/S by taking it to the new point 7 o [(p]. This gives us an anti-representation of the functor f~(L) on the functor associated to F: hence an anti-representation of the scheme if(L) on F. But 9J/S has a given (~-marking. Hence we also have an anti-representation over S of ~_((5)x S on F. Applying Proposition 2, we get an isomorphism

--40r

f

,(E~xS)|

for some line bundle L on S, unique up to multiplication by elements of F(S, Os). In terms of sheaves, this gives us an isomorphism

n,L,

V~|

82

D. MUMFORD:

for some invertible sheaf K on S, unique up to multiplication by elements of F(S, Os). The fact that the isomorphism of F and (E~ x S ) | L commutes with pair of actions of fg(L) and f9(6) gives a corresponding fact on the sheaf level: for all f: T ~ S, let the ,%structure induce an isomorphism

~(E)=igJ'(L)~- ~r(b) where L' is the induced sheaf on f x sT. Then the action of f~(L') on n , L ' - which is canonically isomorphic to f * ~ , L - goes over under this isomorphism to the action of f#r(5) on V~| It follows, incidentally, that the above isomorphism of n , L and determines the isomorphism of CS(L') and f#r(b) for all f : hence it determines the given isomorphism of ~(L) and _~(5)x S. Now assume L is relatively very ample: this occurs if all di are divisible by 4 for example. Then we get a closed immersion over S:

V~|

i: ~f ~ ,P(rc. L)_~P[V~ @zK]~-P(V~) x S, which is determined by the original J-marking on W. Conversely, this immersion determines the J-marking: the sheaf L, for example, is obtained a) by pulling back O~,(1) viapl o i: : ~ P ( V ~ ) , b) by then normalizing this sheaf on the identity section. In other words

(*)

L~-(p,o i)* fOr(D] | (Pl ~ i o ~ o E)* [ 0 / , ( - - 1)].

And the morphism from .~(5)x S to ~(L) is determined as follows: let K=(ploioe)*(_O_e(-1)). Then the isomorphism (*) determines the isomorphism: •, L g z t , [(p~o i)* (Oe(1))] | K

-~P2, 9 [_Op(1) | Os] | K ~-V~| which, as we just saw, determines the O-structure. Summarizing, the abelian scheme 9E/S with c~-marking determines, and is determined by the closed immersion: i: ~ r x S. Recall that the group structure on s is determined completely by the identity section 8: S--*sr, ([9], p. 117). Therefore, the whole functor ~ is isomorphic to a subfunctor of the functor H i l b ~ , } of all flat fa-


83

milies of closed subschemes of P(Va) with distinguished section (cf. [9], Ch. 0, w 5). In particular, we have defined, from &r/s with 6-marking, the S-valued point of P(V~): Pl o io ~: S--*P(V~). It is not hard to verify that this defines a morphism of functors:

d/l~ t ) hp(v~). The next step is to describe the image. First define a closed subscheme

Mn ~P(Vo) by RIEMANN'S theta relations: let Q(a)~ Vo be the function which is 1 at a, and 0 elsewhere; let Z 2 c K ( 6 ) be the subgroup of points of order 2; embed K(6)c K(26) as before. 1. For all elements a, b, c, deK(26) such that they are all congruent modulo K(6), and for all l~Z 2, set

[ ~ l(q). Q(a+b+tl). Q(a-b+r/)] x tleZ2

• [ ~ l(tl). Q(c+d+q). Q(c-d+tl)]t/~Z2

- [ ~ l(rl). Q(a+d+~l). Q ( a - d + q ) ] • t/eZ2

x [ ~ l(q). Q(c+b+~l).Q(c-b+tl)]=O; ~/~Z2

2. For all aeK(6), set

Q(a)-Q(-a)=O. The main result can now be stated: Theorem. If dr, ..., dg are all divisible by 8, then there is an open subset Ma of Ma such that t defines an isomorphism of Jt4~ with the functor of points of the subscheme M~ of P(V~). We can go even further: not only can we identify the moduli scheme M0, but we can write down the universal projective family of abelian varieties over Mn obtained by the immersion i. Define

.~= v(v~)• ~ by the relations: let X~ be the same function as factor:

Q(a), but

in the copy of V~ in the first

I. For all a, b, c, d, I as before, set

[ ~ l(q). Q(c+d+q). Q ( e - d + t / ) ] . [ ~ l(q).Xa+b+ .. Xa_b+,]-t/eZ2

qEZ2

- [ ~ l(q). Q(e+b+tl). Q(e-b+q)]. [ Z l ( q ) - X , + a + , . X ~ _ a + , ] = 0 . T/eZ2

r/eZ2

84

D . MUMFORD :

Thus via the projection P2, A6 is a projective scheme over M~. Moreover, the diagonal A is a section e: M~--~A~

of the projective scheme .~//~r~. Further Theorem. Let A~ =p-; 1(M~). Then AdM~ with section A is an abelian scheme over Ma and it is equal to the universal abelian scheme, with its identity section, embedded in P ( V~) x M~ by the morphism i defined earlier in this section. The remainder of this section will be devoted to proving these theorems. Step 1. Without passing to any subset of M6 at all, ,4dMo has a kind of 0-structure. In fact, let M be the sheaf induced on .4~ by Or(1 ). Then for all morphisms f : T---*M6,

let M ' be the induced sheaf on A~ • ~,T. We get a canonical homomorphism: group of T-automorphisms {T-valued points of ~to)/--o ]plus /

[r

isomorphisms M' )Z*M'

And whenever -46 x ~ , T is the subscheme of P(V,O x T induced from an abelian scheme with f-marking, these homomorphisms give us back the /~-structure. To define this, assume for simplicity that T is connected, and that

(~, x, l) ~ fgr(,b, x = ( a t , ... ,a~), l = (/t, "'", lg). Then we get a projective transformation/~ of P(V~) x T by the linear map

v~ | r(T, or)--o v~ | r ( r , or), fw, f*, f * ((b,, ... , b,)) = ~ . y[ l~' . f ( a z + bl .... , a, + b,)

(regarding V~| particular,

OT) as F(T,__Or)-valued functions on K(6)). In ib~ - aq Xb @ l--~ Xb-x @ (a . r I .~ ,.

One checks immediately that the equations defining .4~ are invariant under this substitution. Therefore the projective transformation # re-


85

stricts to an automorphism ~, of -46 x ~ T . But the linear map defining # also defines an isomorphism ~J: [0p(1) | 0T] "~ '/~* [0p(1) | 0T]. And ~ restricts on .40 • ~ T to an isomorphism (p: M' "" ~2*M'.

Step 1L In the first step, we have used a little bit the actual structure of the equations defining M~: now we shall use this structure in detail. We want to define first a canonical morphism M2~, " ~M~ which will be used in Step V. Let M(resp. M') be the invertible sheaf on _M~ (resp. M2~) obtained by restricting O_Qr(1) from the ambient P(V~) (resp. P(V2 ~)). By definition, we shall have 7~* ( M 2) ~ M' 2

and we define lr by its effect on the sections of the very ample sheaf M 2 :

(*)

n*[Q(a+b).Q(a-b)]= ~ Q'(a+tl).Q'(b+~l), t/~Z2

where Q and Q' are the sections of M and M ' defined above, and a, be K(26) satisfy a+beK(6). To see that this defines a morphism ~, we must check that when these values for n* are substituted, the following are zero: n* [Q(a). Q ( b ) J - n * [Q(b). Q(a)] (1)

n*[Q(a). Q(b)]- re* [Q(c) 9Q ( d ) ] - ~ * [ Q ( a ) . Q(d)]. x*[Q(c). Q(b)]

n*[Q(-a). Q(b)]-n*[Q(a). Q(b)] for all a, b, c, d e K(5),

l(tl) . n*[Q(a + b +~) . Q(a-b+~l)] x t/EZ2

x ~ l(tl), rc*[Q(c+d+~l). Q(e-d+~l)]~I~Z2

(2)

- E l(tl)" ~*[O(a+d+tl)" Q(a-d+tl)] x ~EZ2

x ~ i(tl), rc*[Q(e+b+~l). Q(e-b+~l)] ~eZ2

for all a, b, e, d e K(2 fi), I e Z2 such that a, b, e, d are congruent modulo K(~). The first 2 expressions in (1) are 0 by virtue of the relations imposed on the Q"s; the last expression in (1) and the expression (2) are identically

86

D . MUMFORD ;

zero by virtue of the commutativity and associativity of multiplication in the Q"s. It is clear from the definition of ~ that if L is an invertible sheaf of type 6 on an abelian variety and we choose symmetric ~-structures on (L, L2), then the corresponding geometric points xl, x2 in M~, M2~ are related by n ( x 2 ) = x 1. The main result that we will need is: Hardest Lemma. Let a geometric point x in M2~ be the theta-null point assigned to some abelian variety with 23-marking. Then rc is ~tale in a neighborhood of x.

Proof. Let y=rc(x). By the infinitesimal criterion for a morphism to be 6tale ([3], w 3, Cor. 3.2), we must check the following: let A be an artin local ring with residue field k, the field of definition of x. Assume that tr: S p e c ( A ) ~ / ~ is an A-valued point extending y: S p e c ( k ) ~ Ma. Then there exists one and only one A-valued point z: S p e c ( A ) ~ M 2 , such that tr = 7t o z, and such that z extends x. In down-to-earth language, suppose that at x, the coordinates Q' (a) have values ~'(a) in k; and that at tr, the coordinates Q(a) have values q(a) in A. Then q(a), ~'(a) satisfy the Riemann theta relation and symmetry, and if q(a) is the image of q(a) in k, then ~'(a), ~(a) are related by (.). We must show that there are elements q'(a)~A such that q'(a) lifts -q'(a), and is still a point of M2~ and such that q'(a) and q(a) are related by (,). We have one more thing to help us: the values ~(a) and ~'(a) come from an abelian variety. The first thing to observe is that (.) nearly determines q'(a): in fact, we get ~, l(q). q(a+b+q), q(a-b+tl) (,),

.~

= ~, lO1).q'(a+q). ~ l(q).q'(b+q). r/r

r/EZ2

Since the ~ " s come from an abelian variety and since 4ld~ for all i, we also have For all I~Z 2 and aeK(3), there is an element bea+K(6) such that Z l(q). ~'(b+r/):~O q~Z2

(cf. w4, Proof of Theorem 1). Set

U(a,b,l)= ~ IQl). q(a+b+q), q ( a - b + q ) , ritZ2

x(a,l)= ~ l(~).q'(a+~), ~/eg2

x ( a , / ) = Z l(r/), q'(a +r/).


87

For each l~Z2, and eeK(26)/K(6), choose an element aoeK(26) lifting such that X(ao, l) #0. Then U(ao, ao, l) is a unit in A, and x(ao, l) is to satisfy: i) X(ao,l)2=U(ao,ao,l),

x(ao,I)~Y~(ao,1) in k. Since char(k)#2, this determines one and only one x(a o, l). For any other aeK(26) lifting e, set ii) x(a, l) = U(a, ao, l)

x(ao,l)

From the x's, we determine the q"s by summing over 1. This proves the uniqueness of the values q'(a), i.e., of the point z, and shows that n is unramified at x. It remains to show that if {x(a, l)}, hence {q'(a)} are determined by i), ii), then they satisfy all the requirements. First of all, x(a, l) lifts 2(a, t) since 2(a, l) satisfies (ii) with bars in it. Hence q' (a) lifts ~' (a). Secondly, x(a, l) and U(a, b, l) satisfy (*)': let a, beK(26) both lie over ~. Then

x(a, l). x(b,/)= U(a, ao, I). U(b, ao, l) X(ao, 0 2 U(a, ao,l). U(b, ao, l) U(ao, ao, l) = U(a, b, l) by RIEMANN'S theta relation for the q's. Therefore, the corresponding q'(a)'s satisfy (,). Thirdly, q' is even. In fact, 1 claim x ( - a , l)=x(a, l). This follows from the equation U(-a, ao,/)= U(a, ao, 1) which follows from the evenness of q. Fourthly, we must check that q' satisfies RIEMANN'S theta relation. This is really rather tricky. It is convenient to use a different form of these relations: let

H = K (2 6) x Z4 In. b.: Z4 is the discrete group Horn(Z4, A*)].

If bi=(ai, l~)eH, i = 1, 2, then set

C(bl, bz)=

~

ll(rll) Iz(rlz)q'(al+th)'q'(az+rl2)

I/1, r/z ~ Z 4 t/1 + t/2 ~ Z 2

=-~r

(ll + lz)(q) " x(al + r/, [1)" x(az + q, 12)

[here i i = restrictions of l i to Z2].

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D. MUMFORD:

The relations are:

C(bl, b2)" C(ba, b4) = C(bl +fl, b2 +fl)" C(ba + fl, b4 + fl) for all bl, b 2 , b3 ,b4, t e l l such that

(**)

bl+b2+b3+b4 = - 2 f t . To check that these relations are equivalent to the ones we have been using involving U " s is quite straightforward and we omit it. [The method is the same as the one we will use in the next step.] We also need a third form of these relations, that is, the Riemann form itself involving the x's (cf. w3, right at the end):

x(al, 11)" x(a2 ,12) " x(aa , /3) " x(a4,/4) 1 = 22 g ~ k(2r/), x(al +b+~l, 11+k). x(a2+b+tl, 12+k) x keZ2 ~l e Z 4

x x(az + b+ tl, 13+ k) . x(a4 + b +rh 14+ k ) (***)

for all al, az, aa, a4, b eK(2fi)

11, 12,13,14eZ2 such that

al +a2 +a2 +ar = - 2 b , 11+12+1a+14=0. This is how to go back and forth between (**) and (***): 1) take expression (**), substitute ls +2k~ for 1t, 13+2k3 for la, and

m - k 1 - k a for m, [here bi=(ai, It), and fl=(et, m)]. Sum over all choices of kl, ka, rearrange the right-hand side and you get (***). 2) take expression (***), substitute al + ( for al, a2 + ( for a2, a3 + ~' for aa, a 4 q- ~' for a4, b - ( - ~' for b, multiply by (l I + 12) (~)" (/3 -Jr/4) (('), and sum over all (, ~'eZ4. Rearrange the right-hand side and you get(**). The point to notice here is that you don't need all of the relations (**) to get a particular one of the relations (***), or vice versa. Next, there are some relations (**) that we do get immediately. In fact, suppose b~+b2e2H. Then al+azeK(6) and li=12 on Z 2. Using (,)', one finds that C(bl, b2) splits into a product:

C(b,, b 2 ) = - ~1 [ ~ (/1+/2)(~)" q(a, + a 2 + 2 ~ ) ] x ~eZ4

• [ ~ (l, -- /2) (() " q(al--a2+2()]. s


89

One gets similar equations for C(b3, b4), C(bl +[3, b 2 W fl) and C(b 3+fl, b4+[3): substituting, one gets (**) from the associative law. Next, we can show that the relations (**) also hold if bl +bae2H. We prove this in 3 steps using the precise description of which relations (**) are needed to prove which relations (***) and vice versa. i) relations (**) for all values of b~, b2, b 3 , b4, fl such that bl -t-b2e2H imply relations (***) for all ai, li, b such that al + azeK(fi), ll =/2 on Z 2 . ii) relations (***) are symmetric under permutation of the variables. So we also get relations (***) when al + as eK(tS), l 1 = 13 on Z 2 . iii) Using the fact that Z 4 c K(fi), if a 1 + a 3 ~ g(tS), then a 1 + a 3 + ( + (' e K(fi) and we can go back: we get relations (**) whenever bl + b a ~ 2 H . Next, we can show that the square of relation (**) is always true:

C(bl, b2) 2- C(ba, b4) 2= C(bl +[3, b2+fl) 2. C(b3 +[3, b4+fl) 2. In fact, for all b~, b2eH,

C(b~, b2)2 = C(bi, b2)" C(b:t, - b2) = C(O, b 2 - bl). C(O, - b 1 - b2)

by relation (**) with [3= - b l

=C(b 1- b 2 , 0 ) " C(bl + b 2 ,0). Therefore:

C(bl +[3, b2 +[3)2. C(b3 "1-[3,b4 +[3)2 = C(bl + b2 +2[3, 0). C(bl - b2,0). C(ba + b,,+213, O) x x C(b3-b4, 0). But

b l + b 2 + 2 [ 3 = - b a - b 4, and

ba+b4+2fl=-bl-b2:

so weget

= C(bl + b2,0). C(b 1 - b2,0). C(b3 + b4., 0). C(b3- b4, O) =C(bl, b2) 2" C(b3, b4) 2. But the Riemann relations definitely hold when you take the image of C(bl, b2), etc., in k. Therefore, if C ( b l , b2)" C(ba, bg)aeO

in k,

it follows that C(bl, b2). C(b3,b4) and C(bl+[3, b2+fl). C(b3+[3, b4 + [3) are square roots of the same number, are units, and have the same images in k: hence they are equal.

90

D . MUMFORD :

Now suppose

bt + b 2 + b 3 + b 4 = - 2 f l bi +bi +b; +b'4= - 2 f l ' where b, + b~e 2 H, all i, and fl + fl' e 2 H. Assume moreover that

C(bl, bl). C(b'3, b'4)

is a unit.

Then we can also show that

C(bl, b2). C(ba , b4)=C(bl + fi, b2 + fl) . C(b3 + fl, b4+fl). In fact, we know this equality, when there are primes everywhere. Hence it suffices to show:

C(ba, bD. C(bl, bl). C(b3, b4). C(b'3, b'4) (00

=C(b~+fi, b2+fl)- C(bi +/~', bi+/~')- C(b3+fl, b4+)~) x x C(b'3+fl', b'4+fl').

But assume b i + b ~ = - 2 7 i . Then using the relations (**) that we know to be true, we get:

C(bl, b2). C(bl, - bl) =C(~, -72 + b l - b 2 , ;:1-~2)" C(~1 +;:2+ bl + b ; , ;:t +;:2)

C(b3, b,). C(b;, -b;) =C(~a-;:4 + b 3 - b 4 , ;:3-;:4)" C()'3 + ;:4+ b'3 + b'4 , ;:3+74) C(b, + 9, b2 +9)" C(bl + fl', - b i - K ) = C ( b l - b2 +;:1 - ; : 2 , ; : 1 - 7 2 ) x

x C(~t + ;:2.+ bl + b; + / ~ ' - / L ~1 +Y2 - / ~ - / r )

C(b3 + fl, b4 + fl) . C(b'a + fl', -b'4-fl') = C (b a -

b 4 -I-;:3 -

;:4,73

- ;:4) •

x C(73 + y 4 + b'3+b'4+fl'-fl, ;:3 + 7 4 - f l - f l ' ) . We can assume that y l + ; : z + y a + y , = f l + f l ' by altering one of the ;:'s by an element of order 2. Using the symmetries C(A, B)= C(A, --B)= C(B, A) one checks that the product of the 1st two right-hand sides equals the product of the 2nd two. Hence the same for the left-hand sides, hence (a) is true.


91

It remains only to check that for any b~, b2, b 3, b4, fl such that bl +b2+b3+b4 = - 2 f l , there exists b], b~, b;, b~, fl' differing from the bi's and fl by 2H, such that

b'~ + b'2 + b'a + b'4= - 2 fl'

C(b~,hl). C(b3 , b'4)+O. - -

r

-

'

Now we know that the functions ~, ~' come from an abelian variety. Therefore choosing a compatible 46-marking on this abelian variety, there is a null-value function ~" on K(46) with values in k, such that

-q'(a+b). ~ ' ( a - b ) = }-'. ~"(a+t/). ~"(b+t/). q~Z2

^

Now if a~K(4tS), l~Z4, set: ~ ( a , / ) = ~ t(t/)~"(a+v/). t/eZ4

Then if bl = ( a + b, l), b2 = ( a - b , m) are 2 elements of H, with a, b~K(46), l, m s Z4, one checks immediately that: C(bl, b2)= z(a, l + m). -z(b, l - m ) . Therefore, translating the question into one involving s we have to check: for all al, a2, a3, a 4 s K ( 4 6 ) , congruent mod K(26), for all 11, 12,13,14 ~Z4,

congruent mod 2Z4,

there exists ~ , ~2, a3,0~4 e K(26), congruent rood K(6) and k~2Z4, such that 4

I ~ ( a i + 0 q , I~+ m) +O. i=1

But using the hypothesis that Zs = K(6), this follows from: for all a~K(46), f e Z 4 , there exists a~Zs such that ~(a+~, I)4:0. This is a special case of the result stated following Theorem 1, w4. Phew! The same analysis also gives the result: If (~'(a)~} are the coordinates of a geometric point x~ in M2 a which corresponds to an abelian variety with 26-marking, then the coordinates {q'(a)2} of any other geometric point x2 such that n(xl)=n(x2) are determined by:

l(rl)-q'(a + rl)2=y(a , 1). ~ l(~l) . q'(a+r/)l ~/~Z2 7

Inventiones math., Vol. 3

I/~Z2

92

D . MUMFORD :

where 7 is a quadratic character: 7: K(26)/K(6) x Z2--* { +__1}. We leave it to the reader to check this change from the ~ ' ( a ) [ s to the ~'(a)2's also comes about by a suitable modification of the 26-marking. Therefore:

if a geometric point x of M2~ corresponds to an abelian variety with 2f-marking, so do all geometric points in n-l(n(x)). Step IIL Let 3;/S be an abelian scheme with 6-markings. i: 1~ r x S be the closed immersion defined above. Then

Let

i) Pl o io ~ is actually a morphism j of S to M~, ii) i(5~) equals the subscheme y of P(Vo)x S obtained as the fibre product:

~-

l ~i~

.P(V6) x S

Iid• ,P(~)x~.

In order to establish this, it clearly suffices to take the case S = Spec (A), where A is an artin local ring with algebraically closed residue field. Let p be the characteristic of the residue field k. Then there are considerable technical simplifications: a) for all positive integers n not divisible by p, A contains exactly n n-th roots of 1, which lift the n n-th roots of 1 in k, b) if ~]S is an abelian scheme, then for all positive integers n not divisible by p, the subscheme ~ , of points of order n is the disjoint union of n 2 g subschemes, each being the image of a section of •r/s: i.e., all points of order n are rational over S.

Cot. of (a), (b). Let YC/S be an abelian scheme, and let X/k be its fibre over the residue field. Let L be a relatively ample sheaf of degree d on ~ . Then H(L) is a disjoint union of the images of d 2 distinct sections of ~/S, and there is a one-one correspondence between 0-structures for L and for the induced sheaf L on X. In this case, we can deal entirely with the group of rational points (gs(6) instead of with the combersome group schemes. Thus ~s(6) is just the (discrete) group extension: A

0----,A* ___~~s(b)---, K(6) x K (6)---, 0 A

and where K(6) is just a (discrete) group isomorphic to K 0 ) . And the representation theory boils down to:


93

There is a unique Aqinear representation of ~s(6) on a free A-module of rank d, in which the subgroup A* acts by homotheties. In particular, a O-structure on L induces an isomorphism: F ( ~ , L ) ~ V~|

A

unique up to multiplication by elements of A*. To get down to the proof itself, we are given ~T/S, plus a totally symmetric relatively ample L, plus a symmetric isomorphism f l of N(L) and f~s(3). By the results of w2, there is a 0-structure j~: f#(L2) ~ , f#s(26) such that the pair f l and fz of 0-structures on X is symmetric. Lift fz to a 0-structure f= for L =. Then fa and fz induce isomorphisms:

r(~', L)~ V~| r ( ~ , L2)~

V26|

where the first is the isomorphism used to define i: ~ ~--+P(V~)x S. Also, evaluation of sections at e defines particular A-valued functions qL and qL2 on K(fi) and K(2b) via F ( ~ , E)= V/0 | A

=

A

A

Here qL(a) is just the value of Q(a) at e: so {qz(a)} is a set of homogeneous coordinates for the point i| e: S~P(V~)• S. This means that {qL(a)} are the homogeneous coordinates of j(S), and these are exactly the numbers which are to satisfy RIEMANN'Stheta relations. The only thing that must really be checked is that the pairing

F (~, L)x F(~, L)---, r ( ~ , L2) given by multiplying sections is defined by the same multiplication formula as in w3. But the proof given in w3 of this formula goes over without any change to this more general case: simply replace the ground field k by the ground ring A. Once the multiplication formula is known, RIEMANN'S relations on {qL(a)} follow formally, as in w3. This proves (i). To prove (ii), the first point is that ~ c ~J. This follows immediately from the multiplication formula. In fact, on 5f, the coordinates Q(a) have the constant values qL(a) and the coordinates Xa induce the sections of L on ~ denoted by ~a previously. The identity which we need comes out immediately by reducing the quadratic expressions in the tSa's to 7*

94

D. MUMFORD:

functions on K(2~) and using the relation between qL and qL2. Secondly, if Y is the fibre of a# over k, then X = Y. This follows from Theorem 2, w4, and the fact that the equations on Xa are a complete set of quadratic relations. In fact, for fixed l~Z2, a~K(26), these relations assert that all the quadratic expressions { ~ l(q) Xa+b+~" Xa-b+, [ b E a + K(6)} ~/EZl

are proportional. Since, in fact, on X each of these sets consists in sections of L 2 corresponding to the function Y~,~: {~( Y~,t(b) =

b-a)

if b - a C Z 2 if b - a ~ Z 2 ,

and since the Y~.t's are linearly independent, these do exhaust the quadratic relations on the Xa. Thirdly, we note the obvious: Lemma. Let U be a closed subscheme of V, both being noetherian schemes over Spec(A). Then if 1. U is flat over A, 2. the fibres U and V over Spec(k) are equal, then U= V. This proves (ii). Step 1V. Suppose that N is a connected subscheme of M~ such that if A~ induces a subscheme B over N, then B is smooth over N. Suppose that some geometric point x of N is in the image of ~'~ under t. Then the N-valued point IN of N is in the image of ~'~, i.e., there is an abelian scheme ~I/N with g-marking such that i(~ B. By assumption, one geometric fibre B of B/N is an abelian variety. Therefore, by Theorem 6.14, Ch. 6, [9], B is an abelian scheme over N with identity section A. By construction, BcP(V~) • N. Let __Or(1) induce the sheaf M on B. Normalize it on the identity section, i.e., set L = M | p'~ A * ( M - ' ) . I claim that the inverse ~for B is given by restricting the projective transformationj to B, where j: P(Vo)x N----~P(V6)x N

j* ( x~)= x _ . . But the inverse is given by j on B (cf. w3). Since t = j on one geometric fibre of BIN and on the section A of B/N, t = j everywhere by the rigidity lemma ([9], Ch. 6, w1). Since j is a projective transformation, this shows that z*L ~ L , i.e., L is symmetric. Now j has 2 disjoint subspaces of fixed


95

points: Ll

defined by

X~=X_~,

all a ~ K ( 6 )

L2

defined by

Xa = - X - a ,

all a ~ K ( 6 )

and the identity section A is completely contained in L1. To say that L is totally symmetric is the same as saying that B2, the kernel of multiplication by 2, is completely contained in L1. But g z c L ~ u L 2 since j = id on B2; and 22, the points of order 2 on B, is contained in L1 since the projective embedding of B is assumed to come from a b-marking. Since S is connected, this implies that B2~L~, and hence that L is totally symmetric. Moreover, by Step I, we get a h o m o m o r p h i s m

2: f~(b) x N - - - ~ ( L ) of group schemes over N such that the actions on P2, 9 L and V~ match up. Since 2 induces an isomorphism of the group schemes i f ( b ) x Spec(k), ff__(L,)corresponding to the situation in the geometric fibre B, 2 is an isomorphism everywhere 2. Therefore, 2 is a/)-structure. Also 2 is symmetric at one point, hence it is symmetric everywhere. Therefore, BIN has a 0-marking which induces the given embedding in P(V~) x N. Q.E.D. Step V. Recall the main result on flattening stratifications ([10], Lecture 8): Given a projective morphism f : X ~ Y of noetherian schemes, Y can be decomposed into a disjoint union of locally closed connected subschemes Y~, such that if g: Y ' ~ Y is a morphism from a connected noetherian scheme Y' to Y, and we look at the fibre product: X'

~' ~Y'

X

r

Y,

then f ' is flat if and only if g factors through one of the subschemes Y,. Apply this to ,4~/_Ma: if Z c / ~ a is one of the pieces, and some fibre of .4~ over Z is obtained from an abelian scheme with b-marking, then we must show that Z is open. If we do this, both theorems are proven. The openness of Z follows from: Key Lemma. Let X/Spec(k) be an abelian variety with 6-marking over an algebraically closed field k. Let this define a geometric point x of Mb. z In fact, since ~, is an isomorphism of the subgroups Gra,sin any case, it is only a question of whether 2 is an isomorphism of the quotients/-/(6), H(L). Since these are etale over S, a homomorphism is an isomorphism if it is so at one point.

96

D . MtrMrORD :

Let A be any artin local ring with residue field k, and let f: Spec(A)---~ M~ be a morphism extending x. Then the scheme over Spec(A) induced by A6 is ftat over Spec(A). Proof of Lemma. Let -46 define, over A, the projective scheme ~ P ( V ~ ) x Spec(A). Let L be the invertible sheaf on ~ obtained by restricting Or(l) to X. We shall show: (,)

F ( s r, L2n)

is a free A-module, for all large n.

This certainly is enough to prove the lemma, in view of the following elementary observation: Sublemma. Let A be an artin local ring with residue fieM k, and let R be a graded A-algebra generated over A = R o by R t. Assume that R1 is a finite A-module, and that Rn is a free A-module for an infinite set of integers n. Then Rn is free for all but a finite set of n's. Proof. Without loss of generality, we can assume that k is infinite. Let R = R | In/~, let (0)=Q1

. . .

n

Q,,

P,---1/Q,

be the primary decomposition of (0). Let Pt be the irrelevant ideal n~t

Since k is infinite, there is an element g e R , such that u-.- ups.

Let no be an integer such that Q~ Dp~o, i.e., Q~ DR, for all n > n o . / c l a i m that Rm is a free A-module for any integer re>no. Choose f~, ...,f, in am such that y,, ..., ]~ are a basis of ~=. Note that for all integers p, gr'fl , ..., g,t'f, eR,.+v are independent over k. In fact, if

then ~6P,, for i > 2 implies ~ o~iftr large enough, E o~,f, eQ1. Hence E ~

i~_2. And since the degree is ,=0, which is a contradiction


97

unless all the a's are 0. Now suppose that R m is not a free A-module. Then k

~ a,f~=0 i=1

for some elements al . . . . , ak~A, not all 0. Let p be a positive integer such that Rm+p is a free A-module. Then since Rm+p has a basis of the form -gPfl, ..., gPfk, i~1, " . , fit, it follows that Rm+p has a basis of the form g~ f l, ..., gPf k , hi . . . . , hz. But k E ai(A

gP)=0.

1=1

Therefore, the a's are all 0 which is a contradiction. Q.E.D. Once we know that r ( s r, L*) is a free A-module for all large n, then s is flat over Spec(A) ([5], Ch. 3, 7.9.14). To prove (*), let In c S* (V~) | A be the ideal generated by the quadratic defining relations of , ~ . Then F ( ~ , L')~-(S" V~ | a ) / ( l , ) ,

for all large n, and it suffices to prove: (Ia)2(**)

is a direct summand of

(S z" Va)| A

for all n =0, 1, 2, ....

We shall prove (**) by induction on n, assuming at each stage that it is known for n < n o and all ft. To start things off, (I~)1 equals (0), for (**) is always true for n = 0 . To get from (**) for n = n o to (**) for n = n o + 1, we shall simply show for all A-valued points of M~ as above, there is an A-valued point of M2 ~ with the same properties such that: r

(***)

((s

v,) |

n=O

*

|

T

In particular,

((s

v,) |

v2, |

so (**) will be completely proven. To lift the A-valued point of M~, we use the morphism re: M2~---~ M~

defined in Step II. The underlying k-valued point of f corresponds to an abelian variety with cS-marking. Choose a compatible 2b-marking on this abelian variety. This defines a k-valued point y of M2~ over x. By Step II, n is &ale at x. Therefore, there is one (and only one) A-valued point g: Spec(A)---~ M2 ~ lifting f with underlying k-valued point y.

98

D . MUMFORD"

Let these A-valued points be given by homogeneous coordinates

q(a), a~K(6), and q'(b), beK(26), where q(a), q'(b)~A. These satisfy the usual equations (cf. Step II). Then in concrete terms:

S* V, | a/I~

-- { (~ l(ll)q(c + d +tl) " q(c--d +ll)) " (Z lQ1)Xa+b+'Xa-b+~)} --(Z l(q)q(c + b+tl) q(c-b+q)). (L" t(tl)X,+a+,X,_a+,)~ S* V2~| A/I2~ A [ .... X" ....

-- ~ (Xl(tl)q',(c+d+'l)'q'(c-d+tl))'(ZlO1)X~i+b+~X'-b+~) ~ ( - ( Z l(n)q (c+b+q). q'(c- b+q)) (Z l(,l)X~+a+~X'_a+,)) We set up the isomorphism (***) by requiring that

r[xa+b. Xa-b] = ~ q'(b+~l).X'+~ rteZ2

for all a, beK(25) such that

a+bEK(5). This implies:

r[

Y. r/~Z2

l(,)q'(b+n).

r/r

for all a, beK(25), I~Z 2 such that

r/r

a+b~K(5). Since

Z l(~)q(a+b+~l)q(a-b+~)=S l(q) q'(a+~). Z l(tl) q'(b+~l), this second equation implies that the quadratic relations in the ring.of X's go to 0 in the ring of X"s. Moreover, since for all a~K(25), leZ2, there is an element b~a+K(3) such that ~(l(t/)q'(b+~l)) is a unit, this also shows that the map is surjective. Similarly, since the elements lO1)X'+~ are linearly independent it is clear from this second equation that every quadratic expression in the X's which goes to 0 via T is a combination of expressions of the form: Ab{

b~a+K(6)

~r

And, in fact, the expressions

[Z lO1)q'(b2 + r/)]. [27 I(tl) X,,+b,+,~"X,,-t,, +,t] [Z l(~) q'(bl + t/)] 9 [Z l(,1)X~+b2+,Xa-b, +,] -

will span the kernel. But if b~ = b, b2 = d and we multiply this expression by a unit of the form ~ l(r/) q'(c+~), cea+K(5), we get the typical quadratic relation on the X's. Therefore, T sets up an isomorphism:

S 2 Va| A/(Ia) 2

'

st V2~(~ A/(I2,01"


99

At this stage in the proof, we have already shown that for all 6, (I~)2 is a direct summand of S 2 V~| A. Now try to extend T to a homomorphism of the whole ring. Let U be the free A-module S 2 VaQA/(Io)2. Let Y,.b be the element X, Xb in U. Then, as observed in the proof of Theorem 2, wIV, ( oo O) S 2" Va | A/(Ia)2._~ S* U ~ideal generated by~ .=o / [ Y , , b Yc,a - r,., Y~,bJ"

/

Therefore, all we have to check is that when T is extended to S 2 U, the A-module generated by the expressions T(Y.,b)'T(Yc, e)-T(Y.,d)• T(Y~,b) is the same as the A-module of quadratic relations in the X " s , and the proof will be complete. Call the first module NI, and the second N2. It is easy to check that Nz c N 2 , simply by calculating out the expressions T(Y~, b)" T(Y~, e) - T(Ira, d)" T(Y~, b)" Consider the diagram

N1 ~ ( Nz|

'

N2

~ ,N2| k

"

;S2V2a| l

,S2V2a|

Since N2 is a direct summand of S 2 V2~ | A according to the first part of this proof, it follows t h a t / i s injective. But in Theorem 2, w4, we showed that N1 = N2 in case A = k. Therefore the images of N t | k and N2 | k in S 2 V2 ~ | k are equal. This implies that j is surjective, hence that i is surjective, and hence N1 = ArE. Q.E.D. w 7. The 2-Adic Limit Up to this point, we have been studying pairs (X, L), consisting of abelian varieties X, and ample invertible sheaves L of separable type. To push the theory further, however, it seems almost essential to make it freer of variations within one isogeny type. The simplest way to do this is to study simultaneously a whole tower of abelian varieties. All the simplifications that occur however have to do with dividing by 2, so it doesn't seem necessary or fruitful to look at all isogenies: instead we look only at isogenies of degree 2", some n. This is economical, too, because our moduli will then be the values of functions on 2-adic vector spaces, rather than of functions on adelic spaces. As always, we assume char(k) :F 2. The basic definition is:

Definition 1. A 2-tower of abelian varieties (or tower) is an inverse system {X~},~s of abelian varieties, i.e., S is a partially ordered set such that V ~x, ~2eS, S ticS, fl>~l, ~2, and whenever ct, ticS, ~>fl, we are

100

D. MUMFORD:

given an isogeny of degree 2n:

p,,#: such that: 1) If ct>fl>y, the diagram of isogenies

commutes,

2) If ~>/~1,/~2 and Kt=kernel {X~-*X#,}, then Kt---K2c~'fll >f12. 3a) For all ~eS, and all isogenies ? ( ~ Y, of degree 2n (some n), 3tieS such that ct>fl, and one has a diagram: Y

3b) For all eeS, and all isogenies Y ~ X ~ of degree 2" (some n), 3tieS such that f l > e and one has a diagram:

Note that, starting with one abelian variety 2-, we can generate in a canonical way a 2-tower of abelian varieties by taking coverings and quotients of degrees 2 ~, starting with X.

Definition 2. If X is an abelian variety, let tor 2(X) denote the group of closed points x s X Of order 2 ~, some n. If i f = {X~} is a tower of abelian varieties, we get a derived inverse system of discrete groups {tor2 (X~}. Let V(X) = i!jN_tor2 (X~). If X"is one abelian variety, we also let V(X) denote V(X), where X is the 2-tower generated by X. This is the usual 2-Tate group of X. For all e~S, there is a canonical surjection p~: V(X)----~tor2 (X~) 9


101

Denote the kernel by T(ct). Each T(ct) is an inverse limit of finite groups, so if we topologize V(X) by taking the T(~)'s as a basis of open neighborhoods of the origin, then V(X) becomes a locally compact group, and each T(cc) becomes a compact, open subgroup. From well-known structure theorems, we know that there are topological isomorphisms: 0

.~ T(~)

0

>(Z2)2g

, V(X)

~(Q2)2g

, tor2(X~)

,0

subgroup of] 2g Q / Z of elts.[ ~'lofl order 2~ [ ...... ~0

[(some .)

/

(where g = dimX). In particular, s ~ p a r t i a l l y ordered set of compact,~ =~open subgroups T=(Q2) 2z J" When X is generated from one given X, the kernel of the canonical homomorphism V(X)~tor2 (X) is usually denoted T(X), and T(X) is also called the 2-Tate group of X. A final point: when k = C, one can associate to a tower of abelian varieties a common universal covering space. In fact, for all ct>_fl,p,~: X ~ X p induces an isomorphism p,p: X~

.,

between the universal covering spaces. Therefore let ~ stand for a complex vector space canonically isomorphic to each X~. In the classical theory, ~ plays a role analogous to V(X). For all ~, there is a canonical surjection q~: X ~X~. Let L(~) denote its kernel. Then {L(~)} is a family of commensurable lattices in X, and ~t>_fl if and only if L(~)c_L(fl). Since X, ~-X]L (~), the whole tower can be generated by starting with the complex vector space ~ and dividing by these lattices L(~). ^

Definition 3. A polarized tower of abelian varieties is a tower {X~}, ~eS, plus a set of totally symmetric ample invertible sheaves of degree 2", some n: L, on X~, for ~ in a subset So c S, plus isomorphisms p~*.p(L,) "" ,L~ whenever ct. fleSo. ~>fl. We require: 1) If ~, tieS, and ct>fl, then fl~So=~ct~So . 2) If ct>fl>?, yeSo, then the isomorphisms of L~,,Lp, L r pulled up to X~ - are to be compatible,

all

102

]).MUMFORD:

3) If ~>fl, ~ S o , and if there exists a totally symmetric sheaf M on 2"p such that p*~M~-L~, then fl~So too, [in which case, L~ will have to be M too]. The object of this section is to generalize the theory of w1 to a polarized tower of abelian varieties. So from now on, let's suppose given one such tower 3-={X~, L~}. The first important observation is: " 4 > 2 " Lemma. For all x ~ V ( X ) , there exists an ct~So such that p~(x) eH(L~).

Proof. Start with any a~ ~So. Let 2~ be the order of the point p~, (x). Let a2~So be the element such that a2>a~ and such that the isogeny p ~ , ~ is 2 ~ fi, i.e.,

q

~X~I

Since (2~ ,5)*L~(L~,,)-z~ , H((2~fi)*L,~) contains all points of v of order 22~. Therefore H ( L ~ ) contains all points of Jr of order 22L But 0=2 n p,t(x)=2 n p.... ,(p~(x))=2

2n

[q(p~2(x))]

so p,~ (x) has order 2 2". Q.E.D. Now let S~={cteSo]p,(x)EH(L~)}. Like So, it has the property: fl>a, a e S o tieS o. Suppose that for one ct~S~, you choose an isomorphism

~o~: L, ~ , T*(x)(L,). Then I claim this determines canonically isomorphisms q~: L~ ~

T*(,)(L~)

for all fleSh. This is clear - first suppose fl>ct, and then choose (p~ so that:

LB

'P~

, (Lp) >Tp~x)

commutes. In general, first determine q~r for some 7 e S~ such that ? > ct, fl, then go backwards to determine q~p from q~r- Call such a system of q~'s, all ~eS~, a compatible set of isomorphisms L~-~T* x (L~)9 pC)


103

Definition4. ( r of pairs (x,{q~,}), where x ~ V ( X ) , and {q~,}, ~ S ~ , is a compatible set of isomorphisms.

This forms a group in the usual way. Given (x, {r choose any ~,~ S~ n S~. Then form the composition: L ---~ T*,,(L~)

~

(y, {~,})~fg(Y'),

*

Call this p~ and generate with it a compatible set of isomorphisms {p,}, all c~~ S~ +Y. Then let (x, {r

(y, {~b=})=(x+ y, {p=}).

Moreover, we get an exact sequence:

o

,k*

,~(:)

'~

,V(X)

,0

as usual. Notice the simplification here over the theory of w1 : the finite group H(L) that depended on L has been replaced by the big group V(X) depending only on the tower and not on the polarization. It is not hard to interpret if(Y-) as a simultaneous direct and inverse limit of the ~(L,)'s, with respect to the connections induced between them whenever > fl, as in Prop. 2, w1. In particular, for all a~ e So, if we look at the subgroup:

~. (.~-) = n-1 [p~ '(H (L~,))] N then for all

(x, {m,})~I,(:),p,,(x)~(L,,)

so m~, is defined, and

(x, {q)=})~ (p~,(x). q~,) defines a surjection: 9 (~,~(:-)

qal

,(r

,0.

Let K ( ~ ) denote the kernel of this map. We get the picture: 0 K(oO

,k*

,~r

0 ,

" ,p21(l-I(L~))

l,o 0

,k*.

T(~)

,(r

Ip" ,

0

,0

H(L,) 0

,0

104

D . MUMFORD :

L e m m a 2. For all a ~ S 0, fq*(~Y-) is the centralizer of K(~) in fq(Y), and k* . K(a) is the center of f~* (~-).

Proof. If x~K(~) and y~fq*(~--), then y . x - y - 1 is still in K(a), and it has the same image in T(cr as x has, since p~- 1(H(L,)) is a commutative group. Therefore x = y . x . y-a, i.e., K ( ~ ) c center [fr Since k* is even in the center of fq (3-), k*- K(a) c center [~'* (~'-)] too. But if x ~ center [fr q~(x)~center [f~(L~)] and we know k* is the whole center of f~(L,). Therefore k* 9K(a) is exactly the center of f~* (oj-). Now suppose y ~ ff (Y-) centralized K(a). y is certainly in f~' (~--) for some fle S, (fl > ~). Then the image qp(y) in fg(Lp) commutes with the image qa[K(~)] in ~(L~). Now qa[K(a)] is a subgroup of ff(La) lying over the subgroup Ker(pp,) of H(La). Therefore by Prop. 2, w qp(y) is an element of fr (La) whose image in H(Lt~) is in p~. ~ [H(L,)], i.e., the image of y in V(X) is inp~,~(H(L~)), or yefg*(Y-). Q.E.D. Corollary. k * = center of fg (J ) . As in w we can describe the non-commutativity by a skew-symmetric form ex: V(X) x V(X) ~ k * :

e~(zcx, Try ) = x . y . x -x 9 y-1 all x, yEfg(Y'). Then the lemma tells us that: 1) p~ 1(H(L~)) is the group of elements x~ V(X) such that ea (x, y) = 1, all y~T(~), i.e., T(~) • 2) T(~) is the group of elements x e V(X) such that ex(x, y ) = 1, all y~ps i.e., the degenerate subspace for the pairing ex on p ; ~(H(L,)). 3) For all x e V ( X ) , there is a y e V ( X ) ex is non-degenerate.

such that e x ( x , y ) 4 1 , i.e.,

In particular, for all aeSo, (a) e x = l on T(a), (b) I(a)-~]T(a)~H(L,), and (c) the pairing induced on T(a)• by e~ corresponds to the old pairing eL, on H(L,). Using the symmetry of the L,'s, we obtain an automorphism

5_1: g(~-)--,fa(er) exactly as in w2. But now since 2- V(X)= V(X), 5_ 1 is much more convenient than before. In fact, it induces a canonical section of ~ ( 9 - ) over V(X):

Definition 5. Let x e V(X). Let z ~ fq (J-) satisfy rr (z) = x/2, 5_ 1(z) = z - a : there are exactly 2 such elements in ~ ( J - ) , z and ( - 1) 9 z (here - 1 is an


105

element of k* and we multiply in ~($'-)). Let

G(x)=z 2 (which is independent of the choice of z). Therefore, f9(3-) decomposes as a set:

~ ( : ) ~ - k* x v ( x ) if

~. ~(x),-+ (~, x). Let's compute what happens to the group law: Lemma 3. For all x, yE V(X), ~(x). a(y)=ez(x, ):/2). a(x + y).

Proof. Let z, wEff(~") lie over x/2, y/2 respectively and satisfy ~-1 z = z - 1 , a-1 w=w-1. Let s = e z ( - x / 4 , y / 2 ) , z . w. Then s lies over (x+y)/2 and satisfies:

a-1 s = e a ( - x / 4 , y / 2 ) , a-1 z . 6-1 w = e a ( - x / 4 , y / 2 ) . [z - 1 . w-1 . z . w-I-(z, w)- i = c a ( - x/4, y/2). c a ( - x/2, - y/2). ( z . w)-1 -1

Therefore

ea(x, y/2) a ( x + y ) = ca(x, y/2). s 2 =e~(x/2,y/2). z . w. z . w =e~(x/2,y/2). z2 (z - I . w. z . w -1) w 2

=o(x). ~(y). Q.E.D. In other words, the group law, carried over to k* x V(X), is (a,x). (fl, y ) = ( ~ 9ft. ca(x, y / 2 ) , x + y ) . Let's give a complete structure theorem for ~(~-) and V(X). Let Z:

Q2--*k*

be an additive character with kernel Z z .

Definition 6. A symplectic isomorphism of V(X) and Q$ x Q$ is an isomorphism ~ such that if ~01x, q~2x are the W g and 2"dg components of ~(x), then ea(x, y) =Z ['q~ x- {o2Y-tcP~ Y" r x]

106

D. MUMFORD"

for all x, y e V(X) (here t denotes the transpose vector, and 9 is multiplication of 1 x g and g x 1 matrices). We leave it to the reader to check that such an isomorphism always exists: for example by constructing it inductively via some cofinal series al < ~ 2 < - - - in S.

Definition 7. fig = k * x Q~ x Q~, with group law

y+v).

u,

D _ I : fgg~ffg is the automorphism 2;: Q~ x Q~ -~ ~g is the section

D_l((~,x,y))=(e,-x,-y).

Definition 8. A full O-structure for J - is an isomorphism c: fr (J-)--%fqg, which is the identity on the subgroups k* and such that c o 6_ 1 = D_ 1 o c. It follows immediately that a full 0-structure c induces a symplectic isomorphism ~: V ( X ) ~ Q ~ , , and also that 2 ; o ~ = c o a . Therefore c is even determined by ~. This is a simplification which was foreshadowed in the discussion at the end of w2. In fact, given a symplectic isomorphism ~, we can define the unique full 0-structure c extending ~ by:

for all 2 e k * , xe V(X), and one checks all the requirements easily. In particular, one full 0-structure always exists and we have a structure theorem for fr and its maps. So far, in our polarized tower, we have considered only the totally symmetric L~'s that can be put in an inverse system on the X,'s. F o r a few more a's, however, we may be able to find a symmetric invertible sheaf L~ on X,,, such that for some element tieS o for which fl>a,p*#L~ is isomorphic to the L~ that we already have: such an L~ will be said to be compatible with the given polarization. If we define as before ~* (o~) = n-l(ps

(H (L=))),

then every element ({xp}, {q@)efg*(:-) is compatible with a unique isomorphism @~: L, ~ ,T*(L~) and we obtain a h o m o m o r p h i s m q,: ff*(~Y-)--*ff(L,)just as before. Let K(~) be its kernel: a subgroup of ~(~J-) isomorphic under n with r(co.

O n the Equations Defining AbeIian Varieties. II

107

Lemma 4. I f L, is a symmetric sheaf on X, compatible with our polarization as above, then K&) is the set of elements

e c(p,(89

.

xET(a). In particular, if L, is totally symmetric, then K(a)=a[T(a)]. Proof. To unwind the definition of a(x), let fl> 0~be the element of S giving us a diagram: Xfl

p a,

x, Then p~., L= is totally symmetric, hence isomorphic to La, and u =pa (89x) is in H(L~). Choose an isomorphism

O: LI~ ~ , T* Lp such that b_ l((u, ~b))-- (u, @)-I. Let (u,~/)2=(2u,~o~). Then a(x) is represented by the elements (2u, q~)eff(L~). Since xeT(a), 2uEKer(p~,~) and there is a scalar i such that: L# 9

I1

~

~~

* ~T2.L~ ~*

If

*

g

pp, ~L~ commutes. This means that 2 - 1 . a(x) is compatible with the identity map from L~ to L~, i.e., is in the kernel of q,. The lemma boils down then to checking that 2=eL,~(pp.,(u)). Throwing out irrelevant notation, we can restate this fact as: Lemma 5. Let L be a symmetric invertible sheaf of separable type on an abelian variety X. Let ueX4 and let (u, cp)eff((26)*L) satisfy 6_ l((u, ~o)) =(u, ~0)-1. Let (u, q~)2=(2u, ~). Then the composite isomorphism: (26)*L

~ ~T*,(Z6)*L~-(2~)* T*uL=(Z6)*L

is multiplication by e,L(2u). Proof. Since L is symmetric, (26)*L~L 4. Let v=2u and let (v, p) be the element q2 ((u, cp)) in if(L2). Since ~/2o 6_ 1 = 6_ 1 o ~/2, 6_ l((v, p)) = (v, p)- 1. But eL,2(v)= [e,L(v)] 2 = 1, so by Prop. 3, w2, 6_ a((V, p)) = (v, p). 8 Inventiones math., Vol. 3

t08

D. MUMFORD:

Therefore (v, p ) 2 = l in f~(L2). Therefore by Prop. 6, w

qz((V, p))=

(0, eL,(v)) in f#(L). Explicitly, this means that: L* o__e~T* L* (28)* L --- -~ T* (23)* L molt-bYeL*(O~ (2 ~)* T2*vL

\

II

(28)*L

commutes. But (v, p2), as an element of N(L4), is ~z((v, p)). And

~ ((v, p)) = ~z(q~ ((u, ~))) = ~2 ((u, ~o)) = (u, ~o)3 9 ~_ l((u, ~o)) = (v, ~). Therefore the dotted arrow is ~k and the lemma is proven.

Q.E.D.

Conversely, suppose we start with any a ~ S such that e = l on T(a) • T(a), i.e., T(~) is isotropic, and try to make a subgroup of fg(oq') via K ( a ) = {e, (89x). a(x) lx e T(a)} where e, is a function from 89 any function satisfying

to {_+ I}. This works if we take for e,

e, (x + y) = ex(x, y)Z. e, (x). e, (y). In particular, e, (x)= 1 if xeT(a). Let tieS be such that 2T(a)=T(fl). Then tieS o, and K(fl)= a [T~)] =K(a). Let L B be the totally symmetric sheaf on Xp defined by our polarized tower. Then K(a)/K(fl) is a level subgroup of ~(La) lying over the subgroup pp [T(~)] of H(Ltj ). As in w1, it provides descent data for Lp in the isogeny pa~: Xa~X~, since Ker(pp,)=p~[T(a)]. Let it define L~ on X,. This L~ is easily seen to be symmetric, compatible with the polarization, and satisfying eL,~(p,(x)) = e, (x),

all x e 89T(a).

Condusion. Symmetric sheaves L, on some X~, compatible with the polarization, are in 1 - 1 correspondence with all possible "level" subgroups K = f#(~3-) such that

l) Knk*={1), 2) for all xeK, a ( ~ ( x ) ) = + x . This ends our discussion of the groups that are involved in a polarized tower of abelian varieties. Next, we turn to their representations. The family of vector spaces

{r(X:,L,)},

~So


109

forms a direct system, and we define: r(~) = ~

r ( x ~ , L~).

ct~$o

Just as in w1, f~(~d-) is represented on the vector space F(3-). Moreover, as in w1, we check that we can recover F(X~, L~) from F(3-) since : L ~ )'elements of F ( ~ ' ) F(X~, ~, ~ (invariant under K(~)}" [The same holds for any symmetric L~ compatible with the polarization and the corresponding K(~).] What representations does a group like f# (J') have ? We make the following restriction: Definition 9. An admissible representation ~ ~ U~ of f~ (37-) [resp. c~g] in a vector space V is one in which the subgroup k* acts via its natural character (i.e., if ~ek*, U , = ~ 9 (id)v), and such that for all x~ V:

{a~f~(~'), resp. @g I U,x=multiple of x}, is the inverse image in f9(3-) [resp. f~g] of an open subset of V(X) [resp. Q]g]. Prop. 3 of w1 generalizes easily to: Theorem. @g, and hence f#(J-), has one and only one irreducible admissible representation. All other admissible representations break up into direct sums of the irreducible one with itself. The proof of this is roughly as follows: choose a maximal open subgroup U c Q22g on which the skew-symmetric form e((xl ;Yl), (x2 ;Y2))=Z [txt " Y 2 - ' x 2 " Yl]

vanishes identically, such as Z 2g. Then construct eigenvectors in V for the subgroup k * x U of f~g. All other dements in the group permute these eigenvectors and we show that this permutation can be described simply, (independent of V). We leave the details to the reader as they are similar to those in the proof of Prop. 3, w1. This irreducible representation can be written down like this: Let ~ . )'vector space of k-valued, locally constant~ g = ~functions f on Q~, with compact supportJ" [U(.... y) f ] (z) = ~. )~(ty. z ) . f ( x + z)

if f e ~ , 8*

(~,x,y)ef~g, zeQg2.

110

D. MuMFORD:

Notice that the only elements fGo~g invariant under the operations U(:,x,~), all x, y~Z~, are the multiples of the characteristic function:

zeZ~. Therefore ~ is irreducible (i. e., in view of the Theorem, if any subspace canonically attached to the representation is one-dimensional, the representation must be irreducible). On the other hand, we have: Theorem. F(oq-) is an irreducible admissible representation for f# (3-).

Proof. It is an admissible representation since every x s F (3-) is in some F(X,, L,), hence is an eigenvector for k* 9K(~). H is irreducible since if K ( a ) c f g ( J - ) corresponds to a symmetric L, on X~ of degree 1, then the subspace of K(e)-invariants is isomorphic to F(X,,L,) and this is 1-dimensional.

Q.E.D.

It follows that if we choose a 0-structure c: ~(~--)~ fgg, we get a unique isomorphism:

r(:)

# ,~

such that fl(Uz(s))= Uc(~)(fi(s)), all zGN (~'-), sGF(J-). The isomorphisms fl extend the isomorphisms fl in w1 in the following way: a. let a e S o and let T ( c 0 c V(X) be a compact open subgroup such that ~(T(a)) in Q~ x Q~ is of the form U x V. b. Then pgl(H(L,)) is the orthogonal subgroup, i.e., V • l (if V • = {xe Q~] ~ (tx. y) = 1, all y 9 g}, U 1 similar). Therefore ~ induces isomorphisms:

n(L,): p2 ~(n(L~)) _~(V~IV)x (U~lV). r(a)

If we choose an isomorphism

(ri/U)

~ ,K(~),

then this gives

(UI/V)~-(V-~/U)'~K(~), hence /N

H(L~) ~ K(~) x K(~).


111

c. Thus we get an isomorphism: (L~) ~-

f~*

K(.)

k* x (V•

x

(g~/v)

---k* x K(5) x K(5)=~q(5). d. On the other hand, fl restricts as follows: /'(3"). ~ , ~ [functions ~o on Q~ ~support in V1 s (~)m~___ + |constant w. r. t. If [translations by

F(X~, L~)

{functions on V•

SII

{functions on K(5)} = V(5). The induced isomorphism of F(X~, L~) and V(5) is exactly the isomorphism fl of w1 corresponding to the 0-structure on ~q(L,) occurring in c. In short, fl is just the union of the isomorphisms fl obtained on a finite level previously. Next choose consistent isomorphisms (L~)o | ~:(0)-%k, for all aeSo. These induce "evaluation at 0" maps: F(X,, L~) -ok, for all c~, which fit together into one big "evaluation at 0" map:

A: r(3-)--,k. To describe this piece of information in closed form, we need to describe the dual space o~**:

Dual of ~f~g: o~, is spanned by the characteristic functions q~v of compact, open subsets U c Q , 2, with the obvious relations: Now let ~ be the Boolean algebra of compact open subsets of Qg. Then a linear functional on ~ is determined by its values on the q)v'S, and relations which follow from (,) make this set function into a measure. In other words, if we let o~, [vector space of k-valued finitely additive = ~measures # defined on the Boolean algebra N f

112

D . MUMFORD"

then the pairing: )..

,V(X)

,0

, v(x)

,0

1

, k * - - ~ e ( 5 - ~4))

, v(x)

,0-

1

,k*

,v(x)

,o,

, ~(5-)

and everything commutes with 6_ 1 and a. In terms of our standard groups, this diagram goes over to: 1

,k*

1

,k*

9 , ~g

, e 2 ),.

1----~ k* - - - - ) N~

,Q~*

,0

,el'

.,0

,Q22g

,0

120

D. MUMFORD:

where ~2) and resp.

~4) are both equal to k* x Q2g x Q2,g but with group laws:

(o:,x,y) . (o:, x', y ' ) = ( a , a'. Z(2'x . y'),x + x', y + y')

= ( a - ~ ' . Z(4tx 9y ' ) , x + x', y + y' and

E2(~, x, y) = & , x, y), F(~, x, y) = (a, 2x, 2y).

Thus given one symplectic isomorphism V(X)--~ Q[ • Q[, x ~ (~olx, ~o2x) we get symmetric theta structures CI: ~ ( " ~ )

'(~g,

r

~(,~"(2))

,'v )~(2)

and c 4 : & ( 5 (*)) ~ ,f~4).

Via the theta function representation, we obtain injections: ffunctions) O: F(~')--*to n V(X)j~' 9~:~-(2), (functions) 0(2): " t ~ ' )--~ V(X) f and it is easy to check that, for all sl, s2~F(X~, L~) ~(2)(S1( ~ S2)=~(S1) 9 O(S2),

i.e., tensor product of sections becomes pointwise multiplication of theta functions (compare Property II above). Now, define actions of both f~(~-) and @(~-(2)) on the vector space of all k-valued functions on V(ff) by: (U,,(tP)) Y=o~ " e;, ( 2 , Y) " q~(y- x)

if w = ~

.

a(x)~fr

(U~2)(q0) y = a . ez (x, y) . tp ( y - x) if W=tZ. O'(X) ~ ~ ( f f ( 2 ) ) .

Then according to Property I of algebraic theta functions, Image(S) is an irreducible ~(Y)-space, and Image(8 (2)) is an irreducible ~(~-(2))_ space. Moreover, Image(9) must be generated by the various functions


121

ioe~

and Image (~q(2)) must be generated by their various products:

Y~-'~e~(xt+x22 'Y)'~[~](Y-Xl)'~[O] (y-x2)" Now it is a non-trivial condition that this second family of functions spans an irreducible (g(~q'(2J)-space. In particular, let K~N(~ r(2)) be the subgroup: c21[-((1, x,y) lx ~:r 1 Z2, , y~Z~}]. Then in an irreducible fg (~J-(2))-space, K has a one-dimensional space of invariants. Now it is easy to check that all the functions

(any x e V(X)) are invariant under c~t(1 x Z~ x Z~). Let V, c V(X) be the subgroup ~ , (Z~ x {0}). Then the functions:

r~ ~ 42 V l / V t

are all invariant under K. It follows that they are all proportional to one function r Therefore, there are constants, depending on x - call them c(x) - such that r 42 V 1 / V t

Interchanging x and y in this expression, using the evenness of 0 [ ~ ] and its periodicity with respect to elements of left-hand side is symmetric in x and y. Thus

c(x). ~(y)=c(y). ~(x),

all

IT,, you

check that the

x, ye V(X).

Since neither c nor 4) can be identically 0, this implies that e(x)=c~ 9 ~(x), for all x and some eek*. Replacing 9 by V ~ . ~, we get c=~, or

{ s 42gl/V,

This is equation (**)' in the proof of Lemma 2, so referring to this proof, we see that we have proven that/) [00] satisfies PdEMANN'Stheta relation. Because of the central significance of this result, I want to give a second proof, following the lines of the proof in w3 in the finite case.

122

D. MUMFORD:

First, introduce the maps:

G

\,ff -,cC s( (on v(x)J'

~ d~,~functions 1

(o~ v(x)J"

These induce a pairing

o: ~ x ~ - ~ such that equivalently

/~(st) o l~(s~)=/~2)(s~|

s~),

s~eg(g')

or

T(A)" T(f2)= T(2)(A o A ) , Notice that the algebraic tensor product ~ | cular, the map A, f2 ~ f t o f2(0)

~e~. ~

is just ~2g. In parti-

is a linear functional on ~ | ~ , hence it is represented by a finitely additive measure 2 on O~ x Q~ : i.e., ft o f2(0)=

f ft(u) "f2(v)'d2.,v"

Since for all st, s 2 e F ( f ) , a e f f ( f ) , U~ sl | U~ s2 = U,2(~)(s1| we find that for all fl,AeG,~eG, GAoGf~=G2~(ftof~). Let a=(1, Yl, Y2). Then: ft o f2 (Yl) = (U(t, ,,, y2)(fl o f2)) (0) = [V~l,y,, y2)fl o e(t, y,, y,)f2] (0)

=

~ Z(tY2"(u+v)).fl(u+Y,)'f2(v+Yt).d~,o

for all y~, y2eQ*2. Taking combinations of these equations for various Y2'S, it follows that

g(u+v)" ft(u+ yD" f2(v+ Y2)'d2..~ =0 for all locally constant functions g such that g(0)=0. This shows that

h(u,v)d2,,~=O whenever h E ~ g, and h (u, - u ) = 0, all u. This implies that 2,,, is given by a measure on the set {(u, -u)}, i.e., there is a ~ e ~ * such that

h(u,v)d2~,~= ~ h ( u , - u ) d ~


123

all h ~Jrfz g. Therefore: f t 0 f 2 ( y ) = ~ fl(y+u)f2(y-u)

d~ u.

Now use the fact that sl | 9s2(0): therefore if g and v are the null-value measures for 9- and ~d-r we find

I fx(u)dl~t~" ~ f2(v)d#~= I A~

e~

Q~

Q~

=

g•

Qz Hence if

~(x,y)=(x+y, x - y ) Q~*

~ g fl(w+t)f2(w-t)dvw'd-2,. Q2

as usual, we find

F.d(#x#)= ~ ~*F.d(vx~)

i.e.,

Q~g •

foralIfEgf'2g,

= v • ~.(~- ~ U)

for all compact open sets U c Q2g. Using the evenness of/1, it follows from this equation that v • 2=A x v, hence v and 2 are proportional Changing v by a constant, which is permissible, we may assume v=~. Then if v' is the measure v'(tJ) = v (89 t 0 ,

it follows that p x # ( U ) = v'• v'(~ U), all U. This is condition (A) of Lemma 2.

w 9. The 2-Adic Moduli Space We will now put the results of w 8 in a moduli-theoretic form, and relate these to the finite-level results of w6. Once we have done this, we will be able to go further and determine the structure of the boundary of the moduli space. The whole moduli problem for abelian varieties looks very different when viewed from an isogeny invariant point of view. The difference between polarization types disappears because any abelian variety is isogenous to a principally polarized abelian variety (as is easily proven by generalizing some of the results of w1 to inseparably polarized abelian varieties). The natural thing is then to view the classification in 2 steps: first one has the totality of polarized towers; second, within each tower one has a huge system of variously polarized abelian varieties. We will not treat this entire moduli problem since the study of all inseparably polarized abelian varieties within an isogeny type is a subject in itself. In order to a) restrict to separably polarized abelian varieties, while 9 Inventiones math., VoL 3

124

D. MtnarORD:

b) constructing moduli schemes simultaneously in as many characteristics as possible, we shall consider only the polarized 2-tower inside each full polarized tower, and at the same time exclude only char. 2. Analogous results would be obtained if we restricted ourselves to all characteristics p, pXd (d a fixed even integer), and to isogenies within a tower of degree dividing d n, N>>O. As far as the category of sets is concerned, we have the following sets and canonical maps to consider:

.

[set of polarized 2-towers ~--= (X,, L,}] /plus symplectic isomorphisms 1 [up to isomorphism

_\ \ \ ]

./.f

\\Y\

set of abelian varieties X, | ample totally symmetric sheaves ~'n' = / L of type 61, and symmetric [theta-structures fl: f~(L) ~ 'f#(61) [ up to isomorphism

set of abelian varieties X, plus de~ = ample totally symmetric sheaves L ~ type 61 uP t~ x h ~ ~

......... ~other0's)

............ (other~'s)

[ set of polarized 2-towers]

e-={x.,L.} up to isomorphism

IJ

Here ~51 is any g-tuple (2"~, 2"L ..., 2"*), nl =n2~_ "'" ~ng>= 1. The various arrows arise as follows: (I) g takes (X, L, fl) to (2, L), (II) h takes (X, L) to the 2-tower generated by (X, L). (III) The ft's are given by choosing a compact, open isotropic subgroup U=Q~ ~ such that U• and a symmetric isomorphism

fl0:

k* x ~ ( U • Z(U)

~~(~1) 9


Then for all (~-', r there is a unique level ~eS such that r and f ( ( ~ , rp)) should be (X,, L,, fl) where fl is the composite N(L,)

, ~io ~

k* x ~;(U') Z(U)

125

U,

,N(6,). #o

It is apparent from this diagram that the various moduli sets ~'~ treated in w6 are inter-related in a rather complicated way: Given ~1, 62, one can choose any of an infinite number of f l , f 2 in the diagram:

and relate ~'01, "A/~2via the, in general, many-many correspondence so obtained. Each Jt'~ is related to itself in this way by the well-known Hecke ring of correspondences. The whole set-up is much easier to visualize starting from .~'~. Note that G = S p ( 2 g , Q2) acts on ~t'~, if we let a e G act as follows: ~ ( ~ r , e ) = ( ~ r , a o q,). Then dr is nothing but the quotient dQ~/G, and the dr and dd~ are quotients dQo/F where F ~ G is a suitable subgroup commensurable with Sp (2g, Z2). The different maps from JQ o to dt~ are simply the compositions of (a) action of some aeSp(2g, Q2) and (b) the canonical map from dQ o to dl~o/F. Clearly the most basic sets to get ahold of are dt_~o and dQo. We have seen that the dt~'s "are" varieties. Thus the dQo is an infinite covering of a variety and dr is an infinite quotient. As far as I know there is no sensible object whose underlying point set is d t ~ . But dt~o is an inverse limit of varieties and will turn out to be a perfectly upstanding (though non-noetherian) scheme. This moduli space (and its adelic generalizations) seem to be the most important ones for the entire moduli theory of abelian varieties. The next step is to define the scheme an open subset of which will represent dQ o . We will work over the following ground ring R: Definition 1. R = Z [89, ~i , Q , ~3 . . . . ]where ~ = Q - 1 if n > 2 , r -1. The multiplicative group generated by the ~'s is isomorphic to Q2/Z2 and we define: X: Q 2 / Z 2 ~ R * via z(m/2")=(Q) m.

Actually, adjoining the r is not essential, but it makes life easier and seems to be quite natural. 9*

126

D. MUMFORD:

Next, to R adjoin independent transcendentals X~, one for each aeQ~ g. Then divide out by the following relations: all (x~Q~,g 3 e Z ~ g.

2) x_,=x~. 4

4.

3) I-I x , , = 2 - ~ i= 1

Z

)c(t71",lz-'y2"ql)'~-Ix,,+,+~

q E 89 z t ~ / z l g

i= 1

4

all el, a2, a3, (x4e Q~ g, where 7 = - 89~ ai. i=1

In what follows, it will be convenient to abbreviate the characters in these formulas as follows: Definition 2. e(a, fl) = ~(' al . f12 - ' f i t " a2) , e,(a) = ~(2'a 1 9a2),

if e~ 89 2g.

Definition 3. A=R[...,X,

. / (ideal generated by) .... Jl~relations 1, 2,3 ~" M~ = Proj (A).

In order to get a preliminary idea of how big M~ is, introduce the subrings: Definition 4. A, = subring of A generated by {X=I 2"aeZzZ~}. Lemma 1. A is integrally dependent on A 2. Proof. By induction, it suffices to check that X, is integrally dependent on A,_ 2, when 2"- ~eZ~ g and n > 2. Use relation (3) with el = ez = ~3 = ~, a 4 = 7 = - ~ . Then since X _ , = X , , we find that X 2 e A , _ I . Q.E.D. Corollary. There are integral affine morphisms : Moo

I"a

Proj (A2)


127

and Moo is the category-theoretic li_m_mof the algebraic schemes Proj (A.), i.e.,for all schemes S,

Hom(S, Moo)_--~ Hom(S, Proj(A,)). Proof. Cf. EGA, Ch. 4. The X~'s will be nothing but the values of the function ~9/~] when L,~.I

we connect _Mooto the moduli problem. It is also convenient tointroduce a second set of generators of the ring A, whose values will be thevalues of the measure/~, in the moduli problem: For all compact open sets U c Q~, let Yv = 2 - " ' ~,, if

Z

Z

X(,, , a)

s

u= U [~+2"z~] and c ~ i ~ i (mod2nZ~). i=1

Using Lemmas I and 2, w8, the relations on the X's go over to the following relations in the Y's: 1. Yv, + Yv, = Yv,~ vz + Yv,,~ vz .

2. Y_v=Yv, r~=o. 3. If we define quadratic polynomials, Zv, for all compact open sets g U c Q zg• by relations (1.) and Zv, xv2=Yv, Yvz, and if ~ ( x , y ) = (x+y, x - y ) as usual, then: Z~(v, • v2)" Z~(v3 x v4) = Z~(v, x v4)" Z~(v3x tt2) 9 In particular, let n > 1 and let l: 2n-' Z~---~ {• 1} be a homomorphism. Define Z~,p=

Y',

l(tl). Y,+~+n+z,z~" Y~-p+~+z,z~.

Then the general relations imply: Z~,p. Z~,~=Z~,~. Z~.p for all a, fl, ~, 6eQ~. Conversely, these relations, for all n's, and l = + 1, imply all the quartic relations.

128

D. MUMFORD:

Moreover, the subring A. generated by X~, with ~ 2 -n Z 2g is just the subring generated by

{Y~I v= u+2nz~, v=2-nz~} or by

{r~+2oz~1~2-nz~}. The group G = S p ( 2 g , Q2) acts on M~o in the following way:

Definition 5. For all (*) U~(X~)----

~

o'~G, let

# e z ~ ~ / z i g ,~~-~(z~ s)

e,(fl/2) e(fl/2, ~). e(712, a--fl). X~_~#_~7,

where y~Q2 g is some fixed element satisfying e. (/3/2) 9e, (o///2) = e(y,/~) for all ~ e Z ~ g n a - l ( Z ~ ) . Concerning this definition, one verifies by mechanical calculation the following: 1. If the 7 in the definition is varied, it must change by an element of Z~ s + a - ~(Z~ s), and, if so, the operator Ur is only changed into a constant multiple of itself: U,~(X~,)=c. Ur all ~. We shall assume that for each (r, some fixed V is chosen. Note that - I ~ S p ( 2 g , Q 2 ) = G , and U_~(X~,)=X_~,. 2. For all triG, ~t~Q 2~ and fl~Z~ s,

U,~(X~ +#) = e, (fl/2) e (fl/2, ~) . Ur (X~). Therefore, if we let M be the free R-module spanned by the X~'s modulo only relations (1.), each Ur defines an R-module homomorphism from M t o M. 3. For all o, "reG, there is a non-zero constant c,,~ such that

U~oV,=c,.,.U,.,, i.e., a ~ Uo is a "projective" representation of G in M. 4. For all a e G , one computes easily that there is a sign e~= + 1 such that U_~o U~=eo U~o U_~. Unfortunately, it does not appear to be easy to show directly that e~ = + 1 for all ~. For example, if g (Z22 g)= Z22 g, e~= e ,(y). However, using (3.), one checks that

and since G is well known to be its own commutator subgroup, this implies 8~---+ 1, all o. But now the submodule A~ of A of elements of degree 1 is just M modulo the span of the elements {X~-X_,}, and this

On the Equations DefiningAbelian Varieties. II

I29

proves that the U.'s induce homomorphisms from A1 to A 1. We will let U~ denote this homomorphism too. 5. The last step is to cheek that all the U.'s induce ring homomorphisms from A to A. Frankly, I balked at directly applying U. to the relations (3.) and seeing what comes out. But, in 2 special cases, it isn't too bad. Suppose first that a is in the subgroup F = Sp(2g, Z2). Then a(Z2g)=Z 2g, and U~ reduces t o : U~(X~)= e(]z[2, ~) . X~ ~_~ ~. In this case, it's not hard to check that U. takes a relation of type (3.) to another relation of the same type, so that U. induces a map from A to .4. Suppose second that a is in the sub-semi-group:

H+

~ [ A I 0 ~ A E G L ( g , Q2)~ c -[\01*A-I/ A(Z[)~Z~ j G.

For such a, Ur reduces to: U.(X.) =

~

e(-/?t[2, ~)-X.~+.a,

# 9 Z~l,t-t Z~

(where/~t is the 2g-vector (/~, 0)). Now, referring back to the proof of Lemma 2, in the last section, we see that an equivalent form of the relations (3.) is: Y~.~ 9 Y~.,= L , , 9 Yr,a, all ~,/?, ?, 6eQ~ ~ where Y~,a = ~ e(-~lt, cOX~+a+~,'X.-a+,,. . ~ z~/z]

For a ~ H +, Y~,a behaves very nicely. One computes easily that: Uo(Y~,a)=

~

e(-nx, a~). e ( - ~ I , o/~)- Y~~+,,,*a+:, 9

From this it follows immediately that U~[Y~.p 9 Y~,a-Y~,6" Yy,~] is an R-linear combination of expressions Y~,,~, 9 Yr " Y~',~'Finally, it follows from the paper of IWAHORI-MATsUMOTO[13] that all the double cosets in F\G/F are represented by matrices: o

A = /\[2a~0

" . "2"0)

O>__al>~a2>= ... >a~.

130

D. MUMFORD:

Since there are in H +, G = F- H + 9 F and our 2 calculations suffice to prove that for every aeG, U, maps A to A. Putting all this together, we conclude that Definition 5 defines a projective representation of G on A, and an action of G on the scheme Moo. In fact, let V~: Moo ~ Moo denote this action, then, by definition, 11"

= u,_,

(the a-1 makes it an action of G instead of the opposed group). Now we can connect Moo to the moduli problem. For all algebraically closed fields k (char(k)#2), there is a map: O: ~r

~ Set of k-valued~ (points of Moo J

(~'oo denotes the set defined at the beginning of this w which assigns to a tower 5 and a q~: V(X)~--~Q~g, the point with homogeneous coordinates X~=8 [00] (ep-1~),

or

Yv=#(U)

(cf. Lemma l and 2, w8). Recall that G acts on ~/4oo(k) as follows:

Definition 6. Let (oq-, ~0)eJCoo(k) and (reG. Then let Uo((~J", q~)) be the pair (J', a o q~), i.e., modify the symplectic isomorphism a to: V(X) ~ , Q22s__~_~Q2~*. Theorem 1. Under O, the actions of G on Moo(k) and on the set of k-valued points of M~ are compatible.

Proof. Recall t h a t 8 variant satisfying a)

[00] is the unique element f of 8 ( F ( 3 - ) ) i n -

f ( x + a ) = e.(a/2), ez(a]2, x).f(x)

for all aeq~-l(Z~g), with e.(~p-l(a))=~(2 t ~1" a2), all ae 89Z~*. If we use the symplectic isomorphism ao q~, the new function 8 ]~1 L,a$ be the unique f in 8(F(~-)) satisfying: a')

will

instead

f ( x + a) = e. (a/2) 9e~(a/2, x) .f(x)

for all aeq~-l(a-~(Z2*)), with e,(tp-~(a-~(ct)))=Z(2 t a~. ~2) all ae89 Let O" [ O] denote this new 8 [~]. Then

=

[~ (q,-


131

To find O~ [0], recall that the functions ~ ~ e;, (fl/2, ~z) . O

span V(X). Therefore, it suffices to find some linear combination of these functions satisfying a'). If we make all these into functions on Q2g via aoq~: V(X)-%Q 2g, we find that

is that linear combination f of the functions gp(o:)=e(fl/2,~), g(a-1 ~ - a - 1 fl),

f l e Q 2g

g ( " ) = ~t [00] (tp- 1 00 such that

a")

f ( e +/6) = e, (/6/2)- e (/712, ~)" f ( e ) ,

for all fleZ~ g. One solution of (a") is the function: g~(~) =

~,

e , ( f l / 2 ) , e(/6]2, o0 9 e(y/2, ~-/6)

tJeZ]g/zigc~crz~g x

g ( a -1 ~ - a - I f l _ ~ - i

~,)

where 7 satisfies e. (fl/2) 9e. (0-1 i l l 2 ) = e ( ~ , fl), all fleZ2, g n a Z ~ g. If this function is not O, it must equal 0~[01o(0-1o~ -'. But on the other hand, X~ has values g(u) at the geometric point x corresponding to ( ~ , ~0). Hence X~ has the values at V, (x):

x=(v,(x))=v: X:(x) = U,_, x , (x) =

~,

e , ( f l / 2 ) , e(fl/2, ~) x

# ez~g/zig n azig

x e (7/2, ~ -/6) X~_ , , _ ~_, a _, _, ~(x)

= g" (~t). Therefore, g ' ~ O , so g~(cr is also the value of X, at the geometric point corresponding to ( ~ , a o tp). Q.E.D.

132

D. MUMFORD:

Now every moduli space is supposed to represent a functor. In our case, instead of making a big fuss over defining afarnily of 2-towers of abelian varieties, over a scheme S, it is simpler to observe that ./g~ (k) is an inverse limit of some of the Jg~(k)'s introduced in w6, and then to define clioo(S) as the corresponding limit of these ~g~(S)'s. Given a polarized 2-tower J-=({X~}, {L~}) and q~: V(X)-%Q~ ~, for all n__>1, we get a) an abelian variety X, = X~., where

T(~) = ~- '(2-" z~ ~), b) a totally symmetric ample invertible sheaf

L.=L~.

on

X.,

of type

6=(22~,22" ,...,22"), c) a symmetric 0-structure: * 2,: ~(L~)___- ~"(~r K(a.) This is a map

... k * x 2 - " Z ~ z v,-~o {1} x 2"Z~ * ~ ( 6 . ) .

( : , q,),--,(X.,L.,&) ./K~(k)---~..Cl6.(k). Note that all these X.'s are canonically isomorphic, e.g., to X I, via diagrams:

x~

xl and that under these isomorphisms, L~ is just L~~- ~. On the other hand, between the functors . ~ , +, and . ~ . , we have a natural transformation: zr~: M~.+ ~~ M ~ . In fact, given f / S , L, and 2 in ~/6. +~(S), we get a) a second ample sheaf M on ~ , by descending L with respect to the isogeny 26: .~ ----~~


133

and the descent data 2-~(K2), where K2~fg(6n+l)x S is the subscheme representing the subfunctor of triples (1, x, l), 2 x = 2 l = 0 . b) Since

~(M)~

normalizer of 2-1(K2) in f#(L) 2-1(K2)

_~ normalizer of K 2 in f~(~n+ 1) ~fq(6n) K2 we get a #-structure #: ~(M)--~(~5.). It can be checked that (W/S, M,/a)e~6~(S ), so we call this rr~((~?/S,

L, 2)). Going back to k-valued points, we have a diagram:

and it is clear that this induces an isomorphism ,/#~o(k) ~ ~

.,r162(k).

Definition 7. For all schemes S, let ~go~(S)= i~_~_.ga~(S). Next, let's translate the results of w 6 on the representability of ./t'a. into the discussion. The results there show that there is an open set: M~ ~ Proj (A*)---M6.

A* = R

I / ) ' M o d u l o Q("(a)=Q("(-a) ['..., Qtn)(a) .... J / ~ a n d certain quartic relationsJ

a e K (ft,)

such that M ~ represented .//~ (over the ring R). Moreover, in Step II of the proof, a canonical morphism

7z: M2a---~M6

134

D. MUMFORD:

was introduced. Iterated, this defines a morphism from M4~ -~ M,, e.g., from M~,§ to M ~ . If you work it out, it is just the projection:

N

II

Proj (A*+ l)

Proj (A*),

n*(Q(")(a))= ~, Q("+')(b),

all

aeK(6.)

2b=a

b ~ K(6r,+ D

(here we identify K(6,) with a subgroup of K(6n+l) as before). In particular, n, is a finite morphism. Moreover, as we saw in w6, (Step II). (a) n, is 6tale at all points of Mb.+, and nZI(M,.)=Mo.+,. In fact: (b) The morphism of schemes ~,: Mo~ transformation of functors n,: ~ . + , ~ . proven in w6.

corresponds to the via the representability

(This is easy to check and we omit the proof.) Now, passing to the limit over the homomorphisms n*, note that the direct limit of the rings A* is just A itself. In fact,

Z/22"Z,

K(6.)= 9 g times

and let

Q(")(al .... , %) ~ Y(2--"a...... 2-"a.)+2"Z~ (ai~Z/22nz). It is easy to check that the relations imposed on the Q(n)(a)'s give us exactly the defining relations on the Yv's. Moreover, A, is just the image of A* in A. (It may very well happen that each n* is injective, so that A -~ U A* and An = A*: b u t / h a v e no proof for this.) Geometrically, this shows that

Moo~_m_m~ . n

Definition 8. Let M~o be the inverse image in Moo of M~. in M0. (independent of n by (a)). 1. Moo ~--~Lm M a. 2. ~oo(S)~- i]jm.tln.(S) 3. The Ma's represent ~tt'~. (compatibly as n varies). Hence: Theorem 2. The scheme M~o represents .t{oo.


135

Recapitulating the discussion, the basic set which we are classifying is:

) f set of polarized 2-towers J ' = (X~, L~) vh'~o(k)= ~ plus symplectic isomorphisms cp: V(~-) ~ > Q2g~

=

set of abelian varieties X, totally symmetric ample L on X of type (4, 4 .... ,4), and symmetric O-structures 2. on if((2" 6)* L), for all n, which are "compatible" as n varies

Quite clearly, we can also say: /r

[ set of abelian varieties X, totally symmetric ] (k) = { ample L on X of type (4,4, ..., 4), and symplectic ~, [ isomorphisms q~: V(X) ~ , Q~g such that q~(T(X)) = 2 Z~ gJ

or: [ set of abelian varieties X, symmetric ample L on X of] " "k" /degree 1, and symplectic isomorphisms: q~: V(X)---~Q~gl ~oo(" ) = / s u c h that r g, and eL,(c.p-l(o0)-Z(2'~l. ~2) alll" In any case, the principal result, on the level of geometric points, is:

Corollary. For all algebraically closedfields k, the map 6) is a bijection between ~[~o(k) and the set of k-valued points of the open set Moo. In other words, the whole tower ~J", plus q~: V ( X ) ~ Q ~ g, is determined by the theta function

(or the measure #) and the theta functions that arise in this way are those which satisfy some finite set of inequalities. Our next task is to determine these inequalities, and hence Moo, explicitly. Department of Mathematics Harvard University Cambridge, Massachussetts

(Received February 20, 1967)