michael harris 1. moduli

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ARITHMETIC OF THE OSCILLATOR REPRESENTATION Michael Harris §1. Moduli 1.1. Let V = Q 2g , and let <, > be the standard symplectic form on V , represented by the matrix 0 I g -I g 0 . Let T be the lattice Z 2g V . We extend <, > to the finite adeles V A f . Let G = Sp(V,<,>). For any additive character ψ : A f μ G m , the pairing (x, y) < x, y > ψ def = ψ(< x, y >) is a bilinear form on V A f , with values in μ . Moreover, T ˆ Z is a maximal subgroup U V A f with the property that <t 1 ,t 2 > ψ =1 t 1 ,t 2 U. If ψ : A f μ is another character, then ψ(z )= ψ(az ), for some a A f,× . On the other hand, the action of Gal(Q ab /Q) on μ G m defines an action on Hom(A f , G m ). There exists σ Gal(Q ab /Q) such that ψ = ψ σ . If we denote the reciprocity isomorphism ˆ Z -→ Gal(Q ab /Q) a σ a normalized so that p goes to the inverse of Frobenius (mod p), then evidently (1.1.1) ψ(az )= ψ σ -1 a (z ) ψ Hom(A f , G m ). 1.2. Let N 3 be a positive integer, and let M g N be the moduli space of g- dimensional principally polarized abelian varieties A with level N structure α N : A[N ] -→ N -1 T/T V A f /T. We require that the restriction to N -1 T/T of the bilinear form <, > ψ on V A f /T correspond via α, up to a scalar multiple in (Z/N Z) × , to the Weil pairing on A[N ] induced by the given polarization. Choose ψ Hom(A f / ˆ Z, G m ), and let M g N,ψ ⊂M g N be the connected component for which α is a symplectic isomorphism, with respect to <, > ψ on the right hand side. Since N 3, there exists a universal abelian scheme A g N /M g N , with principal polarization [Λ] g , and a symplectic similitude of finite group schemes α g N : A g N [N ] -→ (N -1 T/T ) M g N . Typeset by A M S-T E X 1

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Page 1: Michael Harris 1. Moduli


Michael Harris

§1. Moduli

1.1. Let V = Q2g, and let <,> be the standard symplectic form on V , represented

by the matrix(

0 Ig−Ig 0

). Let T be the lattice Z2g ⊂ V . We extend <,> to the

finite adeles V ⊗Af . Let G = Sp(V,<,>). For any additive character

ψ : Af → µ∞ ⊂ Gm,

the pairing(x, y) 7→< x, y >ψ

def= ψ(< x, y >)

is a bilinear form on V ⊗Af , with values in µ∞. Moreover, T ⊗ Z is a maximalsubgroup U ⊂ V ⊗Af with the property that

< t1, t2 >ψ= 1 ∀t1, t2 ∈ U.

If ψ : Af → µ∞ is another character, then ψ(z) = ψ(az), for some a ∈ Af,×.On the other hand, the action of Gal(Qab/Q) on µ∞ ⊂ Gm defines an action onHom(Af ,Gm). There exists σ ∈ Gal(Qab/Q) such that ψ′ = ψσ. If we denote thereciprocity isomorphism

Z ∼−→ Gal(Qab/Q)

a 7→ σa

normalized so that p goes to the inverse of Frobenius (mod p), then evidently

(1.1.1) ψ(az) = ψσ−1a (z) ∀ψ ∈ Hom(Af ,Gm).

1.2. Let N ≥ 3 be a positive integer, and let MgN be the moduli space of g-

dimensional principally polarized abelian varieties A with level N structure

αN : A[N ] ∼−→ N−1T/T ⊂ V ⊗Af/T.

We require that the restriction to N−1T/T of the bilinear form <,>ψ on V ⊗Af/Tcorrespond via α, up to a scalar multiple in (Z/NZ)×, to the Weil pairing onA[N ] induced by the given polarization. Choose ψ ∈ Hom(Af/Z,Gm), and letMg

N,ψ ⊂MgN be the connected component for which α is a symplectic isomorphism,

with respect to <,>ψ on the right hand side.Since N ≥ 3, there exists a universal abelian scheme AgN/M

gN , with principal

polarization [Λ]g, and a symplectic similitude of finite group schemes

αgN : AgN [N ] ∼−→ (N−1T/T )MgN.

Typeset by AMS-TEX


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1.3. Let π : AgN → MgN be the canonical map. With ψ as in 1.2, let AgN,ψ =

π−1(MgN,ψ). We let

Mg = lim←−N

MgN , M

gψ = lim←−



define Ag to be the pullback toMg of AgN/MgN , for any N , and define Agψ likewise.

1.4. Let S be any scheme over Spec(Q), and let A be an abelian scheme of relativedimension g over S. The family T = T (A) of homomorphisms B/S → A/S, withkernel finite and flat over S, is an inverse system for which

[n] = (multiplicationbyn) : A → A

forms a cofinal subset. The inverse limit over T exists in the category of schemes;we denote it A. Lacking a better name, we call A the isocompletion of A. We let

A(t) = lim←−B∈T (A)


A full level structure on A (or on A) is an isomorphism over S

(1.4.1) α : A(t) ∼−→ V ⊗AfS

Obviously any finite flat isogeny A/S → A′/S induces an isomorphism A∼−→ A′.

Following Deligne, we define the category of abelian varieties up to isogeny, or a.v.i.,to be the category whose objects are abelian schemes over S and in which isogeniesare isomorphisms. The functor A 7→ A takes the category of abelian varieties overQ to the category of schemes over Q; the essential image of this functor is equivalentto the category of a.v.i.

More generally, if d ∈ Z, we can consider the family Td (resp. T d ofB/S → A/S ∈T whose kernel is of order equal to a product of primes dividing d (resp of orderprime-to-d); we can define (A)(d) (resp. A(d) to be the inverse limit over ′CalTd(resp. T d). If we define the category of abelian schemes over S up to d-primaryisogeny (resp. prime-to-d isogeny) to be the category of abelian varieties with iso-genies of degree equal to a product of primes dividing d (resp. prime to d ) thenA 7→ A(d) and A 7→ A(d) take the category of abelian schemes over Z[d−1] (resp.over the localization of Z at d) to the respective categories of abelian schemes upto isogeny.

For any abelian variety A, let NS(A) be the Neron-Severi group of A. ThenNS(A) ⊗ Q depends only on the isogeny class of A. A polarized a.v.i. is an a.v.i.A together with an element of (NS(A)⊗Q)/Q∗.

1.5. We apply these remarks to the situation considered in 1.2, 1.3. There is auniversal full level structure on Ag:

(1.5.1.) αg : hatAg(t) ∼−→ V ⊗AfMg

Let ψ ∈ Hom(Af ,Gm), and let Agψ be the restriction of Ag to Mgψ. Now, Mg

ψ isthe moduli space for polarized a.v.i. with full level structures (1.5.1) of type ψ, inthe obvious sense, and the morphism Agψ → M

gψ may be viewed as the universal

polarized a.v.i. with full level structure of type ψ.

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1.6. Let GSp = GSp(V,<,>) be the group of symplectic similitudes of V . Thegroup GSp(Af ) acts naturally on the moduli space Mg

ψ through its action on fulllevel structures. In fact, a geometric point of Mg

ψ is given by a triple (A,Λ, α),where A is an a.v.i., Λ is a polarization on A, and α is a full level structure (1.4.1).Thus if γ ∈ GSp(Af ) and x = (A,Λ, α) ∈Mg

ψ, then γ(x) = (A,Λ, γ α).Similarly, a geometric point of Agψ is given by a quadruple (A, a,Λ, α), where

a ∈ A. Thus the action of GSp(Af ) onMgψ lifts to an action on Agψ : γ(A, a,Λ, α) =

(A, a,Λ, γ α), for all γ ∈ GSp(Af ).

2. Linear systems

Most of the contents of §2 and §3 are copied verbatim from Mumford’s series ofarticles “On the equations defining abelian varieties.” Not all of our assertions areliterally stated as such by Mumford, but the reader will easily supply the missingdetails.

The following definitions and results are mostly in §1 and §6 of [M1]. Let Sbe a scheme over SpecQ. The basic object in this section is a pair (A,L), whereπ : A → S is an abelian scheme of relative dimension g, and L is a relatively ampleinvertible sheaf (= line bundle) on A. One should think of S as the moduli spaceof polarized abelian varieties with level structure, although eventually more generalcases will be considered.

Let T be a scheme over S, and let H(L)(T ) be the group of sections α : T → ATsuch that, if Tα : AT → AT is translation by α, then

T ∗αL∼= L⊗ π∗TM

for some invertible sheaf M on T ; here πT : AT → T is the natural morphism. LetH0(L)(T ) be the subgroup of α ∈ H(L) such that

T ∗αL∼= L.

Finally, let G(L)(T ) be the group of pairs (α, ϕ), where α ∈ H0(L)(T ) and ϕ :T ∗αL

∼−→ L is an isomorphism. Then T 7→ H(L)(T ) and T 7→ G(L)(T ) are functorsfrom the category of S-schemes to the category of groups. Mumford shows in [M1],II, p. 76, that these functors are representable by group schemes G(L) and H(L),respectively, flat and of finite presentation over S, which fit into an exact sequencein the Zariski topology:

(2.1) 1 → (Gm)S → G(L) → H(L) → 1.

Moreover, H(L) is a finite flat subgroup scheme of A.The group scheme G(L) is a version of the Heisenberg group. It acts naturally

on the sheaf π ∗ L: If U ⊂ S is an open subset, γ = (α, ϕ) ∈ G(L)(U), andϕ ∈ Γ(AU , L), then

(2.2) ϕγdef= ϕ T ∗α(f) ∈ Γ(AU , L)

This action can be understood most easily when S is the spectrum of a local ringR. Then H(L) is the constant group scheme H(L) × S. In general, the extension

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(2.1) defines a skew symmetric pairing on H(L) with values in Gm: if x, y ∈ G(L)map to x, y ∈ H(L), then

(2.3) eL(x, y) = xyx−1y−1 ∈ Gm

is well-defined and non-degenerate. When S = SpecR as above, then we maydecompose H(L) as a product

(2.4) H(L) ∼= H0 × H0,

where H0 is a maximal isotropic subgroup for eL, and hatH0 = Hom(H0,Gm).On the other hand, we may define the Heisenberg group G(H0) to be the set of

triples(t, h, h) | t ∈ Gm, h ∈ H0, h ∈ H0

with multiplication

(2.5) (t, h, h) · (t‘, h‘, h‘) = (t · t‘ · h′(h), h+ h‘, h+ h′).

An isomorphism as in (2.4) induces an isomorphism

(2.6) G(L)def= G(L)(S) ∼= G(H0).

It follows from a version of the Stone-Von Neumann theorem, and from resultsof Mumford in §1 of [M1], that

Proposition 2.7 (Mumford). The R-module Γ(A,L) is free of rank |H0| =√|H(L)| . The representation (2.1.2) of G(L) on Γ(A,L) is identified, via (2.6),

with the unique irreducible representation of G(H0) on which its center Gm acts bythe identity character

t 7→ (multiplication by t).

Although H0 is not uniquely determined by L, its set of elementary divisors is:

H0 ∼= Z/d1Z⊕ · · · ⊕ Z/dgZ.

for some g-tuple of integers δ = (d1, . . . , dg), with d1 | d2 | · · · | dg. We say L isof type δ; the type of a relatively ample line bundle is constant on an irreduciblescheme S over Spec(Z) provided dg is invertible on S. We let

d = d(δ) =∏i


Then if (A,L) and π : A → S are as above, the sheaf π∗L is locally free of rank don S.


K(δ) =g⊕i=1


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with δ as above; let K(δ) = Hom(K(δ),Gm), and let H(δ) = K(δ)× K(δ). We letG(δ) = G(K(δ)), with multiplication as in (2.5). Following Mumford, we define aθ-structure on (A,L) to be an isomorphism

G(L) ∼−→ G(δ)S

which restricts to the identity on their respective centers Gm. Of course, the alge-braic group G(δ) has a representation ρδ on the free Z[d−1]-module

Vδ = Vδ(Z[d−1]) = functions from Kδ to Z[d−1] :

If f ∈ Vδ, α ∈ Gm(Q), x ∈ K(δ), ` ∈ K(δ), then

(2.8) ρδ((α, x, `))f(y) = α · `(y) · f(x+ y).

We paraphrase a proposition of Mumford ([M1], II, p. 80):

Proposition 2.9 (Mumford). Let (A,L) be of type δ, and let

λ : G(L) ∼−→ G(δ)S

be a θ-structure on (A,L). Denote by ρλ the representation of G(δ)S on πL definedby (2.2) and the isomorphism λ. Let K = K(L, λ) ⊂ πL be the subsheaf on whichρλ(K(δ)) acts trivially. Define an action of G(δ)S on Vδ ⊗K by tensoring ρλ withthe trivial action on K. Then there is an isomorphism of sheaves

(2.10) πL∼−→ Vδ ⊗K

equivariant with respect to G(δ)S, and unique up to multiplication by an element ofΓ(S,O∗S).

2.11. Suppose now that (σ : B → S,M) is another polarized abelian scheme overS, and suppose we are given an isogeny p : A → B over S, with kernel K flat overS, and an isomorphism L

∼−→ p∗M . In [M1], p. 290 ff., Mumford constructs asubgroup scheme K = K(L,M) ⊂ G(L) which fits into a diagram

G(L) −−−−→ H(L) −−−−→ 0

∪ ∪

K∼−−−−→ K

and a canonical isomorphism Z(K)/K ∼−→ G(M), where Z(K) is the centralizer ofK in G(L).

The isomorphism K∼−→ K ⊂ G(L) defines an action of K on L; this action is

the descent datum corresponding to the isomorphism L∼−→ p ∗M . In particular,

the canonical morphism σ∗M → π∗L provides an isomorphism σ∗M∼−→ (π ∗ L)K

(cf. [M1], §1, proof of Theorem 4). On the other hand, in characteristic prime tod, there is a natural projection

P (L,M) : (π∗L)K → (π ∗ L)K ∼= σ∗M.

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Let R(L,M) = KerP (L,M) ⊂ π∗L; R(L,M) is a locally free subsheaf of π∗L.Suppose (B,M) is of type η, and let


λ : G(L) ∼−−−−→ G(δ)S

∪ ∪

Z(K) ∼−−−−→ Z(λ(K))y yµ : G(M) ∼−−−−→ G(η)S

be a commutative diagram where the horizontal arrows are θ-structures and the leftvertical arrow is given by the isomorphism Z(K)/K ∼−→ G(M). We may naturally

identify Vη ∼= Vλ(K)δ . Proposition 2.9 then gives us a commutative diagram of

sheaves over S, equivariant under Z(λ(K)):


π∗L∼−−−−→ Vδ ⊗K(L, λ)y y

σ∗M∼−−−−→ Vη ⊗K(M,µ).

Here the left vertical arrow is P (L,M) and the left vertical arrow is of the formP (δ, η)⊗t, where P (δ, η) : Vδ → V


∼= Vη is the projection and t : K(L, λ) → K(M,µ)makes the diagram commute. Note that the horizontal arrows are determined onlyup to multiplication by an element of Γ(S,O∗S).

Now L and M are both relatively ample, hence define morphisms A → PS(π∗L),B → PS(σ∗M). It follows immediately from the ampleness of M that the image ofA in PS(π∗L) does not intersect PS(R(L,M)). The homomorphism P (L,M) deter-mines a morphism PS(π∗L)−PS(R(L,M)) → PS(σ∗M). Let R(δ, η) = Ker P (δ, η).We obtain a commutative diagram of schemes over S:


A∼−−−−→ PS(π∗L)− PS(R(L,M)) ∼−−−−→ (P(Vδ)− P(R(δ, η))× Sy y y

B∼−−−−→ PS(σ∗M) ∼−−−−→ P(Vη)× S.

3. Polarized towers

We want to work out the adelic analogue of the constructions in §2. AlthoughMumford only concerns himself with the 2-adic theory, there is no difficulty inapplying his methods to the general p-adic case. Indeed, at no point do we makeuse of the deeper parts of Mumford’s theory.

We begin by rigidifying the problem, although this is not essential at this point.Let π : A → S be as in §2.

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Definition 3.1. A (relative) line bundle L over A is symmetric if there is anisomorphism

ι : L ∼−→ (−1)∗L

where (−1) denotes multiplication by (−1) on A. The isomorphism ι induces anisomorphism

ιx : Lx∼−→ Lx

for each point x of order 2 in A. We say L is totally symmetric if ι can be chosenso that ιx is the identity for all x ∈ A[2].

3.2. Remarks. (a) Here we have identified A[2] with the set of geometric pointsof the finite flat group scheme [Ker(2) : A → A] over S, where (2) denotes multi-plication by 2.

(b) If L is any line bundle, then L⊗ (−1)∗(L) is totally symmetric.(c) The most important property of totally symmetric line bundles is the follow-

ing: If L and L′ are algebraically equivalent totally symmetric line bundles on A,then there exists a line bundle M on S such that

L ∼= L′ ⊗ π∗M.

(d) On p. 78 of [M1, II], Mumford defines a canonical involution

δ−1 : G(L) → G(L)

for any symmetric line bundle L. On the other hand, there is an obvious involutionD−1 : G(δ) → G(δ), for any δ:

D−1(α, x, `) = (α,−x,−`), α ∈ Gm, x ∈ K(δ), ` ∈ K(δ).

A θ-structure λ : G(L) ∼−→ G(δ)S is symmetric if

λ δ−1 = D−1 λ.

3.3. A tower (resp. a prime-to-N tower, for some positive integer N) is an inversesystem of abelian schemes over S

T = πα : Aα → Sα∈Σ

indexed by a partially ordered set Σ, and isogenies pαβ : Aα → Aβ (resp. isogeniesof degree prime to N) whenever α > β, satisfying the compatibility conditionpβγ pαβ = pαγ . We also require that T satisfy the following saturation condition:if Aα ∈ T and B is an abelian scheme over S admitting an isogeny (resp. prime-to-N isogeny) p : B → Aα (resp. p : Aα → B) then B is isomorphic to some Aβwith β > α (resp. α > β) in such a way that p corresponds to pβα (resp. pαβ). Toavoid trivialities, we assume deg pαβ > 1 whenever α > β. We also do not requirethat the Aα be distinct; indeed, we require that the isogeny (n) = multiplicationby n:

(n) : Aα → Aα

be an isogeny in the tower (resp. provided (n,N) = 1). Finally, we assume that,if α > β1, β2, and Ki = Ker(pαβi), i = 1, 2, then K1 ⊂ K2 ⇔ β1 ≥ β2. We letT = Aα, for any α ∈ Σ, where Aα is the isocompletion defined in 1.4.

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A polarized tower (T ,P) is a tower T = Aα and a collection Lα/Aα of sym-metric relatively ample line bundles, ∀α ∈ Σ, with isomorphisms

ϕαβ : p ∗αβ (Lβ)∼−→ Lα

satisfying ϕαβ ϕβγ = ϕαγ . We require furthermore that Lα be totally symmetricfor all but finitely many α. We also make the following saturation requirement:Let Aα ∈ T ; let B be an abelian scheme over S admitting an isogeny p : B → Aα(resp. p : Aα → B); one can also place restrictions on the degree as in the previousparagraph. Suppose there exists a line bundle LB/B such that p ∗ Lα ' LB (resp.p ∗ LB ' Lα). As above, we have an isomorphism σ : B ∼−→ Aβ , for some β ∈ Σ,and we assume that LB ' σ∗(Lβ). Given an isogeny pαβ, we let Kαβ = K(Lα, Lβ)be the subgroup scheme of G(Lα) defined in 2.11; then Z(Kαβ)/Kαβ ' G(Lβ).

The symbol P above is used to denote the polarization, or the set Lαα∈Σ. Thehypothesis of saturation implies easily (cf. [M1], I, p. 293) that Lα is of degree 1for (exactly) one α ∈ Σ.

Any pair (A,L) over S, with L totally symmetric and relatively ample, generatesa polarized tower over S in the obvious way. If (A′, L′) is isogenous to (A,L) , inthe sense that there exists an isogeny p : A → A′ such that p ∗ (L′) ' L, then thetowers generated by (A,L) and (A′, L′) are isomorphic. Thus we may speak of anisogeny class of polarized towers.

3.4. If (T ,P) is a polarized tower, then the locally free sheaves πα,∗(Lα) form aninverse system over S:

ϕαβ,∗ : πα,∗(Lα) → (πα,∗(Lα))Kαβ ∼−→ πβ,∗(Lβ)

and we may take the inverse limit

π∗(P) = lim←−απα,∗(Lα).

The fiber at x ∈ S of π∗(P) is just the inverse limit of the spaces of sections of allthe line bundles Lα/Aα.

Let V f (T ) = Aα(t), for any α ∈ Σ, (notation (1.4.1)), viewed as a sheaf in theZariski topology on S. Then V f (T ) is canonically isomorphic to the etale homologygroup

V t(T ) ' (π`R−1πα,∗(Z`))⊗Q

for any α ∈ Σ. Assuming T has a polarization P as above, represent x ∈ V f (T ) asa limit

x = (xα)α∈Σ xα ∈ Aα(tors),

and letΣx = α ∈ Σ | xα ∈ H(Lα).

We define the Heisenberg group scheme of (T ,P) to be the group scheme G(P)whose T -valued points, for T a scheme over S, are pairs

(x, ϕαα∈Σx) x = (xα) ∈ V f (T ),

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ϕα : T ∗xα(Lα) ∼−→ Lα.

We require the isomorphisms ϕα to satisfy natural compatibility conditions, as in[M1], §7. There is an exact sequence of sheaves in the Zariski topology

(3.5) 1 → Gm,S → G(P) → V f (T ) → 1.

For any α ∈ Σ, the lattice K(α) = ker(Aα(t) → Aα(tors)) ⊂ Aα(t) ' V f (T )lifts canonically to a subgroup scheme K(α) ⊂ G(P). In [M1], II, p. 103 ff. (whereour K(α) is denoted K(α)), Mumford shows that Z(K(α))/K(α), where Z(K(α))is the centralizer of K(α) in G(P), is canonically isomorphic to G(Lα).

In order to continue, we provisionally choose an isomorphism

ψ : Af/Z∼−→ lim−→


def.= µ∞.

Let G(g, ψ) be the standard Heisenberg group on the finite adeles:

G(g, ψ) = (α, x, `) | α ∈ Gm, x, ` ∈ (Af )g,

with multiplication

(α1, x1, `1) · (α2, x2, `2) = (α1 · α2 · ψ(tx1 · `2), x1 + x2, `1 + `2).

In analogy with (2.8), there is a natural representation

ρψ : G(g, ψ) → Aut(Sg)

whereSg is the Schwartz space of locally constant, compactly supported Qab-valuedfunctions on (Af )g. (The groups Aut(Sg) and G(g, ψ), as well as the representationρψ, have interpretations in the category of schemes over Q: cf. Appendix to §4.)

3.7. The comparison of ρψ with the actions on finite levels is worked out in detailin [M1], II, pp. 110-111. Let U ⊂ V be two lattices in Qg; let U = U⊗Af ⊂ (Af )g,V = V ⊗Af ⊂ (Af )g. Define Sg(U, V ) to be the space of functions on (Af )g withsupport in V , constant modulo U . We say (U ′, V ′) ⊂ (U, V ) if U ⊂ U ′ ⊂ V ′ ⊂ V ;then we have maps

Res : Sg(U, V ) → Sg(U, V ′); Tr : Sg(U, V ′) → Sg(U ′, V ′),

where Res is restriction of functions on V to functions on V ′, and Tr(f) =∫U ′/U

f(u)du, where du is Haar measure on U ′/U with∫U ′/U

du = 1. In termsof these two maps, we may define

Sg = lim←−U⊂V

Sg(U, V ).

Given (U, V ) as above, we may identify V /U ' K(δ) for some δ as in §2; in termsof this identification, Sg(U, V ) is isomorphic to Vδ. LetK(U, V ) = 1×U××V ⊥ ⊂G(g, ψ), where

V ⊥ = y ∈ (Af )g | ψ(tx · y) = 1∀x ∈ V .

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Then Z(K(U, V ))/K(U, V ) ' G(δ). One sees easily that Sg(U, V ) = (Sg)K(U,V ),and that the corresponding action of G(δ) ' Z(K(U, V ))/K(U, V ) on Vδ ' Sg(U, V )is just (2.8).

Suppose g(U ′, V ′) ⊂ (U, V ) and V ′/U ′ ' K(µ). The map TrRes : Sg(U, V ) → Sg(U ′, V ′)coincides with projection onto the K(U ′, V ′)-invariant subspace. We have an actionof


Z(K(U, V ))/K(U, V ) ' G(g, ψ) (cf. [M1], II, p. 103)

on Sg = lim←−Sg(U, V ) induced from the actions of G(δ) ' Z(K(U, V ))/K(U, V ) on

Vδ ' Sg(U, V ) for all (U, V ); this action is exactly ρψ.As in the case of finite level, we have the following data:

(3.8) A skew symmetric pairing eP : V f (T )⊗ V f (T ) → Gm;(3.9) involutions δ−1 : G(P) ∼−→ G(P), D−1 : G(g, ψ) ∼−→ G(g, ψ).

3.10. A symmetric θ-structure on (T ,P), of type ψ, is an isomorphism

c : G(P) ∼−→ G(g, ψ)

which restricts to the identity on the mutual center Gm and which satisfies

c δ−1 = D−1 c.

As explained in [M1],II, p. 106, there is a one-to-one correspondence betweensymmetric θ-structures of type ψ, and full level structures

β : V f (P) ∼−→ (Af )g × (Af )g,

under which the bilinear pairing eP on the left-hand side corresponds to the pairing

(x1, y1)⊗ (x2, y2) 7→ ψ(tx1 · y2 − tx2 · y1).

Such a level structure β will be called symplectic (of type ψ).In analogy with Proposition 2.9, we have

3.11. Proposition. : Let (T ,P) be a polarized tower, and let

c : G(P) ∼−→ G(g, ψ)

be a symmetric θ-structure of type ψ. There is a unique line bundle K/S, and anisomorphism of G(g, ψ)-modules

π∗(P) ∼−→ Sg ⊗Qab K

where G(g, ψ) acts trivially on K and through ρψ (resp., through the isomorphismc−1) on Sg (resp., on π∗(P). This isomorphism is unique up to multiplication bya scalar in Γ(S,O∗S).

Proof. : One simply takes the limit over the corresponding θ-structures at finitelevel, checking that they are compatible by using 3.7 and the commutativity of(2.11.2). We observe that the existence of a symmetric θ-structure of type ψ overS implies that S is a scheme over Spec(Qab). We will account for this below.

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4. Construction of the bundle of forms of weight 12

Let N ≥ 3 be a positive integer, and define MgN , π : AgN → M

gN , [Λ]g, and αgN

as in §1. The polarization [Λ]g defines at each fiber AgN,x of AgN/MgN an algebraic

equivalence class [Λ]gx of ample line bundles on AgN,x of degree one. For any Λ ∈ [Λ]gx, let LgN,x = Λ⊗ (−1)∗Λ; then LgN,x depends only on the algebraic equivalence class[Λ]gx, and is totally symmetric (cf. 3.2). This construction globalizes to definea relatively ample, totally symmetric line bundle LgN over AgN . A priori, LgN isdetermined only up to tensoring with the pullback of a line bundle on Mg

N . Letε :Mg

N → AgN be the zero section. We normalize LgN by requiring that

ε∗(LgN ) ' OMgN


then LgN is determined uniquely up to isomorphism.Pick a positive integer d ≥ 3, and let L = Lg,d be the pullback of LgN under

multiplication by d. Let δ be the type of L. We assume N large enough so that

(4.1) H(L×2) ⊂ AgN [N ],

and we assume d is the largest integer for which (4.1) holds.Using ψ, we may define an isomorphism, for any d0 ∈ Z:

Fψ : Z/d0Z ∼−→ Z/d0Z; Fψ(a)(b) = ψ(d−10 ab), a, b ∈ Z/d0Z.

In this way, we may define likewise

Fψ : K(δ) ∼−→ K(δ).

We then have a group G(δ, ψ), whose points are given by Gm ×K(δ)×K(δ), withmultiplication induced from 2.5 via Fψ.

The remarks in [M1] I, pp. 317-320 imply that the hypothesis (4.1) allows us todefine a unique symmetric θ-structure

β : G(L) ∼−→ G(δ, ψ)MgN,ψ

which reduces mod centers to the restriction of αgN to H(L) ⊂ H(L2). It thusfollows from Proposition 2.9 that there exists a canonically defined line bundleΘN,ψ overMg

N,ψ, and an isomorphism

(4.2) Sch : π∗L⊗ΘN,ψ∼−→ Vδ ⊗OCalMg


We recall that Sch intertwines the action of G(L) on π∗L with the action of G(δ, ψ)on Vδ, and is uniquely determined up to a scalar in Γ(Mg


), which isisomorphic to Gm except when g = 1 (the case g = 1 will be taken care of ??? whenwe discuss Fourier expansions, in §7 below).

We let LΘ = L ⊗ π∗ΘN,ψ. Then LΘ is relatively very ample, since d ≥ 3, andwe have an imbedding overMg


(4.3) AgN,ψ → PMgN,ψ

(π∗L⊗ΘN,ψ) ∼−→ P(Vδ)×MgN,ψ,

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where the second isomorphism is Sch.Denote by jthe composite of (4.3) with projection on the first factor:

j : AgN,ψ → P(Vδ).


(4.4) (j ε) ∗ O(1)P(Vδ) ' ΘN,ψ.

The content of [M1], §6, is the identification of the image of j ε in P(Vδ). Forus, the important point is a much more elementary fact: namely, that the image ofj ε is a moduli space of a slightly lower level (on the order of N/2). This is notreally difficult to make explicit, but is suffices for our purposes to work in the limit.

Thus, letMg, Ag be as in 1.3, and define Lg likewise. Define Lg,d as above. Thepolarized abelian scheme (Ag,Lg) defines a polarized tower, in the sense of 1.2, inwhich the pairs (g, Lg,d), d = 1, 2, . . . form a set of cofinal objects. We denote thispolarized tower (T g,Pg) and define Ag as in 1.4.

Let Agψ be the restriction of Ag toMgψ. There is a universal full level structure

on Agψ:

(4.5) αg : Agψ(tors) ∼−→ V ⊗AfMgψ

under which the skew symmetric pairings eP and <,>ψ correspond. Let

π∗P = lim←−d


as above. As in 3.10, we have a unique symmetric θ-structure β : G(P) ∼−→ G(g, ψ)compatible with αg. Proposition 3.11 provides us with a line bundle Θψ on Mg

ψ,and an equivariant isomorphism

Sch : π∗P ⊗Θψ∼−→ Sg ⊗OMg


This induces an imbedding

(4.6) Agψ → PMgN ,ψ

(π∗Θψ) ∼−→ P(Sg)×Mgψ

Let ϑψ : Agψ → P(Sg) denote projection on the first factor. The composite of theidentity section ε : Mg

ψ → Agψ, followed by ϑψ, defines a morphism also denoted

ϑψ :Mgψ → P(Sg). The following theorem follows directly from the remarks on p.

82 of [M1], II:

Theorem 4.7 (Mumford). : The morphism ϑψ :Mgψ → P(Sg is a locally closed

immersion; the bundle Θψ = ϑ∗ψ(OP(Sg)(1)).

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4.8. The action of G(g, ψ) on Sg induces a projective representation P(ωψ) ofG(Af ) on Sg, first defined by Weil. For our purposes it is most convenient towork at finite levels. Let

B0(g, ψ) = γ ∈ Aut(G(g, ψ)) | γ restricts to the identity on Gm.

Now B0(g, ψ) is canonically the semi-direct product of G(Af ) and the group ofinner automorphisms of G(g, ψ). (Actually in [W], pp. 180-183, 188, Weil showsthat B0(g, ψ) is the semi-direct product of G(g, ψ) and the pseudosymplectic groupover Af , but since we are working in characteristic zero, the pseudosymplecticgroup and G(Af ) are canonically isomorphic. His calculations make sense in thecategory of algebraic groups over Q, as described in the appendix to §4.)

Thus G(Af ) acts naturally on G(g, ψ), and for γ ∈ G(Af ), we can define arepresentation

ργψ : G(g, ψ) → Aut(Sg) : ργψ(σ) = ρψ(γ−1σ).

This can be interpreted in terms of finite levels as follows. Let U ⊂ (Af )g be alattice, and let Γ(U) ⊂ G(Af ) be the stabilizer of the lattice U×U ⊂ (Af )g×(Af )g.Let S(U) be the Schwartz space of U , and let G(U,ψ) = Gm × U × U⊥ ⊂ G(g, ψ),with multiplication law (3.6). We denote by ρψ,U the natural representation ofG(U,ψ)onS(U).

Now the map G(Af ) → B0(g, ψ) takes Γ(U) to the stabilizer of G(U,ψ). Asabove, each γ ∈ Γ(U) defines a representation ργψ,U on S(U). By an analogue ofthe Stone-Von Neumann theorem (or a generalization of Proposition 2.7), one seesthat ρψ,U and ργψ,U are equivalent irreducible representations. The correspond-ing projective representations are thus canonically isomorphic. We thus obtain aunique morphism P(ωψ,U )(γ) : P(S(U)) → P(S(U)) which intertwines ρψ,U andργψ,U ; P(ωψ,U ) is a projective representation of Γ(U).

On the other hand, Γ(U) = Γ(n−1U), n = 1, 2, . . . . We thus obtain a compatiblesystem of projective representations P(ωψ, n−1U) of Γ(U), n = 1, 2, . . . , where themap S(n−1U) → S(m−1U), for m ÷ n, is given by restriction of functions. In thelimit, we have a projective representation P(ωψ, (U)) of Γ(U) on lim←−n(n

−1U) ' Sg;for any γ ∈ Γ(U), P(ωψ, (U)) intertwines the restrictions to G(U,ψ) of ρpsi and ργψ.

Now every element of G(Af ) belongs to Γ(U) for some lattice U . If γ ∈ Γ(U) ∩Γ(V ), then the projective representations P(ωψ,(U)) and P(ωψ,(V )) both intertwineρψ |G(U∩V ) and ργψ |G(U∩V ), hence coincide. In this way we obtain a projectiverepresentation P(ωψ) of G(Af ) on Sg.

In [W], Weil proves that P(ωψ) (or rather the analogous representation on theSchwartz space) lifts to a genuine representation of a double cover of G(Af ). Ingeneral, we prefer to work with the tautological representation associated with theprojective representation:

ωψ : G(Af ) → Aut(Sg).

Here G(Af ) is an extension of G(Af ) by Gm defined by the projective represen-tation P(ωψ). Thus G(Af ) acts on OP(Sg)(1), extending the tautological action ofGm, given by t 7→ multiplication by t. We explain in the appendix to §4, below,how G(Af ) may be regarded as an extension in the category of group schemes, andhow to define the structure of algebraic variety on P(Sg).

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Proposition 4.9. . The action of G(Af ) on P(Sg) restricts to the canonical actionon Agψ. In particular, there is an action of G(Af ) on Θψ which covers the action ofG(Af ) onMg

ψ. The restriction of this action to the subgroup Gm is the tautologicalaction.

Proof. The proposition follows immediately from the commutativity of the diagram


G(P) −−−−→ Aut(π∗P)

γcy y

G(g, ψ)ργψ⊗1−−−−→ Aut(Sg ⊗Qab K)

for all γ ∈ G(Af ). Here c and the right-hand vertical arrow are as in Proposition3.11, and γ c is c followed by the action of γ ∈ G(Af ) ⊂ B0(g, ψ).

4.10. Remark. It is possible, a priori , that the extension G(Af ) depends onthe choice of character ψ. There is an easy way to see that the isomorphism classof G(Af ) is independent of ψ. Let B0(g, ψ)1 denote the normalizer in Aut(Sg) ofρψ(G(g, ψ)). Let B0(g, ψ) ⊂ B0(g, ψ)1 be the subgroup of elements α for whichthere exists an increasing filtration

0 ⊂ U1 ⊂ U2 ⊂ · · · ⊂⋃i

Ui = (Af )g

such that α stabilizes ker(Sg → S(Ui)) for all i. There is an exact sequence ofgroup schemes

(4.10.1) 1 → Gm → B0(g, ψ) α−→ B0(g, ψ) → 1;

the surjectivity of α is equivalent to the existence of the projective representationP(ωψ). We have G(Af ) = α−1(G(Af )) ⊂ B0(g, ψ); it is canonically determined byG(g, ψ).

But the subgroup ρψ(G(g, ψ)) ⊂ Aut(Sg) is independent of ψ. In fact, givena non-trivial ψ, every non-trivial additive character of Af is of the form ψt(x) =ψ(t−1x), for some t ∈ Af,×. Now the map jt : G(g, ψ) → G(g, ψt), defined in thecoordinates (3.7) by jt(α, x, `) = (α, x, t`), is an isomorphism of groups. Obviously

(4.10.2) ρψt(jt(α, x, `)) = ρψ((α, x, `)),∀(α, x, `) ∈ G(g, ψ).

It follows that B0(g, ψ) and B0(g, ψt) are canonically isomorphic, for every t ∈Af,×. Using the exact sequence (4.10.1), we see that G(Af ) is independent of thechoice of ψ.

Appendix to §4: G(Af ) as an ind-group scheme over Q

In order to explain how to interpret the exact sequence

1 → Gm → G(Af ) → G(Af ) → 1

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as an extension of group schemes, we have first to describe the ind-group schemestructure on the totally disconnected group G(Af ). The procedure described hereis valid for any locally compact totally disconnected group Γ, and provides such aΓ with the structure of group scheme over Spec(Z).

Thus let Γ be any totally disconnected group, and let ZΓ be the ring of locallyconstant compactly supported Z-valued functions on Γ. Multiplication in the groupΓ allows one to define a canonical coalgebra structure on ZΓ, making Spec(ZΓ) intoa group scheme.

Lemma 4.A.1. The set Γ is canonically isomorphic to the set of global sectionsSpec(Z) → Spec(ZΓ). The Zariski topology on the latter induces the usual topologyof locally compact totally disconnected group on Γ.

Proof. The lemma is obvious for Γ finite. Suppose Γ = lim←−i Γi , with Γi finite forall i. If C ⊂ Γ is a closed subset, then C = lim←−i Ci, where Ci ⊂ Γi is the set ofzeroes of the ideal Ji ⊂ ZΓi , say. Then C is obviously the set of zeroes of the ideallim−→i Ji ⊂ ZΓ. Conversely, if J = lim−→i Ji is an ideal in ZΓ with zero set C, then Cis easily seen to be the inverse limit of the zero sets Ci of Ji. The lemma followsfor profinite Γ.

Now let K ⊂ Γ be a profinite open subgroup, and let Σ be a set of coset repre-sentatives for Γ/K. Then we have

(4.A.1.1) Γ =∐σ∈Σ

σK; ZΓ =∑σ∈Σ


where ZσK ' ZK is the ring of locally constant functions on σK. The lemmafollows immediately from (4.A.1.1) and the special case of profinite groups.

4.A.2. In defining the projective space P(Sg) over Spec(Q), we view the infinitedimensional vector space Sg as the inverse limit of finite dimensional subspacesSg(U, V ), as in §3. Then the projective space P(Sg) over Spec(Q) is the projectivespectrum of the (non-noetherian) graded ring


Q[Q[V/U ]] = Q[Q(Af )g]].

4.A.3. We henceforward let Γ = G(Af ), and identify Γ with the group schemeSpec(Q ⊗ ZΓ) over Q. The projective representation of G(Af ) on Sg is deducedfrom actions on finite levels, as in §4.10. It follows from Lemma 4.A.1 and thedefinition in 4.A.2 that

Lemma 4.A.4. The Qab-rational projective representation P(ωψ) of G(Af ) onP(Sg) is a continuous action in the Zariski toplology.

Standard considerations now imply

Proposition 4.A.5. : There exists an ind-group scheme G(Af ) over Q, an exactsequence

1 → Gm → G(Af ) → G(Af ) → 1

in the category of ind-group schemes , and a representation

ωψ : G(Af ) → Aut(Sg),

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continuous in the Zariski topology, such that ωψ(t) = t, t ∈ Gm, and such thatthe induced action of G(Af ) on P(Sg) coincides with the projective representationP(ωψ) defined in §4.

5. Extending to GSp

Let GSp = GSp(V,<,>) be the group of symplectic similitudes of V . Wewrite GSp = GSpg when it is necessary to emphasize the dimension 2g of V . Letν : GSp → Gm be the symplectic multiplier. Let T ⊂ GSp be the subgroup

γ(t) =(

1 00 t

) ⊂ GSp.

For g ∈ G(Af ), ∈ Af,×, we let gt = γ(t)−1 · g · γ(t). We let ψ, ψt, and jt be as in4.10. Now G(Af ) acts on G(g, ψt) for all t. It is easy to see that, for σ ∈ G(g, ψ),g ∈ G(Af ), and t ∈ Af,×, we have

(5.0.1) gt(σ) = (jt)−1(g(jt(σ)))

The equality (5.0.1) and the considerations in Remark 4.10 provide a natural iden-tification of the projective representations g 7→ P(ωψ(gt)) and g ∈ P(ωψt(g)). Inparticular, the action of T (Af ) on G(Af ) by conjugation lifts to a canonical actionon G(Af ). Let GSp(Af ) be the semi-direct product of T (Af ) with G(Af ), definedin terms of this action. Then GSp(Af ) is am extension of GSp(Af ) by Gm in thecategory of algebraic group schemes over Q. Cf. [PSGe], where something similaris worked out in the case g = 1; the general case is worked out by Vigneras in [V],using explicit cocycles.

The G(Af )-bundle Θψ over the connected component Mgψ of Mg apparently

does not extend to a bundle overMg homogeneous under GSp(Af ), essentially forthe same reason that the representation ωpsi does not extend to the larger group.In order to get around this latter obstacle, a number of authors replace ωpsi bythe induced representation from G(Af ) to GSp(Af ). In our setting the naturalanalogue to the induced representation seems to be a certain Q× torsor over Mg,which we now construct.

We begin by noting thatMg is the Shimura variety attached to the pair (GSp,S±),where S± is the union of the Siegel upper and lower half planes, considered as ahomogeneous space under GSp(R). In Deligne’s formulation of Shimura’s theory ofcanonical models, recalled briefly in §7, below, S± is a GSp(R)-conjugacy class ofhomomorphisms of the real torus S def= RC/RGm into GSpR which satisfies certainaxioms.

5.1. When g = 0, GSp = Gm and S± is just the norm homomorphism N :RC/RGm,C → Gm,R. The corresponding Shimura variety, denoted M(Gm, N), isthe profinite scheme


R×Q×\A×/(1 +N Z) ' R×Q×\A×,

all of whose points are rational over Qab. The Q-structure on M(Gm, N) is definedby the obvious action of Gal(Qab/Q) ' Z× on R×Q×\A×: for e ∈ Z×, a ∈

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R×Q×\A×, we have e(a) = e−1 ·a. (Here the isomorphism of Gal(Qab/Q) with Z×is given by the inverse of the cyclotomic character, as in (1.1).) Then Af,× actsnaturally and Q-rationally on M(Gm, N).

Any abstract group ∆ defines a constant group scheme Spec(Z∆) over Spec(∆),where Z∆ is the direct sum of as many copies of Z as there are elements in ∆.(This is a special case of the construction in the appendix to §4.) Let ∆ = Q×,and let M(Gm, N) be Af,×, viewed as a Q×-torsor over M(Gm, N). The actionof Gal(Qab/Q) on M(Gm, N), defined above, lifts in the obvious way to an actionon M(Gm, N), which thus becomes a Q-rational Q×-torsor over M(Gm, N). Theaction of Af,× on M(Gm, N) obviously lifts to M(Gm, N).

We write M , M instead of M(Gm, N), M(Gm, N), respectively. Let M2 be thequotient of M by the action of ∆ = Q×+ ⊂ Q×. Then M2 is a double cover of M ,defined over Q.

The Shimura variety M has a modular interpretation analogous to that ofMg.We let T f (Gm) be the product over all primes ` of the `-adic Tate modules of thecommutative group scheme Gm, V f (Gm) = T f (Gm) ⊗ Q. Then M parametrizes“isogeny classes” of isomorphisms α : Af

∼−→ V f (Gm), where α, α′ are isogenousif α(x) ∼= α′(a · x), for some a ∈ Q×. Clearly, M parametrizes isomorphismsα : Af

∼−→ V f (Gm), and the morphism M → M just takes α to its isogeny class.The actions of Gal(Q/Q) on M and M derive from this modular interpretation.The natural action of Af,× on M is given by α 7→ (x 7→ α(t−1x) def= αt(x)).

Note that α is uniquely determined by the composition

Af∼−→ V f (Gm) → V f (Gm)/T f (Gm) ' µ∞.

If we denote this composition ψ = ψα, then for each α ∈ M we may constructthe group G(g, ψ) = G(g, α) with multiplication law (3.6); the hypothesis that ψ istrivial on Z is irrelevant. Clearly, G(g, α) is the fiber at α of a group scheme G(g)over M , defined over Q. We note that the actions of Af,× on additive charactersand on M correspond: we have ψ(αt) = (ψα)t, in the notation of 4.10.

The action of Af,× on M lifts to an action on G(g), as follows: if (β, x, y)(α) ∈G(g, α), t ∈ Af,×, then let

(5.1.1) t · (β, x, y)(α) = (β, x, ty)(αt)

The map (5.1.1) is an isomorphism r(t) : G(g, α) ∼−→ G(g, αt).Finally, M2 parametrizes isomorphisms Af

∼−→ V f (Gm) up to multiplicationby an element of ∆. The natural map M → M2 obviously defines an isomorphismfrom the subscheme

(5.1.2) M2 def= α ∈ M | ψα : Af → µ∞ is trivial on Z ⊂ M

onto M2. We identify M2 with M2, or alternatively with the set of charactersAf/Z → µ∞, by means of this isomorphism. The existence of M2 implies that the∆-torsor M → M2 is trivial .

5.2. Now let g be arbitrary. One defines in the obvious way a natural homomor-phism of group schemes over M :

(5.2.1) ρ : G(g) → Aut(SgM


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such that the fiber at α ∈ M is just ρψα . We saw in §4 that G(Af ) acts naturally onG(g, ψ), for each ψ, and therefore acts naturally on G(g), preserving the fibers overM . On the other hand, if γ(t) ∈ T (Af ), we let γ(t) act on G(g) as the morphismr(t) of (5.1.1). The equality (5.0.1) shows that these actions of G(Af ) and T (Af )together define an action of GSpg on G(g), covering the action of GSpg on M 'M2

given by ν : GSpg(Af ) → Af,×. We have defined our Q-structure on M in such away as to guarantee that this action of GSpg(Af ) on G(g) is defined over Q.

5.2.2. Similarly, the projective representations P(ωψ) patch together to define anaction P(ω) : G(Af ) → Aut(P(Sg)M ). Again, (5.0.1) implies that this action ex-tends to an action of GSpg(Af ) on P(Sg)M , and this action is again defined overQ.

5.3. As above, GSpg(Af ) acts on M2 through ν : GSpg(Af ) → Af,×, and there isa natural GSpg(Af )-equivariant map ν′ :Mg → M2, taking the connected compo-nentMg

ψ to the character ψ. Composing with the canonical morphism Ag → Mg,we obtain a GSpg(Af )-equivariant map Ag → M2. We let Mg = Mg ×M2 M ,Ag = Ag ×M2 M . Then Mg (resp. Ag) is a Q-rational ∆-torsor over Mg (resp.Ag), and the actions of GSpg(Af ) on Mg and Ag lift to actions on Mg and Ag,which commute with the action of ∆. The identity section Mg → Ag is denotedε, as in §4.

Now the inclusionMgψ ⊂Mg, together with the constant map fromMg

ψ to thepoint ψ ∈ M2 ⊂ M , define a G(Af )-equivariant morphism Mg

ψ → Mg. Similarly,we have a G(Af )-equivariant map Agψ → Ag.

5.4. The representation ρ of (5.2.1) provides us with a (Q-rational) morphism ofschemes over M generalizing Theorem 4.7;

ϑ : Ag → P(Sg)M .

The fiber of ϑ at the point ψ ∈ M is just ϑψ. We let Θ(A) = ϑ∗(OP(Sg)M),

Θ = ε∗(Θ(A)). Then Θ (resp. Θ(A) is a Q-rational vector bundle over Mg, (resp.Ag) whose restriction to Mg

ψ is Θψ. The construction in 5.2.2 shows that theaction of G(Af ) on Θψ , defined by Proposition 4.9, extends to a Q-rational actionof GSp(Af ) on Θ, the pullback via ε of a Q-rational action on Θ(A). We haveproved:

Proposition 5.5. There is a Q-rational, GSp(Af ) -equivariant line bundle Θ overthe scheme Mg. The restriction of Θ to the subscheme Mg

ψ ⊂ Mg is G(Af )-equivariantly isomorphic to Θψ.

5.6. For n = 0, 1, . . . , let Jn(Θ) denote the bundle of n-jets of Θ(A); let

J∞(Θ) = lim←−nJn(Θ).

The action of GSp(Af ) on Θ(A) defines an action on J∞(Θ). Any global sections of Θ(A) defines a global section j∞(s) of J∞(Θ).

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Let D = DAg denote the O = OAg -algebra of finite-order algebraic differentialoperators on Ag. It follows from the definitions (cf. [H,§7]) that

(5.6.1) J∞(Θ) ' (D ⊗O Θ∗)∗,

where we regard D as a right O-module and Θ∗ as a left O-module. Here theright-hand side is viewed as an inverse limit of coherent sheaves, defined by theorder filtation on D.

The isomorphism (5.6.1) makes J∞(Θ) naturally into a right D-module: if f ∈Γ(U, J∞(Θ)) ' Γ(U,Hom(D⊗O Θ∗,O)), and ∆ ∈ Γ(U,D), for U open in Ag, then

f ?∆(g) = f(∆ · g), ∀g ∈ Γ(U,D ⊗OΘ∗).

Let E∞(Θ) denote theOAg subbundle of J∞(Θ) generated by (j∞(Γ(Ag,Θ(A)))?D,and let S∞(Θ) = ε∗(E∞(Θ)). Then the action of GSp(Af ) on Mg lifts to an actionon S∞(Θ). The bundle S∞(Θ), together with its natural GSp(Af )-action, will bethe subject of §7 and §8.

6. Relations with the analytic theory

In order to apply the theory developed in the preceding sections to the arithmeticof the oscillator representation, we must describe the relations between the vectorbundles Θ and the analytic theory of theta functions. This material is in principlewell known, having been covered by numerous authors, including [I, M2, M3]. Ourformulation is somewhat different, however, from those currently available in theliterature. Most of this section will therefore be taken up with definitions; proofswill be brief.

6.0. Notation. In this section, ψA : Q\A → C× will be a continuous character;we denote by ψ (resp. ψ∞) the restriction of ψA to Af (resp to R). We define thegroup G(g, ψA) (resp. G(g, ψ∞)) to be the set C××Ag ×Ag (resp. C××Rg ×Rg)with multiplication law (3.6), where ψ is replaced by ψA (resp. ψ). Note thatψ is necessarily of the form ψ(x) = e2πλx, x ∈ R, for some λ ∈ R. We defineψC : C → C× by the formula ψC(z) = e2πλz, z ∈ C, and define G(g, ψC) (resp.G(g, ψC · ψ) to be the set C× × Cg × Cg (resp. C× × (C×Af )g × (C×Af )g) withmultiplication law (3.6), where ψ is replaced by ψC (resp. ψC × ψ). Then G(g, ψCis a complex Lie group; and Lie(G(g, ψ∞)) is a Lie subalgebra of Lie(G(g, ψC)).When necessary, we let Z(G) = C× ⊂ G(g, ?), ? = ψ, ψA, ψ∞, ψC, or ψC · ψ. Notethat there is an involution D−1 : G(g, ?) → G(g, ?), defined as in §3, with any ? asabove.

We let S be the real algebraic group RC/RGm. A Hodge structure on a realvector space V is a homomorphism h : S → GL(V ) of real algebraic groups. Thenh defines a Hodge decomposition and a Hodge filtration


V p,q, F pV =⊕p′≥p

V p′,q,

where SC acts on V p,q through the character z−pz−q. The Hodge structure h is acomplex structure if and only if VC = V −1,0 ⊕ V 0,−1.

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Let V = Q2g as in 1.1; thus G(g, ψ) is isomorphic as a manifold to V (R)× C×.Let <,> and GSp = GSp(V,<,>) be defined as in §5. Let

S± = h : → GSp(R) | h defines a complex structure on V (R) and

(x, y)hdef= < x, h(i)y > is a positive- or negative-definite symmetric form

on V (R).Then conjugation by GSp(R) makes S± into a homogeneous space for GSp(R),

and it is well known that S± is naturally isomorphic to the union of the Siegel upperand lower half-planes, introduced in §5. In particular, our notation is consistent.We let S+ be the subset of S± for which (x, y)h is positive-definite.

6.1. We will be concerned with the following situation. Let H ⊂ G be real Liegroups such that G/H has a G-invariant complex structure. (In practice, G will bean adelic group, but the reduction to this case is easy.) The group H is assumedto be reductive, but G is not, nor is G even assumed to be algebraic. We writeg = Lie(G), h = Lie(H), and

GC = H = C⊕ q+ ⊕ q−,

where q+ (resp. q−) corresponds to the holomorphic (resp. anti-holomorphic)tangent space at the identity coset in G/H.

Let GC denote a complex analytic Lie group with Lie algebra gC, HC the sub-group of GC corresponding to hC; we assume q+ and q− to be the Lie algebras ofcommutative subgroups Q+ and Q−, respectively, of GC, each normalized by HC.We assume there is a homomorphism G → GC with finite kernel, contained in H;let G′ (resp. H ′) be the image of G (resp. H) under this map. Let Q = HC ·Q−;we assume Q∩G′ = H ′ and that the complex structure on G/H is induced by thenatural open immersion G/H → GC/Q.

We let Γ ⊂ G be a discrete subgroup which acts properly discontinuously onG/H, and we let M = Γ\G/H. A representation ρ : H → GL(Vρ) determines aC∞ vector bundle [ρ] on M :

[ρ] = Γ\G× Vρ/H,

where Γ acts trivially on Vρ and H acts on G× Vρ by the formula

(6.1.1) (g, v) · h = (gh, ρ(h)−1v), g ∈ G, v ∈ Vρ, h ∈ H.

In general Vρ need not be finite-dimensional, but it should be either a direct sum oran inverse limit of finite-dimensional H-modules. If U is an open subset of M and Eis a C∞ vector bundle over M , we denote by Γ∞(U, E) the space of C∞ sections of Eover U . In the cases to be considered below, the bundle [ρ] will have a holomorphicstructure defined as follows. Let p : Γ\G → M be the natural projection. For anyopen U ⊂M , there is a natural isomorphism(6.1.2)Lift : Γ∞(U, [ρ]) ∼−→ f ∈ C∞(p−1(U)) | f(gk) = ρ−1(k)f(g),∀g ∈ p−1(U), k ∈ H.

We denote the right hand side by C∞(U, ρ), and write C∞(ρ) = C∞(M,ρ). Theholomorphic structure on [ρ] will be given by the sheaf associated to the presheaf

(6.1.3) H(U, ρ) = s ∈ Γ∞(U, [ρ]) | X · Lift(s) ≡ 0,∀X ∈ q−.

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Here q− acts by right differentiation on C∞(G). Again, let H(ρ) = H(M,ρ).We assume our space M to be endowed with an algebraic structure compatible

with the holomorphic structure defined above, and assume [ρ] to be an algebraicvector bundle. We let Dan (resp. D) denote the sheaf of analytic (resp. algebraic)differential operators on M ; let Oan (resp. O) denote the structure sheaf of M asan analytic (resp. algebraic) variety. Let U (resp. U ′) be the enveloping algebra ofGC (resp. HC ⊕ q−). Now the vector bundle of analytic differential operators onGC/Q is naturally isomorphic to

GC × (U ⊗U ′ C)/Q,

where C is the trivial U ′-module and the action of H on U is given by the adjointrepresentation. There is thus an isomorphism of analytic vector bundles

Dan ' Γ\G× U ⊗U ′ C/H = [ρU ],

where ρU is the adjoint representation of H on U ⊗U ′ C. More generally, we have

Dan ⊗Oan [ρ]an ' Γ\G× U ⊗U ′ Vρ/H,

where the action of hC, extended trivially to q−, makes Vρ naturally into a U ′-module and where H acts diagonally on U ⊗U ′ Vρ.

Here and in (6.1.6) below, the holomorphic structure is defined by a variant of(6.1.3): dρ is a representation of hCoplusq

− (which does not necessarily integrateto a representation of Q), and the holomorphic sections of [ρ] over an open set Uare those which lift to q−-invariant Vρ-valued functions on p−1(U). For example, in(6.1.4) dρ is the adjoint representation. The same construction works for any finitedimensional HC ⊕ q−-module Vρ.

Of course, D has a filtration O = D0 ⊂ D1 ⊂ · · · ⊂ Di ⊂ . . . by degree ofdifferential operators. The enveloping algebra U(q+) is isomorphic to the symmetricalgebra S(q+); we denote by Si(q+) ⊂ S(q+) the subspace of tensors of degree ≤ i;then Dani ' Γ\G× Si(q+)/H.

Let Jn[ρ] (resp. J∞[ρ]) denote the bundle of n-jets (resp. the bundle lim←−n Jn[ρ]

of infinite jets) of [ρ]. As in (5.6.1), we have

(6.1.5) Jn[ρ] ' (Dn ⊗O [ρ]?)?, J∞[ρ] ' (D ⊗O [ρ]?)?

and J∞[ρ] is a right D-module. In the category of analytic vector bundles, there isthus an isomorphism

(6.1.6) J∞[ρ] ' Γ\G× (U ⊗U ′ V ?ρ )?/H

where the dual (U ⊗U ′ V ?ρ )? is regarded as lim←−(Si(q+) ⊗ V ?ρ )?. The isomorphismsof C∞ vector bundles:

Si(q+) =n⊕i=0


defines a decomposition of C∞ vector bundles


Jn[ρ] 'n⊕i=0

Γ\G×Hom(Symi(q+), Vρ)/H


SymiΩ1M ⊗ [ρ].

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Here H acts by the adjoint action on Symi(q+) and by ρ on Vρ. Note that thelower isomorphism is not holomorphic. Let ρ(i) denote the natural action of H onHom(Symi(q+), Vρ). We have analogously an isomorphism of C∞ vector bundles

(6.1.8) Dan 'n⊕i=0

Γ\G× Symi(q+)/H '∞⊕i=0


where Hacts trivially on Symi(q+) and TM is the tangent bundle to M . In termsof (, the right action of Dan on J ′[ρ] is given by the natural pairing


SymiTM ⊗⊕∞


M ⊗ [ρ]

defined by contraction of tangent and cotangent vectors; here⊕

denotes the com-pleted direct sum.

6.1.10. We let jn : [ρ] → Jn[ρ] denote the differential operator which takes asection of [ρ] to its n-jet, n ≤ ∞. Let s ∈ Γ∞(M, [ρ]), and let E∞(s) denote the O-subbundle of J∞[ρ] generated by j∞(s) ?D. Let p : Γ\G → M be the projection,as above. The fiber of E∞(s) at the point p(g) is canonically isomorphic to thesubspace of Hom(U, Vρ) given by

D 7→ r(X)DLift(s)(g), X ∈ U.

Here X 7→ r(X) denotes the right action of U on C∞(G). If we denote by V (s) theU -submodule r(U)Lift(s) ⊂ C∞(Γ\G), then the fiber of E∞(s) at p(g) is naturallya U ′-submodule of Hom(V (s), Vρ). Of course, if s is holomorphic, then V (s) is aquotient of the “generalized Verma module” U ⊗U ′ V ?ρ

Write ω = Ω1M . and let δi : [ρ] → Symiω⊗[ρ], i = 0, . . . , n be the C∞ differential

operator obtained as the composition of jn with the projection on the i-th factorin (6.1.9). Let ∆i : C∞(ρ) → C∞(ρ(i)) be the homomorphism which makes thefollowing diagram commute:

Γ∞([ρ]) ∼−−−−→Lift

C∞(ρ)yδi y∆i

Γ∞(Symiω ⊗ [ρ]) ∼−−−−→Lift


If f ∈ C∞(ρ), we have

(6.1.11) ∆i(f)(X) = X · f ∈ C∞(G,Vρ) ∀X ∈ Symi(q+)

6.2. In this section, we apply the above theory when G = G(g, ψA), H = Z(G)(notation 6.0). Since ψ is trivial on Q, the set H(Q)

def= 1 × Qg × Qg ⊂ G(g, ψA)

is a subgroup, which serves as our Γ. We let ρ(−1) be the character t 7→ t−1 ofZ(G), and write

(6.2.1) Tψ = H(Q)\G(g, ψA)/C× (= M , as above), Lψ = [ρ(−1)]

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Let K ⊂ (Af )g × (Af )g be an open compact subgroup such that 1 × K ⊂C× × (Af )g × (Af )g ' G(g, ψ) is a subgroup of G(g, ψ). Then we may identify Kwith the subgroup 1×K ⊂ G(g, ψ). Let Tψ(K) = Tψ/K, Lψ(K) = Lψ/K. ThenTψ(K) is isomorphic to the compact torus R2g/K ∩ Q2g and Lψ(K) is a complexline bundle over Tψ(K). Any h ∈ S+ defines a complex structure on R2g and henceon the torus Tψ(K); it is well known that Tψ(K), with this complex structure, isisomorphic to an abelian variety, which we denote Ah,ψ(K). We let Ah,ψ be theisocompletion of Ah,ψ(K), for any K. Then Ah,ψ does not depend on K and istopologically isomorphic to Tψ. Moreover, the isomorphism Ah,ψ

∼−→ Tψ defines afull level structure Ah,ψ(t) ∼−→ V ⊗Af , and is thus an isomorphism of topologicalspaces with G(g, ψ)-action.

We let F •h (V ) denote the Hodge filtration (6.0) on VC defined by h. Then Qhdef=

F 0h (V )× C× is an analytic subgroup of G(g, ψC · ψ), and we have naturally

(6.2.2) Ah,ψ ' H(Q)\G(g, ψC · ψ)/Qh, Lψ = H(Q)\G(g, ψC · ψ)× C/Qh.

Here the subgroup F 0h (V ) of Qh acts trivially on C. This shows that Lψ is naturally

a holomorphic vector bundle over Ah,ψ; with this holomorphic structure we writeLh,ψ instead of Lψ. In fact, it is well known that Lh,ψ is even algebraic.

In the notation of 6.1, we have GC = G(g, ψC ·ψ), Q = Qh. We write u+h = V −1,0,

u−h = V 0,−1, instead of q+, q−. We write H(Lψ) for H(ρ(−1)) (6.1.3), and defineLift : Γ(Ah,ψ, Lψ) ∼−→ H(ρ(−1)) as in (6.1.2).

We define the Schwartz space S(Rg) in the usual way (cf. e.g., []), and letS(Ag) = S(Rg)⊗ Sg, with Sg as in §3. Then G(g, ψA) acts on S(Ag):

(6.2.4) ρψA(α, x, `)Φ(y) = α · ψA(t` · y)Φ(x+ y), α ∈ C×, x, `, y ∈ Ag.

We let dρψC : Lie(G(g, ψC)) → End(S(Rg)) denote the corresponding Lie algebraaction; the action of Lie(G(g, ψC)) on S(Ag) is also denoted dρψC . The followinglemma is well known (cf. [,]):

Lemma 6.2.5. . For any h, there is a unique function Φh ∈ S(Rg) such that (i)dρψC(u−h )Φh = 0, and (ii) Φh(0) = 1.

Now for any Φ ∈ S(Ag), the series

(6.2.6) ΘψA(Φ)(g) =


ρψA(g)Φ(ξ), g ∈ G(g, ψA)

converges absolutely and uniformly on compact subsets to a C∞ function on G(g, ψA)([], p. ), which satisfies

(6.2.7) ΘψA(Φ)(γgt) = tΘψA

(Φ)(g), ∀γ ∈ H(Q), g ∈ G(g, ψA), t ∈ Z(G).

In particular, if Phi = Φh ⊗ Φf , with Φf ∈ Sg, then ΘψA(Φ) ∈ H(Lψ). We thus

obtain a homomorphism

Sψ : Sg → Γ(Ah,ψ, Lψ);Sψ(Φf ) = Lift−1ΘψA(Φh ⊗ Φf ).

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6.2.8. Let K ⊂ (Af )g×(Af )g be an open compact subgroup such that 1×K ⊂C× × (Af )g × (Af )g ' G(g, ψ) is a subgroup of G(g, ψ). Now the involution D−1

of G(g, ψC), mentioned in 6.0, descends to multiplication by −1 on Ah,ψ(K). Thenthe homomorphism D−1×1 : G(g, ψC)×C → G(g, ψC)×C induces an isomorphismLψ(K) ∼−→ (−1)?Lψ(K). One verifies immediately that Lψ(K) is a totally sym-metric line bundle over Ah,ψ(K), for all K. Moreover, if Z(K) is the centralizer inG(g, ψ) of 1×K, then Z(K) acts on the right on Lψ(K), and this action definesa canonical isomorphism Z(K)/K ∼−→ G(Lψ(K)) (cf. [M2], p. 237).

We let Th,ψ be the tower Ah,ψ(K),Ph,ψ the polarization defined by Lψ(K).It follows from the above remarks that the polarized tower (Th,ψ,Ph,ψ) comesequipped with a canonical symmetric θ-structure G(Ph,ψ) ∼−→ G(g, ψ). There isthus a canonical action of G(g, ψ) on lim−→Γ(Ah,ψ(K), Lψ(K)) = Γ(Ah,ψ, Lψ), givenin terms of the realization (6.1.1) by right multiplication. We denote this action ρ′.

Lemma 6.2.9. The homomorphism Sψ : Sg → Γ(Ah,ψ, Lψ) is an isomorphism,and intertwines the representation ρψ of G(g, ψ) on Sg with the action ρ′.

Proof. Evidently, for any Φf ∈ Sg, Φ = Φh ⊗ Φf , we have Lift(ρ′(h)Sψ(Φ))(g) =ΘψA

(Φ)(gh) = ΘψA(ρψ(h)Φ)(g) by (6.2.6). This implies that Sψ(Sg) is a G(g, ψ)-

stable subspace of Γ(Ah,ψ, Lψ). Since Sψ is not identically zero (cf. []) and theaction of G(Ph,ψ) on Γ(Ah,ψ, Lψ) is irreducible (cf. [M1], II, p. 109), it follows thatSψ is an isomorphism.

6.3. Let Oh,ψ denote the structure sheaf of Ah,ψ, and let Dh,ψ denote the Oh,ψ-module of finite order algebraic differential operators on Ah,ψ.

We write C∞,i(h, ψ) for C∞(ρ(−1)(i)), i = 0, 1, . . . ,, (cf. (6.1.10)), and set

Γi(h, ψ)) = Γ∞(Ah,ψ, Symiωh,ψLψ)).

We define Lift : Γi(h, ψ) → C∞,i(h, ψ) as above. Let

∆i : C∞,0(h, ψ) → C∞,i(h, ψ)

be the homomorphism defined in 6.1.10. The following lemma follows immediatelyfrom (6.1.11):

Lemma 6.3.1. Let Φ = Φh ⊗ Φf , with Φf ∈ Sg. Then

∆i(ΘψA(Φ))(X)(g) = ΘψA

(dρψC(X)Φ)(g)∀g ∈ G(g, ψA), X ∈ Symi(u+h ),

where we regard Symi(u+h ) as a subspace of U and where dρψC is the action of U

on S(Rg) determined by (6.2.4).

Corollary 6.3.2. Let S∞(Lψ) denote the Oh,ψ-submodule of J∞(Lψ) generatedby (j∞(Γ(Ah,ψ, Lψ))) ?Dh,ψ. Then S∞(Lψ) = J∞(Lψ).

Proof. Let Jn denote the image of S∞(Lψ) in Jn(Lψ), n = 0, 1, . . . . It suffices toshow that Jn = Jn(Lψ) for all n. For this we use induction on n. When n = 0,this is equivalent to the statement that Lψ is generated by its global sections, i.e.,that Lψ defines a polarization.

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Now suppose Jn−1 = Jn−1(Lψ). It suffices to show that the global sections ofJn generate Symnωh,ψ ⊗ Lψ at every point, under the map defined by (6.1.7). LetΓn(S) denote the image of Γ(Ah,ψ, Jn) in Γn(h, ψ), and let Hn(S) = Lift(Γn(S)).It follows from the Lemma and (6.1.9) that

( Hn(S) = X ∈ ΘψA(dρψC(X)Φh ⊗ Φf ), X ∈ Symn(u+

h ) | Φf ∈ Sg.

Write Ψ(X,Φf ) = ΘψA(dρψC(X)Φh ⊗ Φf ). To complete the induction, we must

show that, for every g ∈ G(g, ψA), the homomorphism

( → Symn(u+

h )? :

Φf 7→ (X 7→ Ψ(X,Φf )(g)), X ∈ Symn(u+h )

is surjective. Suppose not. Then for some non-zero X ∈ Symn(u+h ), we have

Ψ(X,Φf )(g) = 0 for all Φf . It follows that,

Ψ(X,Φf )(g · gf ) = 0 ∀Φf ∈ Sg, ∀gf ∈ G(g, ψ).

But Ψ(X,Φf ) is H(Q)-invariant; it follows by continuity that Ψ(X,Φf ) is identi-cally zero for all Φf ∈ Sg, which is absurd. The lemma follows.

6.4. Let G = Sp(V,<,>) as in §1. Then G(A) acts naturally on G(g, ψA) and theaction extends by linearity to an action on G(g, ψC · ψ). As in §4, we may definea continuous projective representation P(ωψA

) of G(A) on S(Rg) ⊗ Sg which liftsto a representation ωψA

of an extension G(A) of G(A) by C×. The representationωψA


(6.4.1) ωψA(γ)ρψA(σ)ωψA

(γ−1) = ρψA(γσ), γ ∈ G(A), σ ∈ G(g, ψA)

We similarly define the extension G(R) of G(R) by C× to be the pullback of theextension G(A) to the subgroup G(R) of G(A). Then G(R) acts naturally onS(Rg), and satisfies the analogue of (6.4.1).

The imbedding G(Q) → G(A) lifts to an imbedding G(Q) → G(A).Let Kh ⊂ G(R) denote the centralizer of h(S), with h as in 6.0; let Kh be the in-

verse image of Kh in G(R). Thus G(R)/Kh ' S±; this defines a complex structureon G(A)/Kh, and the moduli spaceMg

ψ is naturally isomorphic to G(Q)\G(A)/Kh

[,].Let g = Lie(G(R)). The adjoint representation of GSp(R) on Lie(GSp(R))

leaves g invariant. Thus Ad h defines a Hodge structure on g, which is of type(0, 0) + (−1, 1) + (1,−1) []. Let kh = g0,0, p− = g1,−1, p+ = g−1,1. Then kh =Lie(Kh)C. We let g = Lie(G(R)) = G⊕C, where C is the center of g and [g, g] = g.Let kh = kh ⊕ C ⊂ g⊕ C.

The action of G(A) on G(g, ψC · ψ) pulls back to an action of G(A). Let D(resp. DC) be the semidirect product of G(A) and G(g, ψA) (resp. G(g, ψC · ψ))with respect to this action, and let B0 (resp. B0,C) denote the quotient of D (resp.DC) by the subgroup (t, t) ∈ Z(G)×Z(G(A)) = C××C×. Then B0 acts naturallyon S(Rg)⊗ Sg; we denote this representation B0(ω).

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Denote by dωψC the natural action of U(gC) on S(Rg) or on S(Rg) ⊗ Sg. It isknown ([], ) that

(6.4.2) dωψC(p−)Φh = 0;

(6.4.3) There exists a character [ 12 ] : Kh → C× such that

ωψC(k)Φh = [12](k)Φh.

Let Γ = H(Q) o G(Q) ⊂ B0, HA = Kh ⊂ B0, HA,C = Q− o Kh ⊂ B0,C; thusAgψ(C) is analytically isomorphic to Γ\B0/HA ' Γ\B0,C/HA,C [,].

We may apply the theory of 6.1 with G = B0, H = HA, Γ = Γ, q+ =u+h ⊕ p+, q− = u−h ⊕ p−, M = Agψ(C). Extend the character ρ(−1) of 6.2 toρ(−1)A : HA → C×:

ρ(−1)A |Z(G)= ρ(−1), ρ(−1)A(k) = [12](k)−1, k ∈ Kh.

Following the convention of (6.1.1), we let Θanψ be the bundle [ρ(−1)A]. So far Θan


is defined only as a C∞ vector bundle. We define a holomorphic structure on Θanψ

by (6.1.3). That this determines a holomorphic structure on Θanψ is verified as in [,

p. 40].It is clear as in [M3] that Θan

ψ is the analytic vector bundle associated to thealgebraic line bundle Θψ(A) = Θ(A) |Agψ , with Θ(A) as in 5.4. Just as in 6.2,there is a homomorphism

Sψ : Sg → Γ(Agψ,Θψ(A))

which intertwines the actions of B0(Af )def= G(g, ψ) · G(Af ) ⊂ B0 on both sides.

When g > 1, Sψ is an isomorphism by Proposition 2.9 (or by the direct limit versionof Proposition 3.11). We exclude the case g = 1 from the following discussion; itwill be treated in §7.

Let UA denote the enveloping algebra of Lie(G(g, ψC)) o GC. Let C(1) denoteC with the action ρ(−1)? of HA,C. There is a B0(Af )-equivariant isomorphism ofanalytic vector bundles (6.1.6):

(6.4.4) J∞(Θanψ ) ' Γ\B0 × (UA ⊗U(hA,C) C(1))?/HA

where hA,C = Lie(HA)C and the dual on the right hand side is taken as an inverselimit.

Denote by dΩψ the representation of UA on S(Rg) which coincides with dρψC

(resp. with dωψC) on Lie(G(g, ψC)) (resp. on U(gC)). The function Φh defines amap

UA → S(Rg); D 7→ dΩψ(D)Φh

which factors through UA⊗U(HA,C)) C(1) ((6.2.5), (6.4.2), (6.4.3)). By duality, thismap defines a holomorphic subbundle


def= Γ\B0 × S(Rg)/HA ⊂ J∞(Θan

ψ ).

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Here the action of Kh on S(Rg) is the contragredient of ωpsiCIn the notation of 5.6, let D denote the algebra of differential operators on Aψg ,

and let E∞(Θψ) denote theOAψg -subbundle of J∞(Θψ) generated by j∞(Γ(Aψg ,Θψ(A)))?

D. Let p : Γ\B0 → Aψg (C) be the projection. The fiber of E∞(Θψ) at the pointp(g) is canonically isomorphic (6.1.10) to the subspace of Hom(UA,C) given by

D ∈ dΩψ(X ·D)(Φh ⊗ Φf )(g), D ∈ UA | X ∈ UA,Φf ∈ Sg.

It follows that E∞(Θψ)an ⊂ S∞,anψ . On the other hand, for any point x ∈ Mgψ,

let Ax be the fiber of Aψg over x, and let E∞x (resp. J∞x ) denote the pullback ofE∞(Θψ) (resp. J∞(Θψ)) to Ax. Then Corollary 6.3.2 states exactly that E∞x = J∞x .By dimension considerations, it follows that E∞(Θψ) = S∞,anψ .

Recall that in 5.6 we defined a bundle S∞(Θ) over Mg. Let S∞(Θψ) denotethe restriction of S∞(Θ) toMg

ψ. Let ε :Mgψ → Aψg denote the zero section. The

above discussion may be summarized as follows:

Proposition 6.5. . There is a natural G(Af )-equivariant isomorphism

(S∞(Θψ))an ∼−→ ε ∗ (S∞,anψ )def= G(Q)\G(A)× S(Rg)/Kh

of analytic vector bundles. Here the action of Kh on S(Rg) is the contragredient ofωψC , and the holomorphic structure on S∞,anψ is determined as in 6.1 by the actionof p− given by the contragredient of dωψC .

6.5.1. Remark. The point of the above proposition is that it defines a rationalstructure on the analytic vector bundle G(Q)\G(A)×S(Rg)/Kh. The above argu-ment actually proves that the vector bundle S∞,anψ has a natural rational structure;this may be of some future use.

6.6. It remains to extend the above theory to GSp. Note that the above theoryis only defined when ψ is the non-archimedean component of a global additivecharacter. However, for any ψ, we clearly have

(6.6.1) Mg 'Mgψ ×

G(Af ) GSp(Af ), S∞(Θ) = S∞(Θψ)×G(Af ) GSp(Af ),

where, if H ⊂ H ′ is an inclusion of groups and X is a space with right H-action,then X ×H H ′ = X × H ′/H is the minimal extension of X to a space with H ′-action. Combining (6.6.1) and Proposition 6.5, we obtain a GSp(Af )-equivariantisomorphism

(6.6.2) S∞(Θ)an ' G(Q)\G(R)× GSp(Af )× S(Rg)/Kh

of analytic vector bundles.Let Γ = G(Q)\[G(R) × GSp(Af )]. It follows from (6.6.2) that we have an

isomorphism (cf. (6.1.2))(6.6.3)Lift : Γ∞(Mg,S∞(Θ)) ∼−→ f ∈ C∞(Γ, S(Rg)) | f(gk) = ωψ(k)−1f(g), g ∈ Γ, k ∈ Kh

We would like to realize sections of S∞(Θ) as functions on GSp(A), defined as in§5 as the semi-direct product of G(A) with the torus T (A). This can be done in

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a number of ways, depending on the choice of a character of R×+, as follows: LetGSp(R)+ = ν−1(R×+) ⊂ GSp(R), where ν is the symplectic multiplier, as in §5.Then GSp(R)+ ' G(R)×Z+, where Z+ is the identity component of ZGSp(R), andis isomorphic to R×+. The covering group GSp(R) admits a natural surjective mapto GSp(R). Let G+ ⊂ GSp(R) denote the inverse image of GSp(R)+ under thismap; then

(6.6.4) G+ ' G(R)× Z+ ' G(R)× R×+.

Let f = Lift(s) be a function on Γ in the image of Lift, as in (6.6.3), and letα be a character of R×+. The decomposition (6.6.4) defines an extension of f to a

function fα = Liftα(s) on the subgroup Γ+ def

= G(Q)\G+ × GSp(Af ) ⊂ GSp(A):

(6.6.5) fα(gz, gf ) = α(z)f(g, gf ), z ∈ Z+, g ∈ G(R), gf ∈ GSp(Af ).

Let GSp(Q)+ = GSp(Q) ∪ G+. We say s ∈ Γ∞(Mg,S∞(Θ)) is α-admissible ifLift

α(s) satisfies

(6.6.6) Liftα(s)(γg) = Liftα(s)(g),∀γ ∈ GSp(Q)+, g ∈ Γ+.

We may identify

GSp(Q)+\G+ × GSp(Af ) ' GSp(Q)\GSp(A);

thus if s is α-admissible, then Liftα(s) is naturally a function on GSp(Q)\GSp(A),

which we denote LiftGSp

(s). If s is α-admissible, then α is uniquely determined;thus there is no danger of ambiguity.

More generally, we say s is admissible if is a finite linear combination∑si, where

each si is αi-admissible for some αi. We let


(s) =∑


(si) ∈ C∞(GSp(Q)\GSp(A), S(Rg)).

The subspace Γ∞(Mg,S∞(Θ))adm ⊂ Γ∞(Mg,S∞(Θ)) of admissible sections is sta-ble under the action of GSp(Af ).

It is known [] that the representation ω = IndGSp(A)


is independent of the

character ψA. It is thus clear from (6.6.1) that, for any ψ, the action of GSp(Af )on Γ∞(Mg,S∞(Θ) is given by the restriction ωf of ω to GSp(Af ). Let ωadm,R bethe representation of GSp(R) on the linear space spanned by

f ∈ IndGSp(R)

G(R)ωψ∞ | ∃ α such that f(gz) = α(z)f(g), z ∈ Z+, g ∈ G(R).

Then ωadm,R is independent, up to isomorphism, of the choice of ψ∞. Let ωadmdenote the subrepresentation ωadm,R ⊗ ωf of ω. It follows from the precedingdiscussion that the natural representation of GSp(A) on Γ∞(Mg,S∞(Θ)adm) isgiven by ωadm.

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7. Fourier Expansions

For each integer N ≥ 1, let SN = Sg,N be the split torus over Q with charactergroup BN = Hom(Sym2(Zg), (1/N)Z); let B = B1. Then for each ψ, SN isthe torus corresponding by Mumford’s theory of toroidal compactification to thestandard point boundary component of Mg

N,ψ. Specifically, the positive-definitesymmetric matrices form a cone C ⊂ B ⊗ R which is homogeneous with respectto the natural action of GL(g,R). For each integer N ≥ 1, let Γ`(N) ⊂ GL(N,Z)be the principal congruence subgroup of level N . Ash et al. [AMRT] considerΓ`(N)-invariant decompositions Σ of C into rational polyhedral cones

C =⋃σ∈Σ


satisfying certain axioms of finiteness and compatibility. To each such Σ corre-sponds a locally finite equivariant torus imbedding SN → SN,Σ which forms a localmodel near part of the boundary ofMg

N,ψ.Similar considerations hold for all locally symmetric varieties. However, in the

case ofMg, the local parametrization of the boundary admits an algebraic interpre-tation in terms of degenerating abelian varieties. This interpretation was developedin characteristic zero by Brylinski [B] and over Z by Faltings and Chai [FC]. Thelatter two authors make explicit use of the arithmetic theory of theta functions;their results include in passing a treatment of the Fourier expansions of forms ofweight 1

2 . LetS∞ = lim←−



where the surjective morphism SN → SM , M ÷ N , corresponds to the inclusionBM ⊂ BN of character groups. Then S∞ is a pro-algebraic torus with charac-ter group B∞ = Hom(Sym2(Zg),Q). Fix a GL(g,Z)-invariant rational polyhe-dral decomposition Σ of C, satisfying the axioms of [AMRT]. Then, for each N ,ΣisΓ`(N)-invariant; thus Σ defines a torus imbedding

S∞ → S∞,Σ = lim←−N


Let SN,Σ denote the formal completion of SN,Σ at DNdef= SN,Σ − SN , let S0

N =SN,Σ − DN , and let S∞ = lim←−N S

0N . Mumford’s theory of degenerating abelian

varieties ([M4], reprinted in [FC]) provides a pair consisting of an abelian schemeπ : A0 → S0

1 and a line bundle L0 on A0 defining a principal polarization on A0.The pair (A0,L0) has the following properties:

(7.1) π∗L0 is canonically isomorphic to OS01

(by construction; cf. remarks in [F],pp. 354-355, where L0 is called N).

(7.2) For M÷N , let pN,M : S0N → S0

M be the natural projection. In view of (7.1),any section θ ∈ Γ(S0

N , (p0N,1) ∗ (π∗L0)) can be written as a formal series,

convergent in the topology defined by the sheaf of ideals corresponding toDN :

θ =∑χ∈BN


(Cf. [F], p. 326, [FC,**].

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(7.3) The subgroup p∗N,1(A0)[N ] is canonically isomorphic to µgN × (Z/NZ)g. Inparticular, given an isomorphism ψ : Af/Z

∼−→ µ∞, we obtain a compatiblefamily of canonical level N structures

αN,ψ : (Z/NZ)2g ∼−→ p∗N,1(A0)[N ].

(7.4) Let p∞,1 : S0∞ → S0

1 , π∞ : (p∞,1)∗A0 → S0∞ be the canonical morphisms.

The triple ((p∞,1)∗A0, (p∞,1)∗L0, αN,ψ) defines a morphism

jψ : S0∞ → M



j∗ψ(Θψ) is canonically isomorphic to π∞,∗(p∞,1)∗L0.

7.5. Let

FΘ = ∑χ∈B∞

a(χ)χ | ∃ N > 0 such that a(χ) = 0 for χ /∈ C ∩BN

where C is the closure of C in B × R. The above remarks allow us to define ahomomorphism

F : Γ(Mgψ,Θψ) → FΘ,

except when g = 1, which we consider below. Remember that B∞ consists ofsymmetric bilinear forms on Qg, thus each χ is associated to a symmetric g × gmatrix M(χ) with rational entries. If we view χ as the function e2πiTr(M(χ)Z) onthe Siegel upper half plane, then F becomes the usual Fourier expansion ([F], §9,[FC]), and in particular is injective.

7.6. When g = 1, we can define, for any open subset U ⊂ M1ψ, a homomorphism

F 1(∞) : Γ(U,Θψ) → F1Θ, where

F1Θ =


a(χ)χ | ∃ N > 0 such that a(χ) = 0 for χ /∈ BN.

The homomorphism F 1(∞) is the Fourier expansion at the cusp ∞; for any othercusp c, there is a similar homomorphism F 1,c. Let

Γ(U,Θψ) = f ∈ Γ(U,Θψ) | F 1,c(f) ∈ F1Θ, for all cusps c.

The varietyM1ψ has a natural G(Af )-equivariant compactification j :M1

ψ → M1ψ,

obtained by adding points. Let Θψ ⊂ j∗Θψ be the subsheaf such that Γ(V, Θψ) =Γ(V ∩M1

ψ,Θψ), for an open subset V ⊂ M1ψ. The arguments of §4-§6 go through

also in the case g = 1, provided M1ψ is replaced by M1

ψ and Θψ by Θψ. In thefuture, these modifications for the case g = 1 will be assumed without furthermention.

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[AMRT] A. Ash, D. Mumford, M. Rapoport, Y. Tai, Smooth compactification oflocally symmetric varieties. Lie Groups: History, Frontiers and Applications, Vol.IV. Math. Sci. Press, Brookline, Mass., (1975).[B] Jean-Luc Brylinski, ”1-motifs” et formes automorphes (thorie arithmtique desdomaines de Siegel). Conference on automorphic theory (Dijon, 1981), Publ. Math.Univ. Paris VII, 15 (1983) 43–106.[F] G. Faltings. Arithmetische Kompaktifizierung des Modulraums der abelschenVarietten. (German) [Arithmetic compactification of the moduli space of abelianvarieties] Workshop Bonn 1984 (Bonn, 1984), Lecture Notes in Math., 1111, Springer,Berlin, (1985) 321–383.[FC] G. Faltings; C-L. Chai, Degeneration of abelian varieties. With an appendixby David Mumford. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 22.Springer-Verlag, Berlin, 1990.[H] M. Harris, Arithmetic vector bundles and automorphic forms on Shimura vari-eties II. Compositio Math., 60 (1986), 323-378.[I] J.-I. Igusa, Theta functions. Die Grundlehren der mathematischen Wissenschaften,Band 194. Springer-Verlag, New York-Heidelberg, (1972).[M1] D. Mumford, On the equations defining abelian varieties, I. Invent. Math., 1(1966) 287–354; II. Invent. Math., 3 (1967) 75–135; III. Invent. Math., 3 (1967)215–244.[M2] D. Mumford, Abelian varieties. Tata Institute of Fundamental Research Stud-ies in Mathematics, No. 5 Published for the Tata Institute of Fundamental Re-search, Bombay; Oxford University Press, London (1970)[M3] D. Mumford, Tata lectures on theta. I. With the assistance of C. Musili,M. Nori, E. Previato and M. Stillman. Progress in Mathematics, 28. BirkhuserBoston, Inc., Boston, MA, (1983).[M4] D. Mumford, An analytic construction of degenerating abelian varieties overcomplete rings. Compositio Math., 24 (1972), 239–272.[W] Sur certains groupes d’oprateurs unitaires. Acta Math., 111 (1964) 143–211.