Abelian Subvarieties and the Shimura Construction

Ernst Kani

Preliminary Version, August 2001

1 Introduction

In his fundamental book, Shimura [Sh1] showed that each (Hecke) 𝕋-eigen-function f ∈ S_{2}(N) on Γ_{1}(N) gives to an abelian subvariety A_{f} ⊂ J_{1}(N) on the Jacobian variety of the modular curve X_{1}(N)/ℚ, and in a subsequent paper [Sh2] he explained that such Hecke eigenfunctions give more naturally rise to quotient varieties A′_{f} of J_{1}(N). The purpose of this note is to show that both these constructions (and more) follow from a general “dictionary” that translates statements about subvarieties of abelian varieties into statements about ideals of the associated endomorphism algebras.

To explain this more precisely, let A be an abelian variety over an arbitrary field K, and let Sub(A/K) = {B ≤ A} denote the set of abelian subvarieties B of A (which are defined over K). Then the aforementioned dictionary translates this set into the set Id_{𝔼} of right ideals of the endomorphism algebra 𝔼 = End_{K}(A) ⊗ ℚ of A as follows:

Theorem 1.1 The map B ↦ I(B) := {f ∈ 𝔼 : Imf ⊂ B} defines an inclusion-preserving bijection

I_{A/K} : Sub(A/K) →^{∼} Id_{𝔼}

between the set of abelian subvarieties of A/K and the set of right ideals of 𝔼 = End_{K}^{0}(A). Furthermore, if B_{1}, B_{2} ∈ Sub(A/K) are any two abelian subvarieties, then there is a canonical (functorial) isomorphism

Hom_{K}^{0}(B_{1}, B_{2}) := Hom_{K}(B_{1}, B_{2}) ⊗ ℚ →^{∼} Hom_{𝔼}(I(B_{1}), I(B_{2})).

From this theorem (which is a special case of Theorem 2.4 below) one obtains as a consequence that if V is any faithful (left) 𝔼-module, then there is a natural bijection between Sub(A/K) and certain algebraic subspaces of V; cf. section 3. In the case that V is finitely generated, this yields (via the Morita theorems) the following result.

Corollary 1.2 Let V be a faithful, finitely generated left 𝔼-module. Then the map B ↦ W_{𝔼}(B) := I(B)V defines an inclusion-preserving bijection

W_{A/K} : Sub(A/K) →^{∼} 𝔼̃Sub(V)

between the set of abelian subvarieties of A/K and the set of left 𝔼̃-submodules of V, where 𝔼̃ = End_{𝔼}(V).

Two (natural) examples for which this result can be applied directly are the following.

Example 1.3 (a) If K ⊂ ℂ, then the assignment B ↦ H_{1}(B) ⊂ H_{1}(A) := H_{1}((A ⊗ ℂ)^{an}, ℚ) defines an inclusion-preserving bijection between the set of abelian subvarieties of A/K and the set of left End_{𝔼}(H_{1}(A))-submodules of H_{1}(A).

(b) If K = ℚ, then the assignment B ↦ T_{0}(B) ⊂ T_{0}(A), where T_{0}(A) denotes the tangent space of A at the origin, defines an inclusion-preserving bijection between Sub(A/ℚ) and the set of left End_{𝔼}(T_{0}(A))-modules of T_{0}(A).

Now this last example is actually the dual version of the one that gives rise to the Shimura construction, for the space S_{2}(N, ℚ) ⊂ S_{2}(N) of cusp forms of weight 2 on Γ_{1}(N) with rational Fourier expansions can naturally be identified with the space of homomorphic 1-forms H^{0}(X_{1}(N), Ω_{X_{1}(N)/ℚ}^{1}) ≃ T_{0}(J_{1}(N))^{∗}, which is the dual space of the tangent space T_{0}(J_{1}(N)). We thus require either an identification of T_{0} with its dual space or a dual version of the above corollary. As we shall see, both these methods are feasible and lead to Shimura’s two constructions.

More precisely, Shimura’s first construction (in [Sh1]) may be deduced from the following general statement.

Corollary 1.4 Let A/ℚ be an abelian variety, and suppose that there exists a commutative subring 𝕋 ⊂ 𝔼 = End_{ℚ}^{0}(A) such that both T_{0}(A) and T_{0}(A)^{∗} are free 𝕋-modules of rank 1. Fix a 𝕋-module isomorphism φ : T_{0}(A)^{∗} →^{∼} T_{0}(A). Then for any 𝕋-submodule W ⊂ T_{0}(A)^{∗}, there is a unique abelian subvariety B_{W} ∈ Sub(A/ℚ) such that T_{0}(B_{W}) = φ(W).

As will be explained in section 5, this can be applied directly to the case A = J_{1}(N) (cf. Theorem 5.1); in fact, in this case Shimura[Sh2] constructed an explicit isomorphism φ.

A more natural method is to work with the module Ω(A) = T_{0}(A)^{∗} directly. Since this is a right 𝔼-module, we require a “dual version” of Corollary 1.2 for such modules. In this case, however, one has replace the set Sub(A/K) by either the set Sub(Â/K) of abelian subvarieties of the dual abelian variety Â, or equivalently, by the set Quot(A/K) of all abelian quotient varieties of A/K, i.e. by the set consisting of all (equivalence classes of) pairs (C, p) where p : A → C is a surjective homomorphism of abelian varieties such that Ker(p) is also an abelian variety; cf. section 4 below.

The precise analogue of Corollary 1.2 is given in Theorem 4.4. As a special case of this, one obtains

Theorem 1.5 If A is an abelian variety defined over ℚ and p : A → C is a quotient of A, then the assignment (C, p) ↦ p^{∗}Ω(C) ⊂ Ω(A) := H^{0}(A, Ω_{A/ℚ}^{1}) induces a bijection

W_{Ω} : Quot(A/ℚ) →^{∼} Sub(Ω(A))𝔼̃

between the set of abelian quotients of A/ℚ and the set of right 𝔼̃-submodules of Ω(A), where 𝔼̃ = End_{𝔼}(Ω(A)).

In particular, if there exists a commutative subring 𝕋 ⊂ 𝔼 such that Ω(A) is a free 𝕋-module of rank 1, then for every 𝕋-submodule W ⊂ Ω(A) there exists a unique abelian quotient p_{W} : A → C_{W} such that p_{W}^{∗} Ω(C_{W}) = W. In addition, dim C_{W} = dim_{ℚ} W.

As will be explained in section 5, the above theorem applies in particular to the case that A = J_{1}(N) is the Jacobian of the modular curve X_{1}(N)/ℚ and 𝕋 = 𝕋_{ℚ} is the Hecke algebra of J_{1}(N), and we thus obtain a different characterization (and proof) of the (second) Shimura construction [Sh2].

2 Subvarieties of an Abelian Variety

As in the introduction, fix an arbitrary ground field K and let A, B, … denote abelian varieties defined over K. Throughout, we shall freely use the basic facts about abelian varieties as presented in Milne[Mi] and Mumford[Mu]. Some other standard facts (not explicitly mentioned there) are the following.

Proposition 2.1 (a) If f : A → B is an isogeny of abelian varieties, then there is a unique isogeny f′ : B → A such that

f′ ∘ f = [e(f)]_{A}, (1) where e(f) denotes the exponent of Kerf, i.e., the smallest integer n > 0 such that nKerf = 0. Moreover, we then also have that f ∘ f′ = [e(f)]_{B}.

(b) If λ_{L} : A → Â is the canonical homomorphism defined by L ∈ Pic(A), then for any homomorphism h : B → A of abelian varieties we have

λ_{h^{∗}L} = h^{#}λ_{L} := ĥ ∘ λ_{L} ∘ h. (2)

(c) If λ : A → Â is a polarization of A and h : B → A is a finite homomorphism (i.e. Ker(h) is finite), then h^{#}λ := ĥ ∘ λ ∘ h : B → B̂ is a polarization of B. In particular, ĥ is surjective.

Proof. (a) Since Ker(f) ⊂ Ker([e(f)]_{A}), such a unique factorization exists by the universal property of quotients (viewing (B, f) as the quotient of A by Ker(f)). Furthermore, if we write h = f ∘ f′ : B → B and e = e(f), then h ∘ f = f ∘ f′ ∘ f = f ∘ [e]_{A} = [e]_{B} ∘ f, and hence h = [e]_{B} because f is surjective.

(b) This follows immediately from the definitions of λ_{L}, λ_{h^{∗}L} and of ĥ. (See also Lang[La], p. 130).

(c) It is enough to verify this over \overline{K}, the algebraic closure of K. Then λ = λ_{L} for some ample sheaf L ∈ Pic(A), and hence h^{∗}L is also ample (because h is finite). Thus h^{#}λ = λ_{h^{∗}L} is a polarization. In particular, ĥ ∘ λ ∘ h is an isogeny and hence is surjective, and thus ĥ is also surjective.

The above facts lead to the following result which is of fundamental importance for the proof of Theorem 1.1:

Corollary 2.2 Suppose h : B → A is a finite homomorphism and that λ : A → Â is a polarization. Put:

p_{h} = p_{h,λ} := ĥ ∘ λ : A → B̂ and N_{h} = N_{h,λ} := (λ_{B})′ ∘ p_{h} : A → B, (3)

where λ_{B} = h^{#}λ : B → B̂ is the induced polarization on B and (λ_{B})′ ∘ λ_{B} = [e_{B}]_{B} with e_{B} = e(λ_{B}). Then we have

N_{h} ∘ h = [e_{B}]_{B}, (4)

and so ε_{h} = ε_{h,λ} := \frac{1}{e_{B}} h ∘ N_{h} ∈ 𝔼 = End_{K}^{0}(A) is an idempotent of 𝔼.

Proof. By definition, N_{h}∘h = (λ_{B})′∘p_{h}∘h = (λ_{B})′∘λ_{B} = [e_{B}]_{B}, which proves (4). Thus ε_{h} ∘ ε_{h} = \frac{1}{e_{B}^{2}} h ∘ N_{h} ∘ h ∘ N_{h} = \frac{1}{e_{B}^{2}} h ∘ [e_{B}]_{B} ∘ N_{h} = \frac{1}{e_{B}} h ∘ N_{h} = ε_{h}, which means that ε_{h} is an idempotent.

Remark. The above proposition applies in particular to the case when h = j_{B} : B ↪ A is the inclusion map of an abelian subvariety B ≤ A. In this case we shall write p_{B}, N_{B} and ε_{B} in place of p_{j_{B}}, N_{j_{B}} and ε_{j_{B}}, respectively.

Notation. If A and B are abelian varieties, then the group Hom_{K}(A, B) of K-homomorphisms is a lattice (of finite rank) in the ℚ-vector space Hom^{0}(A, B) = Hom_{K}(A, B) ⊗ ℚ. In particular, if A = B, then the Kendomorphism algebra End_{K}(A) is a free ℤ-algebra and a lattice in the (finite dimensional) ℚ-algebra 𝔼 = 𝔼_{A/K} := End_{K}^{0}(A) = End_{K}(A) ⊗ ℚ.

For any f ∈ Hom_{K}(B, A), its image Im(f) ⊂ A is an abelian subvariety of A. Since Im(nf) = Im(f ∘ [n]_{B}) = Im(f), for any non-zero n ∈ ℤ, we can extend this definition to f ∈ Hom_{K}^{0}(B, A) by setting Im(f) := Im(nf) ⊂ A, where n ∈ ℤ is chosen such that nf ∈ Hom_{K}(B, A) (and n ≠ 0).

If B ≤ A is an abelian subvariety, then put

I(B) = I_{A/K}(B) = {f ∈ 𝔼_{A/K} : Im(f) ⊂ B}.

Note that I(B) is a right ideal of 𝔼 because we have Im(f+g) ⊂ Im(f)+Im(g) and Im(f ∘ g) ⊂ Im(f), for all f, g ∈ 𝔼.

Proposition 2.3 If B ≤ A is an abelian subvariety of A, then

I(B) = j_{B}Hom_{K}^{0}(A, B) = ε_{B,λ}𝔼, (5)

where λ : A → Â is any polarization of A.

Proof. Clearly, j_{B}Hom_{K}^{0}(A, B) ⊂ I(B). Conversely, if f ∈ I(B), then Im(f) ⊂ B, and so we can write f = j_{B} ∘ f′ for some f′ ∈ Hom_{K}^{0}(A, B). Thus f ∈ j_{B}Hom_{K}^{0}(A, B), which proves the first equality of (5).

To prove the second equality, it clearly enough to show that

Hom_{K}^{0}(A, B) = N_{B,λ}𝔼. (6)

Now if f ∈ Hom_{K}(A, B), then by (4) we have f = \frac{1}{e_{B}} N_{B,λ} ∘(j_{B} ∘f) ∈ N_{B,λ}𝔼, and so Hom_{K}^{0}(A, B) ⊂ N_{B,λ}𝔼. This proves (6) since the other inclusion is trivial.

The above proposition is the main tool required for proving Theorem 1.1 of the introduction, as we shall now see. In fact, we shall prove the following slightly more precise version of this theorem:

Theorem 2.4 The above map B ↦ I(B) := j_{B}Hom_{K}^{0}(A, B) = {f ∈ 𝔼 : Imf ⊂ B} defines a lattice-preserving bijection

I_{A/K} : Sub(A/K) →^{∼} Id_{𝔼}

whose inverse is given by B_{A/K}(𝔞) = 𝔞A := ∑_{f∈𝔞} Im(f). In particular, I_{A/K}(B) is a two-sided 𝔼-ideal if and only if B is 𝔼-stable, i.e. if and only if f(B) ⊂ B, for all f ∈ 𝔼.

Furthermore, for any B_{1}, B_{2} ∈ Sub(A/K) there is a unique functorial isomorphism

(I_{A/K})_{B_{1},B_{2}} : Hom_{K}^{0}(B_{1}, B_{2}) →^{∼} Hom_{𝔼}(I(B_{1}), I(B_{2})) (7)

such that for any h ∈ 𝔼 with h(B_{1}) ⊂ B_{2} we have

(I_{A/K})_{B_{1},B_{2}}(h_{|B_{1}})(x) = hx, for all x ∈ I(B_{1}). (8)

Proof. We first show that I_{A/K} and B_{A/K} are inverses of each other. For this, fix a polarization λ : A → Â. Then for any B ∈ Sub(A/K) we have by equation (5) that B_{A/K}(I_{A/K}(B)) = B_{A/K}(ε_{B,λ}𝔼) = Im(ε_{B,λ}) = Im(N_{B,λ}) = B, the latter because N_{B,λ} : A → B is surjective (cf. (4)).

Now we also have I_{A/K}(B_{A/K}(𝔞)) = 𝔞, for all 𝔞 ∈ Id_{𝔼}. To see this, write 𝔞 = ε𝔼, for some idempotent ε ∈ 𝔼; such an ε exists by [CR], p. 44, since 𝔼 semi-simple. Then B := B_{A/K}(𝔞) = Im(ε), and so ε ∈ I_{A/K}(B) = ε_{B,λ}𝔼 by (5), i.e. ε𝔼 ⊂ ε_{B,λ}𝔼. To prove the opposite inclusion, we first show that εj_{B} = j_{B}. To see this, choose an n > 0 such that g := nε ∈ End_{K}(A). Then g^{2} = ng and so g ∘ j_{B} = [n]_{A} ∘ j_{B} because if b = g(a) ∈ Im(g)(\overline{K}) then g(b) = g^{2}(a) = ng(a) = nb. Thus, εj_{B} = j_{B}, and hence εε_{B,λ} = ε_{B,λ}. (Composing both sides with \frac{1}{e_{B}} N_{B,λ}.) Thus ε_{B,λ} ∈ ε𝔼 and so 𝔞 = ε𝔼 = ε_{B,λ}𝔼 = I_{A/K}(B) = I_{A/K}(B_{A/K}(𝔞)). Therefore, the maps I_{A/K} and B_{A/K} are inverses of each other and hence both are bijections.

Note that since B_{A/K}(f𝔼) = Im(f), the above implies that for any f ∈ 𝔼 we have I(B) = f𝔼 ⇔ Im(f) = B, and so

(9) fI(B) = I(f(B)), for all B ∈ Sub(A/K)

because fI(B) = fε_{B,λ}𝔼 and Im(fε_{B,λ}) = f(Im(ε_{B,λ}) = f(B). Thus, if B is 𝔼-stable, then clearly I(B) is a two-sided ideal. Conversely, if I(B) is a two-sided ideal, then f(Im(g)) = Im(fg) ⊂ B, for all f ∈ 𝔼 and g ∈ I(B).

Thus, if we take g = ε_{B,λ} and use the fact that Im(ε_{B,λ}) = B, then we see that B is 𝔼-stable, as claimed.

Now since both I_{A/K} and B_{A/K} are inclusion-preserving, we have

(10) B_{1} ⊂ B_{2} ⇔ I(B_{1}) ⊂ I(B_{2}), for all B_{1}, B_{2} ∈ Sub(A/K),

and so it follows that I_{A/K} is lattice-preserving, i.e. that we have

I(B_{1} + B_{2}) = I(B_{1}) + I(B_{2}) and I((B_{1} ∩ B_{2})^{0}) = I(B_{1}) ∩ I(B_{2}), (11)

because B_{1} + B_{2} (resp. (B ∩ B_{2})^{0})) is the smallest (resp. largest) abelian subvariety containing (resp. contained in) B_{1} and B_{2}.

To construct the map (I_{A/K})_{B_{1},B_{2}}, put

I_{A/K}(h)(x) = j_{B_{2}} ∘ h ∘ y, for x = j_{B_{1}} ∘ y ∈ I(B_{1}); (12)

note that the right hand side is uniquely determined by x (and h) because j_{B_{1}} is injective. Thus, this rule defines an element I(h) ∈ Hom_{𝔼}(I(B_{1}), I(B_{2})).

We observe that I(h) satisfies (8) because if h(B_{1}) ⊂ B_{2}, then h ∘ j_{B_{1}} =

j_{B_{2}} ∘h′ for a unique h′ ∈ Hom_{K}^{0}(B_{1}, B_{2}), and hence I(h_{|B_{1}})x := I(h′)x =^{(12)}j_{B_{2}} ∘

h′ ∘ y = h ∘ j_{B_{1}} ∘ y = h ∘ x, as claimed.

To see that the (additive) map h ↦ I(h) is injective, suppose that I(h) = 0. Then, since j_{B_{2}} is injective, we have h ∘ y = 0, for all y ∈ Hom_{K}^{0}(A, B_{1}). In particular, h ∘ N_{B_{1},λ} = 0 and so h = 0 since N_{B_{1},λ} is an epimorphism. Thus (I_{A/K})_{B_{1},B_{2}} is injective.

To show that (I_{A/K})_{B_{1},B_{2}} is surjective, we shall use the elementary result in ring theory (analogous to Lemma (3.19) of [CR], p. 45) that for any idempotents ε_{1}, ε_{2} ∈ 𝔼, the left multiplication map f ↦ l_{f} := (x ↦ fx) defines (by restriction to ε_{1}𝔼) an isomorphism

ε_{2}𝔼ε_{1} →^{∼} Hom_{𝔼}(ε_{1}𝔼, ε_{2}𝔼).

Applying this here to the idempotents ε_{i} = ε_{B_{i},λ} = j_{B_{i}} ∘ N′_{B_{i}} (where N′_{B_{i}} = \frac{1}{e_{B_{i}}} N_{B_{i},λ}), we see that each φ ∈ Hom_{𝔼}(I(B_{1}), I(B_{2})) has the form φ = (l_{ε_{2}fε_{1}})_{|I(B_{1})}, for some f ∈ 𝔼. From this the surjectivity of (I_{A/K})_{B_{1},B_{2}} follows because by (8) (with h = ε_{2}fε_{1}) we have the formula

I(N′_{B_{2}} ∘ f ∘ j_{B_{1}}) = (l_{ε_{2}fε_{1}})_{|I(B_{1})}, for all f ∈ 𝔼. (13)

Finally, we note that these isomorphisms are functorial in the sense that if h ∈ Hom_{K}^{0}(B_{1}, B_{2}) and g ∈ Hom_{K}^{0}(B_{2}, B_{3}), then we have

I_{A/K}(g ∘ h) = I_{A/K}(g) ∘ I_{A/K}(h). (14)

Indeed, if x = j_{B_{1}}y ∈ I(B_{1}) then we have (I(g) ∘ I(h))(x) = I(g)(I(h)(x)) = I(g)(j_{B_{2}} ∘ h ∘ y) = j_{B_{3}} ∘ g ∘ h ∘ y = I(g ∘ h)(x).

Remark 2.5 (a) The above theorem has a convenient reformulation in terms of categories: it asserts that I_{A/K} defines a canonical isomorphism of categories

I_{A/K} : \underline{Sub}_{A/K}^{0} →^{∼} \underline{Id}_{𝔼}. (15)

Here \underline{Sub}_{A/K}^{0} denotes the full subcategory of the category \underline{Ab}_{/K}^{0} “of abelian varieties up to isogeny’ (cf. [Mu], p. 172) defined by Sub(A/K) (i.e. \underline{Sub}_{A/K}^{0} is the category whose objects are the elements of Sub(A/K) and whose morphisms from B_{1} to B_{2} are given by Hom_{K}^{0}(B_{1}, B_{2})), and similarly, \underline{Id}_{𝔼} denotes the full subcategory of the category \underline{Mod}_{𝔼} defined by the right ideals of 𝔼, where \underline{Mod}_{𝔼} denotes the category of (finitely generated) right 𝔼-modules.

(b) The above theorem is partially connected with the result of Lange[Lan] (cf. [LB], p. 126) that the map B ↦ ε_{B,λ} defines a bijection

Sub(A/K) →^{∼} {symmetric idempotents of 𝔼}

between the set of abelian subvarieties of A/K and the set of idempotents ε ∈ 𝔼 which are symmetric with respect to Rosati involution a ↦ a^{∗} = λ^{−1} ∘ â ∘ λ defined by λ. (Indeed, this result is easily deduced from Theorem 2.4.) However, this bijection is not so useful for module-theoretic analysis below (cf. section 3). In addition, this bijection depends on the choice of the polarization λ (and hence is less functorial).

(c) Note that it follows from equation (7) of the above theorem that if B_{1}, B_{2} ∈ Sub(A/K) are two abelian subvarieties, then

B_{1} is isogenous to B_{2} ⇔ I(B_{1}) ≃ I(B_{2}) (as 𝔼-modules).

From this it follows easily that if B, B_{1}, …, B_{r} ∈ Sub(A/K) then

B ∼ B_{1}^{n_{1}} × … × B_{r}^{n_{r}} ⇔ I(B) ≃ I(B_{1})^{n_{1}} ⊕ … ⊕ I(B_{r})^{n_{r}}. (16)

Note that this statement immediately implies Theorem 1 of [KR].

3 Algebraic Subspaces of 𝔼-Modules

As before, let A/K be an abelian variety and 𝔼 = End_{K}^{0}(A). In this section we suppose that we have a faithful representation

ρ : 𝔼 = End_{K}^{0}(A) → End_{F} (V)

where V is a finite-dimensional vector space over a field F ⊃ ℚ. In other words, V is a left 𝔼 ⊗ F -module which is faithful as an 𝔼-module.

Definition. An F -subspace W ⊂ V is called 𝔼-algebraic if it is of the form

W = Im(ρ(a)), for some a ∈ 𝔼.

The set of all 𝔼-algebraic subspaces of V is denoted by Alg_{𝔼}(V) = Alg_{𝔼,ρ}(V).

Remarks. (a) It is easy to see that the sum and intersection of algebraic subspaces is again algebraic, as will be clear from the proof of Theorem 3.1.

(b) Let 𝔼̃_{F} := End_{𝔼⊗F}(V) = {f ∈ End_{F} (V) : fρ(a) = ρ(a)f, for all f ∈ 𝔼}. Then every W ∈ Alg_{𝔼}(V) is a left 𝔼̃_{F} -submodule of V, i.e.

Alg_{𝔼}(V) ⊂ 𝔼̃_{F}Sub(V). (17)

(Indeed, if w ∈ W = Im(ρ(a)), then w = ρ(a)(v) for some v ∈ V and then we have for f ∈ 𝔼̃_{F} that fw = f(ρ(a)(v)) = ρ(a)(f(v)) ∈ Im(ρ(a)) = W.)

The set Alg_{𝔼}(V) of 𝔼-algebraic subspaces of V is connected to the set Sub(A/K) of abelian subvarieties in the following way.

Theorem 3.1 The map B ↦ W_{𝔼}(B) = W_{ρ}(B) := I(B) ⊗_{𝔼} V = I(B)V = ∑_{b∈I(B)} Imρ(b) induces a lattice-preserving bijection

W_{𝔼} = W_{𝔼,V} : Sub(A/K) →^{∼} Alg_{𝔼}(V);

in particular, the sum and intersection of algebraic subspaces is again algebraic. Furthermore, W_{𝔼}(B) is a (left) 𝔼-submodule of V if and only if B is an 𝔼-stable subvariety.

Proof. By hypothesis, the ring homomorphism ρ : 𝔼 → 𝔸 := End_{F} (V) is injective and hence faithfully flat because 𝔼 is semi-simple. (Use the criterion of [Bou], Prop. I.3.9(b) (p. 33) and the fact that every 𝔼-module is projective and hence flat). Thus, the map 𝔞 ↦ ρ^{∗}𝔞 = 𝔞 ⊗_{𝔼} 𝔸 = 𝔞𝔸 defines an injection

ρ^{∗} : Id_{𝔼} ↪ Id_{𝔸}

because we have ρ^{−1}(𝔞𝔸) = 𝔞, for all 𝔞 ∈ Id_{𝔼} by [Bou], loc. cit.

Next we use the well-known fact that the map 𝔞 ↦ W_{V} (𝔞) := 𝔞V = 𝔞⊗_{𝔸}V defines a bijection

W_{V} : Id_{𝔸} →^{∼} Sub_{F} (V)

between the set of right ideals of 𝔸 and the set of F -subspaces of V. (This is in fact a special case of the Morita theorem(s) which are used below in the proof of Theorem 3.2; note that here End_{𝔸}(V) = F.)

Now by definition we have W_{𝔼}(B) = I(B)V = (I(B)𝔸)V = W_{V} (ρ^{∗}I(B)), so W_{𝔼} = W_{V} ∘ ρ^{∗} is the composition of two injections and hence is injective. Furthermore, since I(B) = ε_{B,λ}𝔼 by Proposition 2.3, we see that

W_{𝔼}(B) = W_{V} (ρ(ε_{B,λ})𝔸) = Im(ρ(ε_{B,λ})), (18)

for any polarization λ of A. This shows that W_{𝔼}(B) is 𝔼-algebraic, so W_{𝔼} maps into the subset Alg_{𝔼}(V). Conversely, if W = Im(ρ(a)) is 𝔼-algebraic (with a ∈ 𝔼), then by Theorem 2.4 there exists B ∈ Sub(A/K) such that I(B) = a𝔼, and then W_{𝔼}(B) = Im(ρ(a)) = W. Thus W_{𝔼} has image Alg_{𝔼}(V).

Note that W_{𝔼} = W_{V} ∘ ρ^{∗} ∘ I_{A/K} is lattice-preserving because each of the maps since I_{A/K}, ρ^{∗} and W_{V} has this property.

To prove the last statement, suppose first that B is 𝔼-stable. Then I(B) is a two-sided 𝔼-ideal (cf. Th. 2.4) and so for f ∈ 𝔼 we have fW_{𝔼}(B) = fI(B)V ⊂ I(B)V = W_{𝔼}(B), and hence W_{𝔼}(B) is an 𝔼-submodule of V.

Conversely, if W_{𝔼}(B) is an 𝔼-submodule, then for any f ∈ 𝔼 we have fI(B)𝔸V = fW_{𝔼}(B) ⊂ W_{𝔼}(B) = I(B)𝔸V, so fI(B)𝔸 ⊂ I(B)𝔸 (by Morita) and hence fI(B) ⊂ I(B) (by faithful flatness). Thus, I(B) is a two-sided 𝔼-ideal and so B is 𝔼-stable by Theorem 2.4.

In general, it is difficult to give a good characterization of the set Alg_{𝔼}(V) of algebraic subspaces of V. However, if the coefficient field can be chosen to be F = ℚ, then such a characterization is indeed possible:

Theorem 3.2 If V is a finitely generated, faithful left 𝔼-module, then a subspace W ⊂ V is algebraic if and only if W is a left 𝔼̃-submodule of V, where 𝔼̃ = End_{𝔼}(V). Thus the map W_{𝔼} induces a (lattice-preserving) bijection

W_{𝔼} = W_{𝔼,V} : Sub(A/K) →^{∼} 𝔼̃Sub(V).

Furthermore, for any two abelian subvarieties B_{1}, B_{2} ∈ Sub(A/K) there is a functorial isomorphism

(W_{𝔼})_{B_{1},B_{2}} : Hom_{K}^{0}(B_{1}, B_{2}) →^{∼} Hom𝔼̃(W_{𝔼}(B_{1}), W_{𝔼}(B_{2})) (19)

such that for any h ∈ 𝔼 with h(B_{1}) ⊂ B_{2} we have

(W_{𝔼})_{B_{1},B_{2}}(h_{|B_{1}})(w) = hw, for all w ∈ W_{𝔼}(B_{1}). (20)

Proof. First note that since V is a finitely generated faithful 𝔼-module and 𝔼 is semi-simple, 𝔼 is a direct factor of V^{n} (for some n). Thus by [CR], Lemma (3.39), V is a (pro)generator of _{𝔼}\underline{Mod} (in the sense of [CR], p. 56, 60). Next, view V as an (𝔼, 𝔼̃^{op})-bimodule (where 𝔼̃^{op} denotes the opposite ring of 𝔼̃ = End_{𝔼}(V)), and put Q = Hom_{𝔼}(V, 𝔼). Then P := V and Q (together with maps described on [CR], p. 59) satisfy the hypotheses of Morita’s Theorem ([CR], p. 60), and so in particular (by [CR], Theorem (3.54)(vi)) the map 𝔞 ↦ 𝔞 ⊗_{𝔼} V defines a (lattice-preserving) bijection

T_{V} : Id_{𝔼} →^{∼} Sub(V)𝔼̃^{op} = 𝔼̃Sub(V).

But by definition T_{V} (𝔞) = 𝔞V = W_{V} (ρ^{∗}(𝔞)) (in the notation of the proof of Theorem 3.1), and so it follows that W_{𝔼} = T_{V} ∘ I_{A/K} yields the desired bijection.

To prove the second assertion, we first recall that Morita’s theorem (in the above context) also yields that the functor T_{V} = ∗ ⊗ V : \underline{Mod}_{𝔼} → \underline{Mod}𝔼̃^{op} = 𝔼̃\underline{Mod} is an equivalence of categories (cf. [CR], Th. (3.54)(v)) and that hence T_{V} induces a (functorial) isomorphism

Hom_{𝔼}(𝔞, 𝔟) →^{∼} Hom𝔼̃^{op}(𝔞V, 𝔟V), for all 𝔞, 𝔟 ∈ Id_{𝔼}.

Note that by viewing the right 𝔼̃^{op}-modules 𝔞V, 𝔟V as left 𝔼̃-modules, we have a canonical identification Hom𝔼̃^{op}(𝔞V, 𝔟V) = Hom𝔼̃(𝔞V, 𝔟V), and so (19) follows by combining the above identification with that of (7). Explicitly, this isomorphism is given by the formula

(W_{𝔼})_{B_{1},B_{2}}(h)(w) = ρ(j_{B_{2}} ∘ h ∘ f)v = ρ(j_{B_{2}} ∘ h ∘ N′_{B_{1}})w, (21) in which h ∈ Hom_{K}^{0}(B_{1}, B_{2}), w = ρ(j_{B_{1}} ∘ f)v ∈ W_{𝔼}(B_{1}) = I(B_{1})V with f ∈ Hom_{K}^{0}(A, B_{1}) and v ∈ V, and N′_{B_{1}} = \frac{1}{e_{B_{1}}} N_{B_{1},λ}. [Indeed, by construction

we have W_{𝔼}(h)(w) = T_{V} (I_{A/K}(h)ρ(j_{B_{1}} ∘f)v) = ρ(I_{A/K}(h)(j_{B_{1}} ∘f))v =^{(12)}ρ(j_{B_{2}} ∘

h ∘ f)v = ρ(j_{B_{2}} ∘ h ∘ N′_{B_{1}} ∘ j_{B_{1}} ∘ f))v = ρ(j_{B_{2}} ∘ h ∘ N′_{B_{1}})w.] Note that this formula immediately implies (20).

Remark 3.3 Note that if we take V = 𝔼 in Theorem 3.2, then the isomorphism 𝔼̃ = End_{𝔼}(𝔼) ≃ 𝔼^{op} induces a canonical identification 𝔼̃Sub(𝔼) = Id_{𝔼}, and so Theorem 2.4 is actually a special case of Theorem 3.2.

The following corollary is of fundamental importance for the Shimura construction.

Corollary 3.4 In the situation of Theorem 3.2, suppose that there exists a commutative subring 𝕋 ⊂ 𝔼 such that V is a free 𝕋-module of rank 1 (or, more generally, such that the image of 𝕋 in End_{ℚ}(V) is a maximal commutative subalgebra). Then for every 𝕋-submodule W ⊂ V there is a unique abelian subvariety B_{W} ⊂ A such that W_{𝔼}(B_{W}) = W and we have a natural ring embedding

θ_{W} : End_{𝕋}(W) ↪ End_{K}^{0}(B_{W})

such that θ_{W} (ρ(t)_{|W}) = t_{|B_{W}}, for all t ∈ 𝕋. In particular, we have an induced embedding θ′_{W} : 𝕋/Ann_{𝕋}(W) ⊂ End_{𝕋}(W) ↪ End_{K}^{0}(B_{W}).

Proof. First note that if V is a free 𝕋-module of rank 1, then V ≃ 𝕋, and so End_{𝕋}(V) ≃ End_{𝕋}(𝕋) = 𝕋. Thus End_{𝕋}(V) = ρ(𝕋), the image of 𝕋 in End_{ℚ}(V), which means that ρ(𝕋) is a maximal commutative subalgebra. But then 𝔼̃ = End_{𝔼}(V) ⊂ End_{𝕋}(V) = ρ(𝕋), and so every 𝕋-submodule is a fortiori an 𝔼-submodule, and every 𝕋-homomorphism is also an 𝔼̃- homomorphism. Thus, the first assertion follows directly from Theorem 3.2, as does the second by taking θ_{W} = (W_{𝔼}^{−1})_{|End_{𝕋}(W)}.

As we shall see presently, most of the interesting examples of faithful 𝔼-modules V arise from faithful (covariant) functors

F : \underline{Ab}_{/K} → \underline{Vec}_{/F}

from the category of abelian varieties to the category \underline{Vec}_{/F} of (finite dimensional) F -vector spaces. Some examples of such functors are the following.

Example 3.5 (a) (Homology functor) Suppose that K ⊂ ℂ. Then we can view A_{ℂ} := A ⊗_{K} ℂ as a complex analytic space, and so homology theory yields a faithful functor H_{1} : \underline{Ab}_{/K} → \underline{Vec}_{/ℚ} which is defined by H_{1}(A) = H_{1}(A_{ℂ}^{an}, ℚ); cf. [Mu], p. 176.

(b) (Tangent space functor) Suppose that char(K) = 0. Then the tangent space T_{0}(A) of A at the origin is a K-vector space of dimension d = dim(A), and we obtain a faithful functor T_{0} : \underline{Ab}_{/K} → \underline{Vec}_{/K}. (To see that T_{0} is faithful, reduce to the case K = ℂ and use [Mu], p. 176(top).)

(c) (Tate module functor) Let K be any field and fix a prime ℓ ≠ char(K). For any abelian variety A/K, its Tate module T_{ℓ}(A) := T_{ℓ}(A ⊗ \overline{K}) a free ℤ_{ℓ}-module of rank 2d and so T_{ℓ}^{0}(A) = T_{ℓ} ⊗ ℚ_{ℓ} is a ℚ_{ℓ}-vector space of dimension 2d. Moreover, the induced functor T_{ℓ}^{0} : \underline{Ab}_{/K} → \underline{Vec}_{ℚ_{ℓ}} is faithful by [Mu], p. 176ff.

Now if we have a faithful functor F : \underline{Ab}_{/K} → \underline{Vec}_{/F}, then the above theorems can be restated in the following manner.

Theorem 3.6 If F : \underline{Ab}_{/K} → \underline{Vec}_{/F} is a faithful (covariant) functor from the category of abelian varieties to the category \underline{Vec}_{/F} of (finite dimensional) F -vector spaces, then the map B ↦ W_{F}(B) = Im(F(j_{B})) = I(B)F(A) ⊂ F(A) induces for each A/K a lattice-preserving bijection

W_{F} = W_{F,A/K} : Sub(A/K) →^{∼} Alg_{F}(F(A))

between the set of abelian subvarieties of A/K and the set Alg_{F}(F(A)) = {Im(F(a)) : a ∈ 𝔼} of F-algebraic subspaces of F(A).

Moreover, if F = ℚ, then for every abelian variety A/K we have the identification

Alg_{F}(F(A)) = 𝔼̃Sub(F(A))

where 𝔼̃ = End_{𝔼}(F(A)) and 𝔼 = End_{K}^{0}(A), and the map h ↦ F(h) induces for any B_{1}, B_{2} ∈ Sub(A/K) a (functorial) isomorphism

(W_{F})_{B_{1},B_{2}} : Hom_{K}^{0}(B_{1}, B_{2}) →^{∼} Hom𝔼̃(F(B_{1}), F(B_{2})), (22)

where we view F(B_{i}) as an 𝔼̃-module via the identification F(j_{B_{i}}) : F(B_{i}) →^{∼} W_{F}(B_{i}).

Proof. By functoriality, the map a ↦ F(a) defines a ring homomorphism

ρ := ρ_{F,A/K} : End_{K}(A) → End_{F} (F(A)).

Since F is faithful, the map ρ is injective, and hence ρ extends to an injective ring homomorphism ρ : 𝔼 = End_{K}^{0}(A) → End_{F} (F(A)).

Since clearly Alg_{𝔼,ρ}(F(A)) = Alg_{F}(F(A)), the first assertion follows from Theorem 3.1 once we have shown that W_{ρ}(B) = W_{F,A/K}(B). Now since F(N_{B,λ}) ∘ F(j_{B}) = e_{B}id_{F(B)}, the map F(N_{B,λ}) is surjective (and F(j_{B}) is injective) and hence by (18) we obtain W_{ρ}(B) = Im(ρ(ε_{B,λ})) = Im(F(j_{B}) ∘ F(N_{B,λ})) = Im(F(j_{B})) = W_{F,A/K}(B), as desired. Note also that I(B)F(A) = W_{F}(B) by Theorem 3.1.

Now suppose that F = ℚ. Then F(A) is a finitely generated faithful 𝔼-module, and so the second assertion follows directly from Theorem 3.2.

To verify the last assertion, first note that the map g ↦ Φ(g):= F(j_{B_{2}})^{−1}∘ g ∘ F(j_{B_{1}}) defines an isomorphism

Φ : Hom𝔼̃(W_{F}(B_{1}), W_{F}(B_{2})) →^{∼} Hom𝔼̃(F(B_{1}), F(B_{2})).

Thus, if we put (W_{F})_{B_{1},B_{2}} = Φ ∘ W_{𝔼}, then by (19) we obtain an isomorphism (22). Moreover, since (W_{F})_{B_{1},B_{2}}(h) = F(h) if h ∈ Hom_{K}^{0}(B_{1}, B_{2}) (because by (21) we have W_{𝔼}(h) = F(j_{B_{2}} ∘ h ∘ N′_{B_{1}})_{|W_{F} (B_{1})} = F(j_{B_{2}}) ∘ F(h) ∘ F(N′_{B_{1}})_{|W_{F} (B_{1})} = Φ^{−1}(F(h))), the last assertion follows.

Remark. Note that Example 1.3 of the introduction is an immediate consequence of Example 3.5 and Theorem 3.6. Similarly, Corollary 1.4 follows immediately from Example 1.3(b) and Corollary 3.4.

4 Quotient Varieties

In the previous sections we saw that the set Id_{𝔼} of right ideals of 𝔼 = End_{K}^{0}(A) naturally corresponds to the set Sub(A/K) of abelian subvarieties of A/K, and that there is a similar assertion for finitely generated faithful left 𝔼-modules V (in place of 𝔼); cf. Theorem 3.2 and Remark 3.3. Here we shall now show that there is an analogous dual statement involving the set _{𝔼}Id of left ideals which then generalizes to an assertion involving right 𝔼-modules. In this case, however, the set Sub(A/K) has to be replaced by either the set Sub(Â/K) of subvarieties of the dual abelian variety Â or by the set Quot(A/K) of all abelian quotients of A/K which is defined as follows.

Definition. If A/K is an abelian variety, then we call a homomorphism p : A → C an abelian quotient if p is surjective and if Ker(p) is an abelian subvariety of A. Furthermore, if p′ : A → C′ is another abelian quotient, then p′ is called equivalent to p if there exists an isomorphism h : C →^{∼} C′ such that p′ = h ∘ p. The set of equivalence classes of abelian quotients of A is denoted by Quot(A/K).

Remark 4.1 (a) Note that if p : A → C is any surjective homomorphism of abelian varieties, then p is faithfully flat and hence (C, p) is the quotient of A by Ker(p).

(b) If p : A → C is any homomorphism of abelian varieties, then it is well-known (cf. [La], p. 216) that

(23) p is an abelian quotient ⇔ p̂: Ĉ → Â is injective.

It thus follows that for any such p there is a finite homomorphism j_{p}:C →A such that p ∘ j_{p} = [e_{C}]_{C}, (24)

for some integer e_{C} > 0. [Indeed, apply Corollary 2.2 to h = p̂: Ĉ ↪ Â and take j_{p} := κ_{A}^{−1} ∘ N̂p̂ ∘ κ_{C}.]

Proposition 4.2 The map (C, p) ↦ Im(p̂) defines a bijection

D = D_{A/K} : Quot(A/K) →^{∼} Sub(Â/K).

Proof. For B ∈ Sub(Â/K) put D′(B) = (B̂, ĵ_{B} ∘ κ_{A}). Then by the above equivalence (23) we see that D′(B) ∈ Quot(A/K) and that D and D′ are inverses of each other.

We can use the above bijection to prove the following dual version of Theorem 2.4.

Theorem 4.3 The map (C, p) ↦ Î_{A/K}(C, p) := Hom^{0}(C, A)p induces a bijection

Î_{A/K} : Quot(A/K) →^{∼} _{𝔼}Id

whose inverse is given by the map C_{A/K}(𝔞) = (C_{𝔞}, p_{𝔞}), where C_{𝔞} = A/r_{𝔼}(𝔞)A, r_{𝔼}(𝔞) = {f ∈ 𝔼 : 𝔞f = 0} is the right annihilator of 𝔞 and p_{a} : A → C_{a} is the quotient map. Moreover, for any pair \underline{C}_{i} = (C_{i}, p_{i}) ∈ Quot(A/K) there is a functorial isomorphism

(Î_{A/K})_{\underline{C}_{1},\underline{C}_{2}} : Hom_{K}^{0}(C_{1}, C_{2}) →^{∼} Hom_{𝔼}(Î(C_{2}, p_{2}), Î(C_{1}, p_{1})) (25)

such that for any h ∈ 𝔼 with the property that p_{2} ∘ h = h′ ∘ p_{1}, for some h′ ∈ Hom_{K}^{0}(C_{1}, C_{2}), we have

(Î_{A/K})_{\underline{C}_{1},\underline{C}_{2}}(h′)(x) = xh, for all x ∈ Î_{A/K}(C_{2}, p_{2}). (26)

Proof. Since f ↦ f̂ induces an isomorphism 𝔼 = 𝔼_{A/K} →^{∼} 𝔼̂^{op} := 𝔼_{Â/K}^{op}, we

have an induced bijection D_{𝔼} : _{𝔼}Id →^{∼} 𝔼̂^{op}Id = Id𝔼̂. Now

D_{𝔼} ∘ Î_{A/K} = I_{Â/K} ∘ D_{A/K}. (27)

because D_{𝔼}(Î_{A/K}(C, p)) = {f̂ : f ∈ Hom_{K}^{0}(C, A)p} = p̂Hom_{K}^{0}(Â, Ĉ) = I_{Â/K}(p̂(Ĉ)) = I_{Â/K}(D_{A/K}(C, p)), and so we see that Î_{A/K} is a bijection since D_{A/K}, D_{𝔼} and I_{Â/K} are all bijections.

Next we show that C_{A/K} : _{𝔼}Id → Quot(A/K) is the inverse of Î_{A/K}. Since Î_{A/K} is a bijection, it is enough to verify that C_{A/K}(Î_{A/K}(C, p)) ≃ (C, p), for all (C, p) ∈ Quot(A/K), and for this it is enough to show that Ker(p) = r_{𝔼}(Î_{A/K}(C, p))A or, equivalently (by Theorem 2.4) that

I_{A/K}(Ker(p)) = r_{𝔼}(Î_{A/K}(C, p)). (28)

Now if f ∈ 𝔼, then f ∈ I_{A/K}(Ker(p)) ⇔ Im(f) ⊂ Ker(p) ⇔ p ∘ f = 0. Thus, if f ∈ I_{A/K}(Ker(p)), then Î_{A/K}(C, p)f = Hom_{K}^{0}(C, A)pf = 0, and so f ∈ r_{𝔼}(Î_{A/K}(C, p)). Conversely, if Î_{A/K}(C, p)f = 0 then in particular j_{p} ∘ p ∘ f = 0, where j_{p} ∈ Hom_{K}(C, A) is as in (24), and hence e_{C}(p ∘ f) = p ∘ j_{p} ∘ p ∘ f = 0, which means that f ∈ I_{A/K}(Ker(p)). This proves (28), and hence that C_{A/K} is the inverse of Î_{A/K}.

We now construct the bijection (Î_{A/K})_{\underline{C}_{1},\underline{C}_{2}}. For this, we first observe that for any two left 𝔼-ideals 𝔞, 𝔟 ∈ _{𝔼}Id there is a canonical isomorphism

D_{𝔼} = (D_{𝔼})_{𝔞,𝔟} : HomÊ(D_{𝔼}(𝔞), D_{𝔼}(𝔟)) →^{∼} Hom_{𝔼}(𝔞, 𝔟)

such that for all f ∈ HomÊ(D_{𝔼}(𝔞), D_{𝔼}(𝔟)) we have

(D_{𝔼}(f)(x))^{}ˆ = f(x̂), for all x ∈ 𝔞. (29)

To define (Î_{A/K})_{\underline{C}_{1},\underline{C}_{2}}, let α_{i} : Ĉ_{i} →^{∼} Ĉ′_{i} := Im(p̂_{i}) ∈ Sub(Â) be the unique isomorphism such that p̂_{i} = jĈ′_{i} ∘ α_{i}, and put for h ∈ Hom_{K}^{0}(C_{1}, C_{2})

(Î_{A/K})_{\underline{C}_{1},\underline{C}_{2}}(h) = D_{𝔼}((I_{Â/K})_{Ĉ′_{2},Ĉ′_{1}}(α_{1} ∘ ĥ ∘ α_{2}^{−1})). (30)

Clearly, (Î_{A/K})_{\underline{C}_{1},\underline{C}_{2}} defines an isomorphism (25) because D_{𝔼} and (I_{Â/K})_{Ĉ′_{2},Ĉ′_{1}} are both isomorphisms (the latter by Theorem 2.4), and one verifies without difficulty (by using (8) and (29)) that (26) holds.

Remark. Note that the above proof shows that even though the ideal Î(C_{i}, p_{i}) only depends on the equivalence class of (C_{i}, p_{i}), the above iso-

morphism (Î_{A/K})_{\underline{C}_{1},\underline{C}_{2}} : Hom_{K}^{0}(C_{1}, C_{2}) →^{∼} Hom_{𝔼}(Î(C_{2}, p_{2}), Î(C_{1}, p_{1})) actu-

ally depends on the choice of the p_{i}’s.

Similarly, we have the following dual version of Theorem 3.2.

Theorem 4.4 If V is a faithful, finitely generated right 𝔼-module, then the map (C, p) ↦ Ŵ_{𝔼}(C, p) := V ⊗_{𝔼} Î(C, p) = V Î(C, p) = ∑_{g∈Hom_{K}^{0}(C,A)} V (g ∘p) defines a bijection

Ŵ_{𝔼} = Ŵ_{𝔼,V} : Quot(A/K) →^{∼} Sub(V)𝔼̃.

whose inverse is given by C_{𝔼,V}(W) = (C_{W} , p_{W}), where C_{W} = A/r_{𝔼}(W)A, r_{𝔼}(W) = {f ∈ 𝔼 : W f = 0} is the right annihilator and p_{W} : A → C_{W} is the quotient map. Furthermore, for any two quotients \underline{C}_{i} = (C_{i}, p_{i}) ∈ Quot(A/K) we have an induced functorial isomorphism

(Ŵ_{𝔼})_{\underline{C}_{1},\underline{C}_{2}} : Hom_{K}^{0}(C_{1}, C_{2}) →^{∼} Hom𝔼̃(Ŵ_{𝔼}(\underline{C}_{2}), Ŵ_{𝔼}(\underline{C}_{1})). (31)

such that the analogue of (26) holds.

Proof. We can deduce this from Theorem 3.2 as follows. Let 𝔼̂ = End_{K}^{0}(Â) and view V as a left 𝔼̂-module via the rule f̂· v = vf, if v ∈ V and f ∈ 𝔼. Then we have the identification (𝔼̂) = End𝔼̂(V) = End_{𝔼}(V) = 𝔼̃ as subrings of End_{ℚ}(V), and this yields the identification _{(𝔼̂)}Sub(V) = Sub(V)𝔼̃. Thus, since W_{Â/K} and D_{A/K} are both bijections (cf. Th. 3.2 and Prop. 4.2), so is

Ŵ_{𝔼,V} = W_{𝔼̂,V} ∘ D_{A/K}. (32)

Moreover, since r_{𝔼}(V Î_{A/K}(C, p)) = r_{𝔼}(Î_{A/K}(C, p)) (because V is faithful), we see by using Theorem 4.3 that C_{𝔼,V} is the inverse map of Ŵ_{𝔼,V}.

Finally, the last assertion can be deduced from Theorem 3.2 by an argument similar to that of the proof of Theorem 4.3.

Just as covariant functors F : \underline{Ab}_{/K} → \underline{Vec}_{/F} give rise to left 𝔼-modules, contravariant functors

G : \underline{Ab}_{/K} → \underline{Vec}_{/F}^{0} ,

give rise to right 𝔼-modules. The following are examples of such functors.

Example 4.5 (a) (Duals of covariant functors) Let D_{F} : \underline{Vec}_{/F} →^{∼} \underline{Vec}_{/F}^{0} denote the (contravariant) duality functor defined by D_{F} (V) = V^{∗} = Hom(V, F). Clearly, if F : \underline{Ab}_{/K} → \underline{Vec}_{/F} is any faithful covariant functor, then its “dual” F^{∗} = D_{F} ∘ F : \underline{Ab}_{/K} → \underline{Vec}_{/F}^{0} is a faithful contravariant functor. In particular, the duals of the functors H_{1}, T_{0} and T_{ℓ}^{0} considered in Example 3.5 are called the cohomology functor H^{1} = (H_{1})^{∗}, the cotangent functor T_{0}^{∗} = (T_{0})^{∗} and the étale cohomology functor H_{et}^{1}(·, ℚ_{ℓ}) = (T_{ℓ}^{0}(·))^{∗}, respectively.

(b) (The functor of holomorphic differentials) Let char(K) = 0 and let Ω : \underline{Ab}_{/K} → \underline{Vec}_{/K}^{0} denote the functor of holomorphic differentials defined by Ω(A) = H^{0}(A, Ω_{A/K}^{1}). Since the map ω ↦ ω_{0} ∈ T_{0}^{∗}(A) defines an isomorphism of functors Ω ≃ T_{0}^{∗}, this functor is again faithful.

In particular, if K = ℚ, then V = Ω(A) satisfies the hypothesis of Theorem 4.4 and so Theorem 1.5 of the introduction follows from Theorem 4.4 together with (the proof of) Corollary 3.4.

For contravariant functors, the dual analogue of Theorem 3.6 is the following result which is proven in a similar manner.

Corollary 4.6 If G : \underline{Ab}_{/K} → \underline{Vec}_{/ℚ}^{0} is a faithful, contravariant functor then for each A/K the map (C, p) ↦ Ŵ_{G}(C, p) := Im(G(p)) ⊂ G(A) defines a bijection

Ŵ_{G} = Ŵ_{A/K,G} : Quot(A/K) →^{∼} Sub(G(A))𝔼̃,

where 𝔼̃ = End_{𝔼}(G(A)) = {f ∈ End(G(A)) : (vf)a = (va)f, ∀v ∈ G(A), a ∈ 𝔼\}. Furthermore, the rule h ↦ G(h) defines for \underline{C}_{i} = (C_{i}, p_{i}) an isomorphism

(Ŵ_{G})_{\underline{C}_{1},\underline{C}_{2}} : Hom_{K}^{0}(C_{1}, C_{2}) →^{∼} Hom𝔼̃(G(C_{2}), G(C_{1})),

where we view G(C_{i}) as a right 𝔼̃-module via the isomorphism G(p) : G(C_{i}) →^{∼} Ŵ_{G}(C_{i}, p_{i}).

5 Applications to Modular Curves

Let X_{Γ,ℂ} = Γ\ℌ^{∗} be the complex modular curve attached to a subgroup Γ ≤ SL_{2}(ℤ) of level N ≥ 1, i.e. Γ_{0}(N) ≤ Γ ≤ Γ_{1}(N). (We could also take Γ = Γ(N).) By Shimura[Sh1], X_{Γ,ℂ} has a “canonical” model X = X_{Γ}/ℚ over ℚ such that the map f ↦ fdz induces a natural identification S_{2}(Γ, ℚ) →^{∼} H^{0}(X, Ω_{X/ℚ}^{1}) where S_{2}(Γ, ℚ) ⊂ S_{2}(Γ) denotes the subspace of all cusp forms of weight 2 on Γ whose Fourier expansion (at the cusp ∞) have rational coefficients; cf. [Sh1], p. 156 and p. 140 or [DDT], p. 35. Thus, if J = J_{Γ}/ℚ denotes the Jacobian variety of X_{Γ}, then we have a canonical identification

Ω(J) := H^{0}(J, Ω_{J/ℚ}^{1}) = H^{0}(X, Ω_{X/ℚ}^{1}) = S_{2}(Γ, ℚ).

By Hecke’s theory, there is a commutative subring (called the Hecke algebra) 𝕋 = ℚ[{T_{n}}_{n≥1}] ⊂ 𝔼 = End_{ℚ}^{0}(J) such that S_{2}(Γ, ℚ)^{∗} ≃ T_{0}(J) is a free 𝕋-module of rank 1; cf. [Sh1], Theorem 3.51 or [DDT], Lemma 1.34. Furthermore, Ω(J) = S_{2}(Γ, ℚ) is also a free 𝕋-module of rank 1, as can be seen either from Atin-Lehner Theory (cf. [DDT], Lemma 1.35) or by constructing an explicit 𝕋-module isomorphism

φ_{Γ} : S_{2}(Γ, ℚ) →^{∼} T_{0}(J)

as in [Sh2], §2; cf. also [Sh1], Theorem 3.51. (Actually, Shimura constructs only a 𝕋 ⊗ ℂ-module isomorphism φ̃ : S_{2}(Γ) = S_{2}(Γ, ℚ) ⊗ ℂ →^{∼} T_{0}(J ⊗ ℂ) = T_{0}(J) ⊗ ℂ, but it is easy to check that φ̃ is G_{ℚ}-equivariant.) We are now ready to prove the following generalization of the Shimura construction:

Theorem 5.1 Let W ⊂ S_{2}(Γ, ℚ) be any 𝕋-submodule. Then:

(a) There exists a unique abelian subvariety B_{W} ∈ Sub(J_{Γ}) such that φ_{Γ}(W) = T_{0}(B_{W}) ⊂ T_{0}(J). Furthermore, dim B_{W} = dim_{ℚ} W and there exists a ring injection θ′_{W} : End_{𝕋}(W) ↪ End^{0}(B_{W}) such that θ′_{W} (t_{|W}) = t_{|B_{W}}, for all t ∈ 𝕋.

(b) There exists a unique abelian quotient p_{W} : J_{Γ} → C_{W} such that p_{W}^{∗} Ω(C_{W}) = W. Furthermore, dim C_{W} = dim_{ℚ} W and we have an injective ring homomorphism θ_{W} : End_{𝕋}(W) ↪ End_{ℚ}^{0}(C_{W}) such that θ_{W} (t_{|W}) ∘ p_{W} = p_{W} ∘ t, for all t ∈ 𝕋.

Proof. (a) The first assertion follows directly from Corollary 1.4, and the second is clear because dim B_{W} = dim_{ℚ} T_{0}(B_{W}) = dim_{ℚ} W. Finally, the last assertion follows from Corollary 3.4 (applied to φ(W) ⊂ V = T_{0}(J)).

(b) This follows directly from Theorem 1.5, together with (the dual analogue of) Corollary 3.4.

The classical Shimura construction is the following special case of the above Theorem.

Example 5.2 (Shimura) Let f ∈ S_{2}(Γ) be a (normalized) 𝕋-eigenfunction, and put W̃_{f} = ∑ ℂf^{σ}, where the sum is over all Aut(ℂ)-conjugates of f. Clearly, W̃_{f} is Aut(ℂ)-invariant and hence is of the form W̃_{f} = W_{f} ⊗ ℂ for a unique subspace W_{f} ⊂ S_{2}(Γ, ℚ). Moreover, dim_{ℚ} W_{f} = dim_{ℂ} W̃_{f} = [K_{f} : ℚ], where K_{f} is the field generated by the Fourier coefficients a_{n}(f) of f. Let λ_{f} : 𝕋 → K_{f} denote the canonical surjective homomorphism defined by f|t = λ_{f}(t)f, for t ∈ 𝕋. (In particular, λ_{f}(T_{n}) = a_{n}(f), where T_{n} is the n-th Hecke operator.) It is then immediate that Ann_{𝕋}(W_{f}) = Ker(λ_{f}), and thus we have a natural injection K_{f} = 𝕋/Ann_{𝕋}(W_{f}) ↪ End_{𝕋}(W). Thus, by the above theorem there exists an abelian subvariety B_{f} = B_{W_{f}} ≤ J = J_{Γ} and an abelian quotient p = p_{f} : J → C_{f} := C_{W_{f}} together with maps θ′_{f} : K_{f} ↪ End_{ℚ}^{0}(B_{f}) and θ_{f} : K_{f} ↪ End_{ℚ}^{0}(C_{f}) such that (B_{f}, θ′_{f}) and (C_{f}, p_{f}, θ_{f}) satisfy the following conditions (which are in fact identical to those of Theorems 1 and 2 of [Sh2]):

(i) B_{f} ∈ Sub(J/ℚ) and (C_{f}, p_{f}) ∈ Quot(J/ℚ).

(ii) θ′_{f} : K_{f} ↪ End_{ℚ}^{0}(B_{f}) and θ_{f} : K_{f} ↪ End_{ℚ}^{0}(C_{f}) are injective ring homomorphisms such that θ′_{f}(a_{n}(f)) = (T_{n})_{|B_{f}} and θ_{f}(a_{n}(f)) ∘ p_{f} = p_{f} ∘ T_{n}, for all n ≥ 1.

(iii) dim B_{f} = dim C_{f} = [K_{f} : ℚ].

(iv) T_{0}(B_{f}) = φ_{Γ}(W_{f}) and p_{f}^{∗}Ω(C_{f}) = W_{f}.

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