Abelian Subvarieties and the Shimura Construction

Ernst Kani

Queen’s University, Kingston, Ontario, Canada, K7L 3N6 Kani@mast.queensu.ca

Abstract. In 1971 Shimura showed that each weight 2 Hecke eigenfunction f gives rise to both an abelian subvariety and an abelian quotient of the Jacobian variety of the modular curve X_{1}(N)/ℚ. The purpose of this paper is to show that both these constructions follow from a general “dictionary” that translates statements about subvarieties and quotients of an abelian variety into statements about ideals of the associated endomorphism algebra. This dictionary is, in fact, a special case of more general dictionary which applies to subobjects and quotients in a general semi-simple abelian category.

Keywords: Abelian variety, abelian category, endomorphism algebra, modular forms, subobject.

1 Introduction

In his fundamental book, Shimura [Sh1] showed that each (Hecke) 𝕋-eigenfunction f ∈ S_{2}(N) on Γ_{1}(N) gives rise to an abelian subvariety A′_{f} ⊂ J_{1}(N) of 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 paper 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/K as follows:

Theorem 1.1 The map B ↦ I(B) := {f ∈ 𝔼 : Imf ⊂ B} defines an inclusionpreserving 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^{0}(B_{1}, B_{2}) := Hom_{K}(B_{1}, B_{2}) ⊗ ℚ →^{∼} Hom_{𝔼}(I(B_{1}), I(B_{2})).

This theorem is deduced in §3 from a general fact (Theorem 2.7) about semisimple abelian categories which is presented in §2. It is interesting to note that both Theorem A of [KR] (cf. Remark 3.2(a)) and a result of Lange[Lan] (Corollary 3.5) are easily deduced from this theorem.

Another consequence of Theorem 1.1 is the following. 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. Theorem 4.1. 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 ↦ I(B)V defines an inclusion-preserving bijection

S_{A/K,V} : Sub(A/K) →^{∼} Sub𝔼̃(V)

between the set of abelian subvarieties of A/K and the set of left 𝔼̃-submodules of V, where 𝔼̃ = End_{𝔼}(V). Thus, for each left 𝔼̃-submodule W ⊂ V there is a unique abelian subvariety B_{W} ⊂ A such that S_{A/K,V} (B_{W}) = W, and we have a ring isomorphism

θ_{W} : End𝔼̃(W) →^{∼} End^{0}(B_{W}).

Two natural examples for which this result can be applied are V = H_{1}(A^{an}, ℚ), the first homology group of the underlying analytic space A^{an} (if K = ℂ) and V = T_{0}(A), the tangent space of A at 0 (if K = ℚ). Now it is the second example which gives rise to Shimura’s (first) construction because the latter may be deduced from the following general statement.

Corollary 1.3 Let A/ℚ be an abelian variety, and suppose that there exists a commutative subring 𝕋 ⊂ 𝔼 = End^{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), and let W ⊂ T_{0}(A)^{∗} be a 𝕋-submodule. Then φ(W) ⊂ T_{0}(A) does not depend on the choice of φ and there is a unique abelian subvariety B_{W} ∈ Sub(A/ℚ) such that T_{0}(B_{W}) = φ(W). Moreover, dim B_{W} = dim_{ℚ} W.

Indeed, as will be explained in section 5, this can be applied directly to the case A = J_{1}(N), the Jacobian of the modular curve X_{1}(N); cf. Theorem 5.1.

As Shimura explained in [Sh2], it is, however, more natural to work with abelian quotients of A in place of abelian subvarieties because then we can work with the dual module Ω(A) = H^{0}(A, Ω_{A/K}^{1}) ≃ T_{0}(A)^{∗} directly. More precisely, we have the following result (which is a special case of Corollary 4.4):

Theorem 1.4 If A is an abelian variety over a number field K and p : A → C is a quotient of A, then the assignment (C, p) ↦ p^{∗}Ω(C) ⊂ Ω(A) := H^{0}(A, Ω_{A/K}^{1}) induces a bijection

Q_{A/K,Ω} : Quot(A/K) →^{∼} Sub𝔼̃(Ω(A))

between the set of abelian quotients of A/K and the set of left 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_{K} 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). We thus obtain a different characterization (and proof) of the (second) Shimura construction [Sh2].

Moreover, we can use the above dictionary to construct other interesting subvarieties and/or abelian quotients of J_{1}(N). These will be used to show how the Atkin-Lehner decomposition of S_{2}(Γ, ℚ) gives rise to an isogeny decomposition of J_{1}(N) of the form

J_{1}(N) ∼ ∏_{f∈N (Γ_{1}(N))/G_{ℚ}} A_{f}^{n_{f}} . (1)

Here N (Γ_{1}(N)) denotes the set of all weight 2 newforms (of all levels) on Γ_{1}(N), A_{f} is the abelian variety attached to (the Galois orbit of) f ∈ N (Γ_{1}(N)) by the Shimura construction, and n_{f} = σ_{0}(N/N_{f}) denotes the number of divisors of N/N_{f}, where N_{f}|N denotes the level of the newform f.

This paper is organized as follows. In section 1 we shall explain the general categorical setting for the above results and prove them in that generality. Thus, all the above results also hold for motives; cf. [Ja]. In sections 3 and 4 we shall specialize them to the case of abelian varieties, and in section 5 we shall apply them to generalize the results of Shimura [Sh1], [Sh2] for Jacobians of modular curves.

2 The categorical background

Let X ∈ ob(C) be an object of a category C, and let Sub_{C}(X) = Sub_{C}(X)/∼ denote the classes of subobjects of X. Here Sub_{C}(X) denotes the class of pairs (Y, f) where Y ∈ ob(C) and f : Y → X is a monic (=monomorphism) in C, and ∼ is the equivalence relation induced by the pre-order on Sub_{C}(X). By definition, the latter is given by (Y_{1}, f_{1}) ≤ (Y_{2}, f_{2}) ⇔ f_{1} = f_{2}g, for some g ∈ Hom(Y_{1}, Y_{2}). Thus, (Y_{1}, f_{1}) ∼ (Y_{2}, f_{2}) ⇔ (Y_{1}, f_{1}) ≤ (Y_{2}, f_{2}) ≤ (Y_{1}, f_{1}) ⇔ f_{1} = f_{2}g, for some isomorphism g : Y_{1} →^{∼} Y_{2} (cf. [Mac], p. 122), and the pre-order on Sub_{C}(X) induces a partial order on Sub_{C}(X).

Similarly, let Quot_{C}(X) = Quot_{C}(X)/∼ denote the class of quotients of X, which is defined by duality: Quot_{C}(X) = Sub_{C^{op}}(X).

In the sequel we shall be particularly interested in the subclass Sub′_{C}(X) = Sub′_{C}(X)/∼ of Sub_{C}(X) consisting of the split subobjects of X, i.e. Sub′_{C}(X) consists of pairs (Y, f) where f : Y → X is a split monic in the sense of [Mac], p. 19: there exists g : X → Y such that gf = 1_{Y}. Analogously, the class Quot′_{C}(X) = Quot′_{C}(X)/∼ of split quotients (or retracts) of X is defined. We observe:

Remark 2.1 If F : C_{1} → C_{2} is a functor, then F maps split monics and split epis in C_{1} to split monics and split epis in C_{2}, respectively. Thus, for each X ∈ ob(C_{1}), we have induced maps

F_{X} : Sub′_{C_{1}}(X) → Sub′_{C_{2}}(F (X)), _{X}F : Quot′_{C_{1}}(X) → Quot′_{C_{2}}(F (X)),

given by the rules F_{X}((Y, f)) = (F (Y), F (f)) and _{X}F ((Y, f)) = (F (Y), F (f)). It is clear that F_{X} is pre-order preserving in the sense that

(Y_{1}, f_{1}) ≤ (Y_{2}, f_{2}) ⇒ F_{X}((Y_{1}, f_{1})) ≤ F_{X}((Y_{2}, f_{2})), (2)

for all (Y_{i}, f_{i}) ∈ Sub′_{C_{1}}(X)), i = 1, 2, because f_{1} = f_{2}g ⇒ F (f_{1}) = F (f_{2})F (g). Similarly, _{X}F is pre-order preserving, and hence F_{X} and _{X}F induces order-preserving maps

\overline{F}_{X} : Sub′_{C_{1}}(X) → Sub′_{C_{2}}(F (X)) and _{X} \overline{F} : Quot′_{C_{1}}(X) → Quot′_{C_{2}}(F (X))

on the equivalence classes. Note that if F is fully faithful, then the converse of (2) holds and hence \overline{F}_{X} and_{X} \overline{F} are injective in this case. Moreover, if F is an equivalence of categories, then it is easy to see that \overline{F}_{X} and _{X} \overline{F} are bijections.

We now suppose that C is a preadditive category, so E_{X} = End_{C}(X) is naturally a ring; cf. [Mac], p. 28. In that case we can identify Sub′_{C}(X) with a certain subset of the set Id_{E_{X}} of right ideals of E_{X}, and similarly Quot′_{C}(X) can be identified with subset of the set _{E_{X}}Id of left ideals of E_{X}, as we shall now see.

Proposition 2.2 Let X ∈ ob(C) be an object of a preadditive category C and put E_{X} = End_{C}(X). Then the rules (Y, f) ↦ I_{X}(Y, f) := fHom_{C}(X, Y) and (Y, f) ↦ XI(Y, f) := Hom_{C}(Y, X)f induce order-preserving bijections

I_{X} : Sub′_{C}(X) →^{∼} SpId_{E_{X}}^{C} and _{X}I : Quot′_{C}(X) →^{∼} _{E_{X}}SpId^{C},

where SpId_{E_{X}}^{C} (respectively, _{E_{X}}SpId^{C}) denotes the set of right (respectively, left) ideals of E_{X} which are generated by split idempotents.

Proof. It is enough to verify these assertions for I_{X} because those for _{X}I follow from this by duality. Now if (Y, f) ∈ Sub′_{C}(X) is a split subobject with splitting g, i.e. if gf = 1_{Y}, then ε_{f,g} := fg is a split idempotent of E_{X} ([Mac], p. 20) and we have

I_{X}(Y, f) = ε_{f,g}E_{X}. (3)

Indeed, I_{X}(Y, f) ⊂ ε_{f,g}E_{X} because h ∈ I_{X}(Y, f) ⇒ h = fg′, g′ ∈ Hom(Y, X) ⇒ gh = gfg′ = g′ ⇒ ε_{f,g}h = fg′ = h, so h ∈ ε_{f,g}E_{X}. On the other hand, since gE_{X} ⊂ Hom(X, Y), we have ε_{f,g}E_{X} ⊂ fHom(X, Y) = I_{X}(Y, f), and so (3) follows. From this we deduce further that

I_{X}(Sub′_{C}(X)) = SpId_{E_{X}}^{C}. (4)

Indeed, one inclusion follows from (3). Conversely, if εE_{X} ∈ SpId_{E_{X}}^{C}, then ε = fg with gf = 1_{Y}. Thus (Y, f) ∈ Sub′_{C}(X) and I_{X}(Y, f) = εE_{X} by (3), and so (4) follows.

We next observe that

(Y_{1}, f_{1}) ≤ (Y_{2}, f_{2}) ⇔ I_{X}(Y_{1}, f_{1}) ⊂ I_{X}(Y_{2}, f_{2}), ∀(Y_{i}, f_{i}) ∈ Sub′_{C}(X). (5)

Indeed, if f_{1} = f_{2}g, then I_{X}(Y_{1}, f_{1}) = f_{2}gHom(X, Y_{1}) ⊂ f_{2}Hom(X, Y_{2}) = I_{X}(Y_{2}, f_{2}). Conversely, if I_{X}(Y_{1}, f_{1}) = ε_{1}E_{X} ⊂ ε_{2}E_{X} = I′_{X}(Y_{2}, f_{2}), where ε_{i} = ε_{f_{i},g_{i}}, then ε_{1} = ε_{2}h for some h ∈ E_{X} and then f_{1} = ε_{1}g_{1} = ε_{2}hg_{1} = f_{2}(g_{2}hg_{1}), so (Y_{1}, f_{1}) ≤ (Y_{2}, f_{2}).

From (5) it follows immediately that I_{X} induces an order-preserving injection I_{X} : Sub′_{C}(X) → SpId_{E_{X}}^{C} which is bijection because of (4).

Remark 2.3 (a) Since the above proof did not make use of the hypothesis that C is a preadditive category, we see that Proposition 2.2 is valid for any category C if we define SpId_{E_{X}}^{C} := {εE_{X} : ε ∈ E is a split idempotent}. Thus, Sub′_{C}(X) and Quot′_{C}(X) are always sets.

(b) Note that the above proof gives the following description of the inverse maps: I_{X}^{−1}(εE_{X}) ∼ (Y, f) and _{X}I^{−1}(E_{X}ε) ∼ (Y, g) if ε = fg with gf = 1_{Y}. In particular, the equivalence classes of (Y, f) and (Y, g) do not depend on the choice of ε nor on the choice of f, g such that ε = fg.

The above maps I_{X} and _{X}I are closely related to the basic representation functors h^{X} : C → \underline{Sets} and h_{X} : C^{op} → \underline{Sets} defined by X. To explain this in more detail, let us first introduce the following notation.

Notation. Let A be an additive category and X ∈ ob(A). For n ≥ 1, let X^{n} = X ⊕ … ⊕ X denote the n-fold direct sum with injections e_{i} = e_{i}^{n} : X → X^{n} and projections p_{i} = p_{i}^{n} : X^{n} → X, 1 ≤ i ≤ n. Furthermore, let \underline{Sub}_{X} = \underline{Sub}_{X,A} denote the full subcategory of A whose objects are subobjects of X^{n} for some n ≥ 1, i.e. ob(\underline{Sub}_{X}) consists of all Y ∈ ob(A) which admit a monic f : Y → X^{n}, for sone n. Similarly, let \underline{Quot}_{X} and \underline{Ret}_{X} denote the full subcategories of A whose objects are quotients and retractions of X^{n} for some n ≥ 1, respectively. Thus, \underline{Ret}_{X} is a subcategory of both \underline{Sub}_{X} and of \underline{Quot}_{X}.

Proposition 2.4 (a) X is a generator of \underline{Quot}_{X} and a cogenerator of \underline{Sub}_{X}.

(b) The restriction of the functors h_{X} and h^{X} to \underline{Ret}_{X} define fully faithful, additive functors

h^{X} : \underline{Ret}_{X} → \underline{Mod}_{E_{X}} and h_{X} : \underline{Ret}_{X}^{op} → _{E_{X}} \underline{Mod}

where \underline{Mod}_{E_{X}} and _{E_{X}} \underline{Mod} denote the categories of right and left E_{X}-modules, respectively.

Proof. (a) Let f_{1}, f_{2} : Y_{1} → Y_{2}, where f_{1} ≠ f_{2} and Y_{1}, Y_{2} ∈ ob(\underline{Sub}_{X}). By definition, there is a monic g : Y_{2} → X^{n} for some n ≥ 1. Then gf_{1} ≠ gf_{2}, and hence p_{i}^{n}gf_{1} ≠ p_{i}^{n}gf_{2} for some i, 1 ≤ i ≤ n. Since p_{i}^{n}g ∈ Hom(Y_{2}, X), this proves that X is a cogenerator of \underline{Sub}_{X} in the sense of [Mac], p. 123. The proof that X is a generator of \underline{Quot}_{X} is analogous.

(b) Since A is additive, it is immediate that h^{X} : A → \underline{Sets} factors over the category \underline{Mod}_{E_{X}} of right E_{X}-modules to yield an additive functor h^{X} : A → \underline{Mod}_{E_{X}}; cf. [Sch], p. 143. (Note that h^{X}(Y) = Hom_{A}(X, Y) is naturally a right E_{X}-module, and for any f ∈ Hom_{A}(Y_{1}, Y_{2}), the map h^{X}(f) : h^{X}(Y_{1}) → h^{X}(Y_{2}) defined by h^{X}(f)(g) = fg is clearly E_{X}-linear.) Similarly, the functor h_{X} : A^{op} → \underline{Sets} factors additively over _{E_{X}} \underline{Mod}.

By (a) we know that X is both a generator and a cogenerator of \underline{Ret}_{X}, and hence the restriction of h_{X} and h^{X} to \underline{Ret}_{X} is faithful. To prove that h^{X} is full, let φ ∈ Hom_{E_{X}}(h^{X}(Y_{1}), h^{X}(Y_{2})) be E_{X}-linear, where Y_{1}, Y_{2} ∈ ob(\underline{Ret}_{X}). We claim that ∃f ∈ Hom(Y_{1}, Y_{2}) such that h^{X}(f) = φ.

For this, we first consider the case that Y_{1} = X^{m} and Y_{2} = X^{n}. Since e_{i}^{m} ∈ h^{X}(Y_{1}) = Hom(X, X^{m}), φ(e_{i}^{m}) ∈ h^{X}(X^{n}), and so φ(e_{i}^{m}) = ∑_{j} e_{j}^{n}f_{ij}, for some f_{ij} ∈ E_{X}. Then ∃!f ∈ Hom(Y_{1}, Y_{2}) such that p_{j}^{n}fe_{i}^{m} = f_{ij}, ∀i, j. Since φ is E_{X}- linear, we have ∀(x_{1}, …, x_{m}) ∈ E_{X}^{m} that φ (∑ e_{i}^{m}x_{i}) = ∑ ∑ e_{j}^{n}f_{ij}x_{i} = f (∑ e_{i}^{m}x_{i}) = h^{X}(f) (∑ e_{i}^{m}x_{i}) . Thus h^{X}(f) = φ, because every g ∈ h^{X}(Y_{1}) has the form g = ∑ e_{i}^{m}x_{i}.

Now assume that Y_{1}, Y_{2} ∈ ob(\underline{Ret}_{X}) are arbitrary, and let f_{i} ∈ Hom(Y_{i}, X^{n_{i}}) be such that g_{i}f_{i} = 1_{Y_{i}} for some g_{i}. For φ as above, define φ̃ : h^{X}(X^{n_{1}}) → h^{X}(X^{n_{2}}) by φ̃(g) = f_{2}φ(g_{1}g), where g ∈ h^{X}(X^{n_{1}}). It is immediate that φ̃ is E_{X}-linear, and so by the above φ̃ = h^{X}(f̃), for some f̃ ∈ Hom(X^{n_{1}}, X^{n_{2}}). Put f = g_{2}f̃f_{1} ∈ Hom(Y_{1}, Y_{2}). Then h^{X}(f) = φ because if g ∈ h^{X}(Y_{1}), then f_{1}g ∈ h^{X}(X^{n_{1}}) and hence h^{X}(f)(g) = g_{2}h̃f_{1}g = g_{2}φ̃(f_{1}g) = g_{2}(f_{2}φ(g_{1}f_{1}g)) = φ(g). This proves that h^{X} is fully faithful. The proof for h_{X} is similar.

Corollary 2.5 If (Y, f) ∈ Sub′_{A}(X) and (Z, g) ∈ Quot′_{A}(X), then

I_{X}(Y, f) ≃ h^{X}(Y) and _{X}I(Z, g) ≃ h_{X}(Z), (6)

as right and left E_{X}-modules, respectively. Moreover, for any (Y_{1}, f_{1}), (Y_{2}, f_{2}) ∈ Sub_{A}(X) we have a (functorial) isomorphism

h̃_{Y_{1},Y_{2}}^{X} : Hom_{A}(Y_{1}, Y_{2}) →^{∼} Hom_{E_{X}}(I_{X}(Y_{1}, f_{1}), I_{X}(Y_{2}, f_{2}))

such that h̃_{Y_{1},Y_{2}}^{X}(f)(f_{1}g) = f_{2}fg, for all f ∈ Hom(Y_{1}, Y_{2}), g ∈ Hom(X, Y_{1}).

Proof. Since f is monic, the map g ↦ fg induces a bijection φ_{f} : h^{X}(Y) →^{∼} fh^{X}(Y) = I_{X}(Y, f). Since φ_{f} is E_{X}-linear, this is the desired isomorphism of right E_{X}-modules. Similarly, the map f ↦ fg defines an isomorphism h_{X}(Z) →^{∼} _{X}I(Z, g) of left E_{X}- modules.

By Proposition 2.4(b), the map h_{Y_{1},Y_{1}}^{X} : Hom_{A}(Y_{1}, Y_{2}) → Hom_{E_{X}}(h^{X}(Y_{1}), h^{X}(Y_{2})) induced by h^{X} is an isomorphism (of abelian groups), and hence the same is true for h̃_{Y_{1},Y_{2}}^{X} which is defined by the rule h̃_{Y_{1},Y_{2}}^{X}(f) = φ_{f_{2}}h_{Y_{1},Y_{2}}^{X}(f)φ_{f_{1}}^{−1}. Thus by definition we have h̃_{Y_{1},Y_{2}}^{X}(f)(f_{1}g) = φ_{f_{2}}h_{Y_{1},Y_{2}}^{X}(f)(g) = φ_{f_{2}}(fg) = f_{2}fg.

Remark 2.6 If (Y_{1}, f_{1}), …, (Y_{1}, f_{r}) ∈ Sub′_{A}(X), then for any s < r we have

(7)  Y_{1} ⊕ … ⊕ Y_{s} ≃ Y_{s+1} ⊕ … ⊕ Y_{r} ⇔
I(Y_{1}) ⊕ … ⊕ I(Y_{s}) ≃ I(Y_{s+1}) ⊕ … ⊕ I(Y_{r}),

where I(Y_{i}) = I_{X}(Y, f_{i}). (Similarly, if (Y, f_{i}) ∈ Quot′_{A}(X), then (7) holds with I(Y_{i}) = _{X}I(Y, f_{i}).) Indeed, put Y := Y_{1}⊕…⊕Y_{s} and Y′ = Y_{s+1}⊕…⊕Y_{r} ∈ ob(\underline{Ret}_{X}). Since h^{X} is fully faithful, we have Y ≃ Y′ ⇔ h^{X}(Y) ≃ h^{X}(Y′). But since h^{X} is additive, we have h^{X}(Y) = h^{X}(Y_{1}) ⊕ … ⊕ h^{X}(Y_{s}) ≃ I(Y_{1}) ⊕ … ⊕ I(Y_{s}), and similarly h^{X}(Y′) ≃ I(Y_{s+1}) ⊕ … ⊕ I(Y_{r}). Thus (7) follows.

In the sequel we shall be interested the following special case of the above results.

Theorem 2.7 Let A be an additive category. If X ∈ ob(A) satisfies the conditions that

(i) every idempotent ε ∈ E_{X} = End_{A}(X) splits,

(ii) E_{X} is a semi-simple ring, then we have order-preserving bijections

I_{X} : Sub′_{A}(X) →^{∼} Id_{E_{X}} and _{X}I : Quot′_{A}(X) →^{∼} _{E_{X}}Id,

and the functor h^{X} induces an equivalence of categories h^{X} : \underline{Ret}_{X} → \underline{Mod}_{E_{X}}^{f} between the category \underline{Ret}_{X} and the category \underline{Mod}_{E_{X}}^{f} of finitely generated right E_{X}-modules. In particular, \underline{Ret}_{X} is an abelian category.

Proof. Let 𝔞 ∈ Id_{E_{X}}. Since E_{X} is semi-simple, 𝔞 = εE_{X} is generated by an idempotent ε ∈ E_{X} ([BA], p. 47). By (i), ε is split in A, and so 𝔞 ∈ SpId_{E_{X}}^{A}. Thus Id_{E_{X}} = SpId_{E_{X}}^{A}, and similarly _{E_{X}}Id = _{E_{X}}SpId^{A}, so the first assertion follows from Proposition 2.2.

To prove that h^{X} is an equivalence, we first note that if Y ∈ ob(\underline{Ret}_{X}), then h^{X}(Y) is a right E_{X}-submodule of E_{X}^{n} for some n ≥ 1, and hence is finitely generated. Thus, h^{X} maps \underline{Ret}_{X} into the full subcategory \underline{Mod}_{E_{X}}^{f} of \underline{Mod}_{E_{X}}. Since h^{X} is fully faithful by Proposition 2.4(b), it is enough to show that if M ∈ ob(\underline{Mod}_{E_{X}}^{f}) is a finitely generated E_{X}-module, then M ≃ h^{X}(Y), for some Y ∈ ob(\underline{Ret}_{X}). Since E_{X} is semisimple, M ≃ 𝔞_{1} ⊕ … ⊕ 𝔞_{r}, for some ideals 𝔞_{1}, …, 𝔞_{r} ∈ Id_{E_{X}} (cf. [BA], p. 47). By the above (and Corollary 2.5) we know that there exist Y_{1}, …, Y_{r} ∈ Sub′_{A}(X) such that h^{X}(Y_{i}) ≃ 𝔞_{i}, for 1 ≤ i ≤ r, and so h^{X}(Y_{1} ⊕ … ⊕ Y_{r}) ≃ M. Thus, h^{X} is an equivalence and hence \underline{Ret}_{X} is an abelian category because \underline{Mod}_{E_{X}}^{f} is abelian.

Remark 2.8 (a) We thus have (in the situation of Theorem 2.7) that for every right E_{X}-ideal 𝔞 ∈ Id_{E_{X}} there is a unique subobject (Y_{𝔞}, f_{𝔞}) ∈ Sub_{A}(X) such that I_{X}(Y_{𝔞}, f_{𝔞}) = 𝔞. Moreover, by Corollary 2.5 we have a ring isomorphism

θ_{𝔞} = (h̃_{Y_{𝔞},Y_{𝔞}}^{X})^{−1} : End_{E_{X}}(𝔞) →^{∼} End_{A}(Y_{𝔞})

which can be characterized by the formula f_{𝔞}θ_{𝔞}((λ_{g})_{|𝔞}) = gf_{𝔞}, ∀g ∈ E_{X} with g𝔞 ⊂ 𝔞, where λ_{g}(f) = gf, ∀f ∈ E_{X}. [To see this, note first that for any φ ∈ End_{E_{X}}(𝔞) we have h̃(θ_{𝔞}(φ)) = φ, where h̃ = h̃_{Y_{𝔞},Y_{𝔞}}^{X}, and so φ(f_{𝔞}g′) = h̃(θ_{𝔞}(φ))(f_{𝔞}g′) = f_{𝔞}θ_{𝔞}(φ)g′, ∀g′ ∈ Hom(X, Y_{𝔞}). Thus, if g′ = g_{𝔞} satisfies g_{𝔞}f_{𝔞} = 1_{Y_{𝔞}}, then φ(ε) = f_{𝔞}θ_{𝔞}(φ)g_{𝔞} where ε = f_{𝔞}g_{𝔞}, and hence φ(ε)f_{𝔞} = f_{𝔞}θ_{𝔞}(φ). Thus, taking φ = (λ_{g})_{|𝔞}, we get f_{𝔞}θ_{𝔞}(φ) = λ_{g}(ε)f_{𝔞} = gεf_{𝔞} = gf_{𝔞}, as claimed.]

(b) The condition (i) of Theorem 2.7 is equivalent to the condition that every idempotent ε ∈ E_{X} has a factorization ε = fg into a monic f and epi g, for the condition ε^{2} = ε implies that fgfg = fg and so gf = 1 as f is monic and g is epi. Thus, (i) holds in any abelian category; cf. [Mac], p. 195. It also holds in any pseudo-abelian category (as defined in [Man]), for if ε ∈ E_{X} is an idempotent, then X = Ker(ε) ⊕ Ker(1 − ε) and we have ε = i_{2}p_{2}, where i_{2} : Ker(1 − ε) → X and p_{2} : X → Ker(1 − ε) denote the canonical injection and projection, respectively. Thus, the following corollary may be viewed as a sharpening of Lemma 2 of [Ja].

Corollary 2.9 If A is an additive category, then the above conditions (i) and (ii) hold for all X ∈ ob(A) if and only if A is a semi-simple abelian category. In particular, every monic and epi splits in such a category.

Proof. If A is abelian and semi-simple, then condition (i) holds by Remark 2.8(b) and (ii) holds by definition.

To prove the converse, we first observe that if X, Y ∈ ob(A), then X = X_{1} ⊕ X_{2}, where X_{1} ∈ ob(\underline{Ret}_{Y}) and Hom_{A}(X_{2}, Y) = 0, Hom_{A}(Y, X_{2}) = 0. Indeed, since X, Y ∈ ob(\underline{Ret}_{Z}) with Z = X ⊕ Y and since \underline{Ret}_{Z} is equivalent to \underline{Mod}_{E_{Z}}^{f} by Theorem 2.7, this observation follows from the corresponding one in \underline{Mod}_{E_{Z}}^{f}.

Now suppose f ∈ Hom_{A}(X, Y), where X, Y ∈ ob(A). Them f has a kernel Ker(f) ∈ ob(\underline{Ret}_{Z}), where Z = X ⊕ Y, because \underline{Ret}_{Z} is abelian. By the above observation, Ker(f) is also kernel of f in A. Similarly, f has a cokernel. In same way it follows that every monic (resp. epi) f : X → Y is a kernel (resp. cokernel) because this is true in \underline{Ret}_{Z}. Thus, A is an abelian category. Moreover, since E_{Z} is semi-simple, every monic and epi in \underline{Mod}_{E_{Z}}^{f} splits (cf. [BA], p. 46 and 32), and so the same is true for any monic and epi in \underline{Ret}_{Z} and hence also in A.

Remark 2.10 (a) Since every map of an abelian category A has a canonical decomposition f = im(f)coim(f) (cf. [Mac], p. 195), we have the following formula for the inverse of I_{X} (and that of _{X}I):

(8) I_{X}^{−1}(fE_{X}) = (Im(f), im(f)), _{X}I^{−1}(E_{X}f) = (Coim(f), coim(f)),

for all f ∈ E_{X}. To see this, recall that m := im(f) : Y := Im(f) → X is monic and p := coim(f) : X → Coim(f) = Y is epi, and so there exist splittings g, h such that gm = 1_{Y} = ph. Then f = m(gm)p = (mg)mp = εf, where ε = mg, and ε = m(ph)g = f(hg). Thus, fE_{X} = εE_{X} and so I_{X}^{−1}(fE_{X}) = I_{X}^{−1}(εE_{X}) = (Im(f), im(f)), the latter by Remark 2.3(b). The proof for the second formula is analogous.

For later usage, let us also observe that since ker(coim(f)) = ker(f), we have

(9) Coim(f) = X/Ker(f) with Ker(f) = I_{X}^{−1}(r_{E_{X}}(E_{X}f)),

where r_{E_{X}}(𝔞) = {g ∈ E_{X} : 𝔞g = 0} ∈ Id_{E_{X}} denotes the right annihilator of a left E_{X}-ideal 𝔞 ∈ _{E_{X}}Id. Here, the second formula holds because by the universal property of kernels ([Mac], p. 188) we have I_{X}(Ker(f)) = ker(f)Hom(X, Ker(f)) = {g ∈ E_{X} : fg = 0} = r_{E_{X}}(E_{X}f).

(b) Note that (8) implies that

hI_{X}(Y, f) = I_{X}(h(Y), im(hf)), ∀h ∈ E_{X}, (10)

where h(Y) = Im(hf) denotes the image of (Y, f) under h (cf. [Sch], p. 134), because hI_{X}(Y, f) = hε_{f,g}E_{X} and im(hε_{f,g}) = im(hf). More generally, we see that each h ∈ Hom_{A}(X, X′) induces maps

h_{∗} : Sub_{A}(X) → Sub_{A}(X′) and h^{∗} : Quot_{A}(X′) → Quot_{A}(X)

by the rules h_{∗}(Y, f) = (h(Y), Im(hf)) and h^{∗}(Y′, f′) = (Coim(f′h), coim(f′h)). We then have I_{X′}(h_{∗}(Y, f)) = hI_{X}(Y, f)Hom(X′, X) and _{X}I(h^{∗}(Y′, f′)) = Hom(X′, X)· X′I(Y′, f′)h, as is easy to verify.

(c) From (10) it follows that I_{X}(Y, f) is a two-sided E_{X}-ideal if and only if (Y, f) is E_{X}-stable in the sense that im(hf) factors over f, for all h ∈ E_{X}. In-

deed, hI_{X}(Y, f) =^{(10)} I_{X}(Im(hf), im(hf)) ⊂ I_{X}(Y, f) ⇔ (Im(hf), im(hf)) ≤ (Y, f) ⇔

im(hf) = fg, for some g : Im(hf) → Y.

(d) Note that Id_{E_{X}} and _{E_{X}}Id are lattices, i.e. that each collection 𝔞_{1}, …, 𝔞_{r} of ideals has a least upper bound ∑ 𝔞_{i} and a greatest lower bound ∩𝔞_{i}. Thus, by Theorem 2.7 and Corollary 2.9, the same is true for Sub_{A}(X) = Sub′_{A}(X) and Quot_{A}(X) = Quot′_{A}(X). We now describe these upper and lower bounds explicitly.

As is explained in [Mac], p. 122, the greatest lower bound of (Y_{1}, f_{1}), …, (Y_{r}, f_{r}) ∈ Sub_{A}(X) is (∩Y_{i}, ∩f_{i}), where ∩Y_{i} = Y_{1} ×_{X} … ×_{X} Y_{r} (which exists in A by [Sch], p.

110) with canonical monic ∩f_{i} = (f_{1}, …, f_{r}) : ∩Y_{i} → X. Moreover, the least upper bound is obtained by factoring the canonical map f : Y := Y_{1} ⊕…⊕Y_{r} → X (defined by fe_{i} = f_{i}) into f = im(f)coim(f); the image (Im(f), im(f)) ∈ Sub_{A}(X) is denoted by ∑(Y_{i}, f_{i}) = (∑ Y_{i}, ∑ f_{i}). In particular, we have unique maps g_{i} : Y_{i} → ∑ Y_{i} and h_{i} : ∩Y_{i} → Y_{i} such that f_{i} = (∑ f_{j})g_{i} and f_{i}h_{i} = ∩f_{i}, for 1 ≤ i ≤ r. Thus:

I_{X} (∑(Y_{i}, f_{i})) = ∑ I_{X}(Y_{i}, f_{i}), I_{X} (⋂(Y_{i}, f_{i})) = ⋂ I_{X}(Y_{i}, f_{i}). (11)

From this we can deduce easily that

I_{X}(Y_{0}, f_{0}) = I_{X}(Y_{1}, f_{1}) ⊕ … ⊕ I_{X}(Y_{r}, f_{r}) (12)

⇔ ∃g : Y_{1} ⊕ … ⊕ Y_{r} →^{∼} Y_{0} such that f_{0}ge_{i} = f_{i}, 1 ≤ i ≤ r,

where e_{i} : Y_{i} → Y_{1} ⊕ … ⊕ Y_{r} denotes the ith inclusion map.

By duality, similar statements hold for Quot_{A}(X) = Sub_{A^{op}}(X) in place of Sub_{A}(X). For example, if (Y_{i}, g_{i}) ∈ Quot_{A}(X), 1 ≤ i ≤ r, then their least upper bound is ∑(Y_{i}, g_{i}) = (Coim(g), coim(g)), where g = (g_{1}, …, g_{r}) : X → Y_{1} ⊕ … ⊕ Y_{r} is the unique map such that p_{i}g = g_{i}.

(e) By using the notation in (d), we can give an alternate definition for the inverse maps S_{X} := I_{X}^{−1} and Q_{X} := _{X}I^{−1}: if 𝔞 is a right (respectively, left) E_{X}-ideal, then

(13) I_{X}^{−1}(𝔞) = ∑_{f∈𝔞}(Im(f), im(f)) and _{X}I^{−1}(𝔞) = ∑_{f∈𝔞}(Coim(f), coim(f)).

Indeed, since each 𝔞 ∈ Id_{E_{X}} has the form 𝔞 = εE_{X} (cf. proof of Theorem 2.7), we have by (8) that I_{X}^{−1}(𝔞) = (Im(ε), im(ε)) = ∑_{f∈𝔞}(Im(f), im(f)), and the second formula is proved similarly.

Example 2.11 Let k be a semi-simple ring, and let A = _{k}\underline{Mod}^{f} be the (abelian) category of finitely generated left k-modules. Then Corollary 2.9 shows that A is a semi-simple abelian category because for every M ∈ ob(A) the ring E_{M} = End_{k}(M) is semi-simple by [BA], p. 47.

In addition, from Theorem 2.7 and Remark 2.10(e), (a) we see that the maps S_{M} : 𝔞 ↦ 𝔞M = ∑_{f∈𝔞} Im(f) and Q_{M} : 𝔞 ↦ M/r_{E_{M}} (𝔞)M induce order-preserving bijections

S_{M} : Id_{E_{M}} →^{∼} Sub_{k}(M) = Sub′_{k}(M) and Q_{M} : _{E_{M}} Id →^{∼} Quot_{k}(M) = Quot′_{k}(M), where Sub_{k}(M) = Sub_{A}(M) and Sub_{k}(M) = Quot_{A}(M) denote the sets of ksubmodules and k-quotients of M, respectively.

Furthermore, if M is a faithful k-module, then k ⊂ M^{n} for some n > 0 (by [BA], p. 26) and so \underline{Ret}_{M} = _{k}\underline{Mod}^{f} (because every M′ ∈ _{k}\underline{Mod}^{f} is isomorphic to a (finite) direct sum of ideals of k). Thus, by Theorem 2.7 we see that the functor h^{M} induces an equivalence of categories h^{M} : _{k}\underline{Mod}^{f} → \underline{Mod}_{E_{M}}^{f} = _{E_{M}^{op}} \underline{Mod}^{f}. Note that since each M′ ∈ ob(A) is projective, we have h^{M} ≃ M^{∗} ⊗_{k} ∗, i.e. h^{M} is equivalent to the Morita functor of [CR], p. 60.

In order to be able to apply the above results to the Shimura construction, it is useful to extend Theorem 2.7 as follows.

Theorem 2.12 In the situation of Theorem 2.7, suppose that M is a faithful left E_{X}-module, and put Ẽ_{X} = End_{E_{X}}(M). Then the rule (Y, f) ↦ I_{X}(Y, f)M defines an order-preserving bijection

S_{X,M} : Sub′_{A}(X) →^{∼} Alg_{E_{X}}(M)

where Alg_{E_{X}}(M) = {aM : a ∈ E_{X}} ⊂ SubẼ_{X}(M). Moreover, if M is a finitely generated E_{X}-module, then Alg_{E_{X}}(M) = SubẼ_{X}(M), and we have an equivalence of categories

h^{X,M} := h^{X} ⊗_{E_{X}} M : \underline{Ret}_{X} → Ẽ_{X} \underline{Mod}^{f}

with the property that h^{X,M}(Y) ≃ S_{X,M}(Y, f), for all (Y, f) ∈ Sub_{A}(X). In particular, for every N ∈ SubẼ_{X}(M) there is a unique (Y_{N}, f_{N}) ∈ Sub_{A}(X) such that S_{X,M}(Y_{N}, f_{N}) = N and we have a ring isomorphism

θ_{N} : EndẼ_{X}(N) →^{∼} End_{A}(Y_{N})

such that f_{N}θ_{N}((λ_{g})_{|N}) = gf_{N}, ∀g ∈ E_{X} with gN ⊂ N, where λ_{g}(x) = gx, ∀x ∈ M.

Proof. Since E_{X} is semi-simple and M is faithful, it follows that M is a faithfully flat E_{X}-module. (Use Exercise 1 of [BCA], p. 49). Thus, the map 𝔞 ↦ 𝔞M defines an (order-preserving) injection μ_{M} : Id_{E_{X}} ↪ SubẼ_{X}(M). (Here we use the obvious fact that each 𝔞M is a left Ẽ_{X}-submodule of M.) It thus follows from Theorem 2.7 that the map S_{X,M} = μ_{M} ∘ I_{X} is an order-preserving bijection onto its image Alg(M) := Im(S_{X,M}) ⊂ SubẼ_{X}(M). Note that Alg(M) = Alg_{E_{X}}(M) because I_{X}(Y, f) = εE_{X} for some (idempotent) ε ∈ E_{X} and then S_{X,M}(Y, f) = εM.

Now suppose that M is finitely generated. Then Ẽ_{X} is again semi-simple ([BA], p. 47) and EẼ_{X} ≃ E_{X} ([BA], p. 50). Thus, by Example 2.11 it follows that the map S_{M} = μ_{M} : Id_{E_{X}} → SubẼ_{X}(M) is a bijection, and so Alg_{E_{X}}(M) = SubẼ_{X}(M). Furthermore, since M is faithful and projective, it follows from Morita’s Theorem ([CR], p. 60) that the functor ∗ ⊗_{E_{X}} M : \underline{Mod}_{E_{X}}^{f} → \underline{Mod}f ; Ẽ_{X}^{op} = Ẽ_{X} \underline{Mod}^{f} is an equivalence of categories, and so it follows from Theorem 2.7 that h^{X,M} := (∗ ⊗_{E_{X}} M) ∘ h^{X} is also an equivalence. Moreover, since M is E_{X}-flat, we have by Corollary 2.5 that S_{X,M}(Y, f) = I_{X}(Y, f)M ≃ I_{X}(Y, f) ⊗_{E_{X}} M ≃ h^{X}(Y) ⊗_{E_{X}} M = h^{X,M}(Y).

Finally, if N ∈ SubẼ_{X}(M), then the existence and uniqueness of (Y_{N}, f_{N}) is clear since S_{X,M} is a bijection. Moreover, since N = 𝔞M ≃ 𝔞_{E_{X}}M, for some (unique) 𝔞 ∈ Id_{E_{X}}, we see from Remark 2.8(a) that θ_{N} = θ_{𝔞}⊗_{E_{X}} M is the desired isomorphism.

As we shall see in section 5, the following corollary can be viewed as an abstract version of the Shimura construction.

Corollary 2.13 In the situation of Theorem 2.12, suppose that M is finitely generated and that there is a commutative subring T ⊂ E_{X} such that M is a free T - module of rank 1. Then for every T -submodule N ⊂ M there is a unique subobject (Y_{N}, f_{N}) ∈ Sub_{A}(X) such that S_{X,M}(Y_{N}, f_{N}) = N and we have a natural ring embedding

θ′_{N} : End_{T} (N) ↪ End_{A}(Y_{N})

such that f_{N}θ_{N}((λ_{t})_{|N}) = tf_{N}, for all t ∈ T.

Proof. Since M is a free T -module of rank 1, we have End_{T} (M) = T_{M}, where T_{M} denotes the image of T in End(M). Thus Ẽ_{X} = End_{E_{X}}(M) ⊂ End_{T} (M) = T_{M}, and so every T -submodule N of M is a fortiori an Ẽ_{X}-submodule. Thus, the assertions follow directly from Theorem 2.12, if we let θ′_{N} be the restriction of θ_{N} to End_{T} (N) ⊂ EndẼ_{X}(N).

In practice, most examples of E_{X}-modules arise via additive functors in the following way.

Corollary 2.14 Let A be a semi-simple abelian category and let k be any ring. If F : A → _{k}\underline{Mod} is a faithful additive functor, then for every X ∈ ob(A) the map F_{X} induces an order-preserving bijection

\overline{F}_{X} = S_{X,F (X)} : Sub_{A}(X) →^{∼} Alg_{F} (F (X)),

where Alg_{F} (F (X)) = {Im(F (a)) : a ∈ E_{X}} ⊂ SubẼ_{X}(F (X)). Moreover, the inverse of \overline{F}_{X} is given by \overline{F} _{X}^{−1}(Im(F (a))) = (Im(a), im(a)).

Proof. By functoriality, the rule a ↦ F (a) defines a ring homomorphism ρ_{X,F} : E_{X} → End_{k}(F (X)) which is injective since F is faithful. Thus, M = F (X) is a faithful left E_{X}-module via ρ_{X,F}, and so S_{X,F (X)} : Sub_{A}(X) →^{∼} Alg_{E_{X}}(F (X)) is a bijection by Theorem 2.12. Since I_{X}(Y, f) = fgE_{X}, where g ∈ Hom(X, Y) satisfies gf = 1_{Y}, we see that S_{X,F (X)}(Y, f) = F (fg)F (X) = Im(F (f)F (g)) = Im(F (f)). (14)

Here the last equality holds because F (g) is a retract (and hence is surjective). We thus see that \overline{F}_{X}(Y, f) := (F (Y), F (f)) ∼ Im(F (f)) = S_{X,F (X)}(Y, f) and that Alg_{F} (F (X)) = Alg_{E_{X}}(F (X)) ⊂ SubẼ_{X}(F (X)). Finally, it is clear from (14) that S_{X,F (X)}^{−1}(Im(F (a))) = (Im(a), im(a)).

On the other hand, contravariant functors can also be used in the following mannner.

Corollary 2.15 If A is a semi-simple abelian category and G : A^{op} → _{k}\underline{Mod} is a faithful (contravariant) additive functor, then for every X ∈ ob(A) the map G_{X} defines an order-preserving bijection

Q_{X,G} := \overline{G}_{X} : Quot_{A}(X) = Sub_{A^{op}}(X) →^{∼} Alg_{G}(G(X)) = \{Im(G(a)) : a ∈ End(A)\},

and Q_{X,G}(Y, p) = G(X)_{X}I(Y, p) = Im(G(p)), for any (Y, p) ∈ Quot_{A}(X). Moreover, if G(X) is a finitely generated E_{X}-module, then Alg_{G}(G(X)) = EndẼ_{X}(G(X)), where Ẽ_{X} = End_{E_{X}}(G(X)) = {f ∈ End(G(X)) : fG(a) = G(a)f, ∀a ∈ E_{X}}. Thus, for each (left) Ẽ_{X}-module W ⊂ G(A), there is a unique (Z_{W} , p_{W}) ∈ Quot_{A}(X) such that Im(G(p_{W})) = W, and we have Ker(p_{W}) = I_{X}^{−1}(r_{E_{X}}(W)).

Proof. Since A^{op} is again a semi-simple abelian category, the first assertion is clear from Corollary 2.14, and the second follows from equation (14) together with the fact that Q_{X,G}(Y, p) = S_{X,G(X)}(Y, p^{op}). Moreover, since G(X) is naturally a left E_{X}^{op}- module or, equivalently, a right E_{X}-module, the third assertion follows from Theorem 2.12. To prove the last assertion, put ε = fp_{W}, where p_{W} f = 1_{Z_{W}}. Then W = G(X)ε and r_{E_{X}}(W) = r_{E_{X}}(E_{X}ε) = (1 − ε)E_{X}, and hence (Z_{W} , p_{W}) = (Coim(ε), coim(ε)). Thus, by (9) we have I_{X}^{−1}(r_{E_{X}}(W)) = Ker(ε) = Ker(p_{W}), as claimed.

3 Subvarieties and Quotients of an Abelian Variety

As in the introduction, fix an arbitrary ground field K and let A/K be an abelian variety defined over K. Throughout, we shall freely use the basic facts about abelian varieties as presented in Milne[Mi] and Mumford[Mu].

Notation. Let Sub(A/K) denote the set of abelian subvarieties B of A/K in the usual sense, i.e. B is an abelian variety over K together with a closed immersion j_{B} : B ↪ A which is a K-homomorphism of abelian varieties. Since closed immersions are monics in the category \underline{Var}_{K} of K-varieties, they are also monics in the subcategory \underline{Ab}_{K} of abelian varieties over K with K-homomorphisms, and so we have an inclusion Sub(A/K) ⊂ Sub_{\underline{Ab}_{K}}(A). However, these sets are rarely equal, as we shall see; cf. Remark 3.2(a).

Similarly, let Quot(A/K) ⊂ Quot_{\underline{Ab}_{K}}(A) denote the set of equivalence classes of abelian quotients (C, p) of A; the latter means that p : A → C is a surjective K-homomorphism such that its (schematic) kernel Ker(p) is an abelian variety.

Furthermore, let \underline{Ab}_{K}^{0} denote the category of abelian K-varieties up to isogeny (cf. [Mu], p. 172): we have ob(\underline{Ab}_{K}^{0}) = ob(\underline{Ab}_{K}) and Hom_{\underline{Ab}}_{K}^{0}(A, B) = Hom^{0}(A, B) := Hom(A, B) ⊗ ℚ. Thus, since Hom(A, B) is torsionfree (cf. [Mi], p. 122), \underline{Ab}_{K} is naturally a subcategory of \underline{Ab}_{K}^{0}. We observe:

Proposition 3.1 \underline{Ab}_{K}^{0} is a semi-simple abelian category, and the embedding i_{K} : \underline{Ab}_{K} → \underline{Ab}_{K}^{0} induces bijections

i_{K,A} : Sub(A/K) →^{∼} Sub_{\underline{Ab}}_{K}^{0}(A) = Sub′_{\underline{Ab}_{K}^{0}}(A),
Ai_{K} : Quot(A/K) →^{∼} Quot_{\underline{Ab}}_{K}^{0}(A) = Quot′_{\underline{Ab}_{K}^{0}}(A).

Proof. The first assertion is well-known, but we shall give a quick proof below using Corollary 2.9. For this, we first observe that

(15) f ∈ Hom(A, B) is a monic ⇔ i_{K}(f) is a monic in \underline{Ab}_{K}^{0}

(and similarly for “monic” replaced by “epi”). Indeed, since i_{K} is an embedding (hence faithful), i_{K}(f) monic (epi) ⇒ f monic (epi). Conversely, if f is monic and i_{K}(f)g = i_{K}(f)h with h, g ∈ Hom^{0}(C, A), then ∃n > 0 such that g[n]_{C} = i_{K}(g′) and h[n]_{C} = i_{K}(h′) with g′, h′ ∈ Hom(C, A), where [n]_{C} denotes the mulipication by n map on C. Then fg′ = fh′, so g′ = h′ and hence g[n]_{C} = h[n]_{C}. Thus g = h because [n]_{C} is an isomorphism in \underline{Ab}_{K}^{0}, and so f is monic. The proof for epis is similar.

Now since every map in \underline{Ab}_{K} factors as f = gh where h is a surjection (hence epi) and g is a closed immersion (hence monic), it follows from (15) that every map in \underline{Ab}_{K}^{0} factors as f = gh with g monic, h epi. Thus condition (i) of Theorem 2.7 holds; cf. Remark 2.8(b). Moreover, \underline{Ab}_{K}^{0} is clearly an additive category, and condition (ii) holds for every A ∈ ob(\underline{Ab}_{K}^{0}) by [Mi], p. 122. Thus, by Corollary 2.9 we see that \underline{Ab}_{K}^{0} is a semi-simple abelian category and that Sub_{\underline{Ab}}_{K}^{0}(A) = Sub′_{\underline{Ab}_{K}^{0}}(A) and Quot_{\underline{Ab}}_{K}^{0}(A) = Quot′_{\underline{Ab}_{K}^{0}}(A), for every A. By (15) we see that the rule (B, j_{B}) ↦ (i_{K}(B), i_{K}(j_{B})) defines a map i_{K,A} : Sub(A/K) → Sub_{\underline{Ab}}_{K}^{0}(A) and similarly we have a map _{A}i_{K} : Quot(A/K) → Quot_{\underline{Ab}}_{K}^{0}(A).

To see that i_{K,A} is injective, suppose that i_{K,A}(B, j_{B}) = i_{K,A}(B′, j_{B′}), i.e. that i_{K}(j_{B}) = i_{K}(j_{B′})h, for some isomorphism h : i_{K}(B) →^{∼} i_{K}(B′). Then ∃n > 0 such that h[n]_{B} = i_{K}(h′), for some isogeny h′ : B → B′, and so j_{B}[n]_{B} = j_{B′}h′[n]_{B}, or j_{B} = j_{B′}h′ since [n]_{B} is epi. But since Ker(j_{B}) = 0, it follows that Ker(h′) = 0, and so h′ : B →^{∼} B′ is an isomorphism. Thus (B, j_{B}) ∼ (B′, j′_{B}), i.e. i_{K,A} is injective.

It remains to show that i_{K,A} is surjective. Let (B, f) ∈ Sub′_{\underline{Ab}_{K}^{0}}(A), and let g be such that gf = 1_{B}. Then ∃n > 0 such that fn = i_{K}(f′), gn = i_{K}(g′) with f′ ∈ Hom(B, A), g′ ∈ Hom(A, B), and so g′f′ = [n^{2}]_{B}. Thus, f′ has finite kernel and and so there exists an isogeny π : B → B′ = B/Ker(f) such that f′ = f′′π, where f′′ : B′ → A is a closed immersion (use [Mu], p. 118). Thus (B′, f′′) ∈ Sub(A/K) and i_{K,A}(B′, f′′) ∼ (B, f) because i_{K}(f′′)i_{K}(π) = f[n]_{B} (and i_{K}(π) and [n]_{B} are isomorphisms in \underline{Ab}_{K}^{0}. Thus i_{K,A} is surjective.

The proof for _{A}i_{K} is entirely analogous. Alternately, we can deduce it directly from what was proved above by observing that the duality functor D_{K} : \underline{Ab}_{K}^{op} → \underline{Ab}_{K} defined by D(A) = Â, D(f) = f̂ induces for every A/K an order-reversing bijection

(16) D_{K,A} : Quot_{\underline{Ab}_{K}}(A) = Sub_{\underline{Ab}}_{K}^{op}(A) →^{∼} Sub_{\underline{Ab}_{K}}(Â).

Indeed, the latter assertion is equivalent to the well-known fact (cf. [La], p. 216) that if p : A → C is any homomorphism of abelian varieties, then p is an abelian quotient ⇔ p̂: Ĉ → Â is a closed immersion.

Remark 3.2 (a) From the above proof we see that f : A → B is a monic in \underline{Ab}_{K} if and only if f = f_{0}g, where (A′, f_{0}) ∈ Sub(B/K) and g : A → A′ is an isogeny. Thus f is monic if and only if Ker(f) is finite. Similarly, f is epi in \underline{Ab}_{K} if and only if f = gf_{0}, where (C, f_{0}) ∈ Quot(A/K) and g : C → B is an isogeny; for later reference let us write (B, f)^{0} := (C, f_{0}). Thus, f is epi if and only if f is surjective.

(b) Since \underline{Ab}_{K}^{0} is an abelian category, every f ∈ Hom^{0}(A, B) has a canonical factorization f = jhp where j : Im(f) → B is a monic, p : A → Coim(f) is epi and h : Coim(f) →^{∼} Im(f) is an isomorphism. Thus, by (a), every f ∈ Hom(A, B) has a canonical factorization f = jhp where j is a closed immersion (and Im(f) is an abelian subvariety of B), p : A → Coim(f) = A/Ker(p) is an abelian quotient and h : Coim(f) → Im(f) is an isogeny.

We are now ready to prove Theorem 1.1 of the introduction, as well as a dual version. More precisely, we shall prove the following result.

Theorem 3.3 Let A/K be an abelian variety and put 𝔼 = End^{0}(A). Then the maps B ↦ I(B) := j_{B}Hom^{0}(A, B) = {f ∈ 𝔼 : Imf ⊂ B} and (C, p) ↦ I′(C) = Hom^{0}(C, A)p define lattice-preserving bijections

I_{A/K} : Sub(A/K) →^{∼} Id_{𝔼} and I′_{A/K} : Quot(A/K) →^{∼} _{𝔼}Id

whose inverses are given by I_{A/K}^{−1}(𝔞) = 𝔞A := ∑_{f∈𝔞} Im(f) and by I′_{A/K}^{−1}(𝔞) = (C_{𝔞}, p_{𝔞}), respectively. Here C_{𝔞} = A/r_{𝔼}(𝔞)A, where r_{𝔼}(𝔞) = {f ∈ 𝔼 : 𝔞f = 0} is the right annihilator of 𝔞, and p_{𝔞} : A → C_{𝔞} is the quotient map. Furthermore, the functor h^{A} induces an equivalence of categories

h^{A} : \underline{Sub}_{A/K}^{0} = \underline{Quot}_{A/K}^{0} →^{∼} \underline{Mod}_{𝔼}^{f}

where \underline{Sub}_{A/K}^{0} (respectively, \underline{Quot}_{A/K}^{0}) is the full subcategory of \underline{Ab}_{K}^{0} consisting of those abelian varieties which are isogenous to a subvariety (respectively, to a quotient) of A^{n}, for some n ≥ 1. In particular, for any B_{1}, B_{2} ∈ Sub(A/K) and C_{1}, C_{2} ∈ Quot(A/K) we have functorial isomorphisms

Hom^{0}(B_{1}, B_{2}) →^{∼} Hom_{𝔼}(I(B_{1}), I(B_{2})), Hom^{0}(C_{1}, C_{2}) →^{∼} Hom_{𝔼}(I′(C_{2}), I′(C_{2})).

Proof. Put I_{A/K} := I_{A} ∘ i_{A,K} and I′_{A/K} = _{A}I ∘ _{A}i_{K}, so by definition I_{A/K}(B) = j_{B}Hom^{0}(A, B) and I′_{A/K}(C, p) = Hom^{0}(A, C)p, for B ∈ Sub(A/K) and (C, p) ∈ Quot(A/K). Note that I_{A/K}(B) = {f ∈ 𝔼 : Imf ⊂ B = Im(j_{B})} by the universal property of an image (viewed as a kernel). From Proposition 3.1 and Theorem 2.7 (with A = \underline{Ab}_{K}^{0}) we see that I_{A/K} and I′_{A/K} are order-preserving bijections and that h^{A} is an equivalence of categories. (Note that since \underline{Ab}_{K}^{0} is semi-simple, we have \underline{Sub}_{A/K}^{0} = \underline{Quot}_{A/K}^{0} = \underline{Ret}_{A/K}^{0} by Corollary 2.9.) Moreover, by Remark 2.10(e) together with formula (9) we see that the inverses are given by the indicated formulae. The last assertion is an immediate consequence of Corollary 2.5.

Remark 3.4 (a) The above results give a new proof of Theorem A of [KR], which may stated as follows: if B_{1}, …, B_{r} ∈ Sub(A/K) and s < r, then

B_{1} × … × B_{s} ∼ B_{s+1} × … × B_{r} ⇔ I(B_{1}) ⊕ … I(B_{s}) ≃ I(B_{s+1}) ⊕ … ⊕ I(B_{r}). (17)

Indeed, via Proposition 3.1, this is just a restatement of (7). [To see that (17) is equivalent to Theorem A, note that B_{i} = Im(ε_{i}) = ε_{i}(A), for some idempotents ε_{i} ∈ 𝔼 (cf. (3)), and then I(B_{i}) = ε_{j}𝔼. Since ε_{1}𝔼 ⊕ … ⊕ ε_{s}𝔼 ≃ ε_{s+1}𝔼 ⊕ … ⊕ ε_{r}𝔼 ⇔ ε_{1} + … + ε_{s} ∼ ε_{s+1} + … + ε_{r}, the equivalence of (17) and Theorem A is clear.]

(b) Recall from Remark 3.2(b) that each f ∈ End(A) has a factorization f = jhp, where h : C → B is an isogeny, (B, j) ∈ Sub(A/K), and (C, p) ∈ Quot(A/K). It then follows from (8) that I_{A/K}(B) = f𝔼 and I′_{A/K}(C, p) = 𝔼f.

(c) Note that it follows from Proposition 3.1 that each closed immersion j_{B} : B → A is split in \underline{Ab}_{K}^{0}, i.e. that ∃h : A → B such that hj_{B} is an isogeny. Such a map can be chosen canonically if we fix a polarization λ = λ_{L} : A → Â. Indeed, let λ_{B} := ĵ_{B}λh = λ_{j_{B}^{∗}L} : B → B̂ denote the induced polarization on B. Then there exists a unique λ′_{B} such that λ′_{B}λ_{B} = [n]_{B}, for some (minimal) n = n_{B} > 0, and N_{B,λ} := λ′_{B}(j_{B})^{}ˆ λ : A → B satisfies N_{B,λ}j_{B} = [n_{B}]. Thus ε_{B,λ} := \frac{1}{n_{B}} N_{B,λ}j_{B} is a canonical generator of I(B) = ε_{B,λ}𝔼.

Using Remark 3.4(c), we see that the above theorem implies the following result of Lange[Lan] (cf. [LB], p. 126, for the case K = ℂ).

Corollary 3.5 (Lange) Fix a polarization λ : A → Â. Then the map B ↦ ε_{B,λ} defines a bijection

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

between the set of abelian subvarieties of A/K and the set SymId(𝔼) of idempotents ε ∈ 𝔼 = End^{0}(A) which are symmetric with respect to the Rosati involution a ↦ a^{∗} = λ^{−1} ∘ â ∘ λ on 𝔼.

Proof. We first show that ε_{B,λ} is symmetric. Indeed, since λ and λ_{B} = ĵ_{B}λh are

polarizations, we have that λ̂κ_{A} = λ and λ̂_{B}κ_{B} = λ_{B}, where κ_{A} : A →^{∼} ˆ Â and κ_{B} : B →^{∼} ˆ B̂ are the canonical identifications. Thus (λ′_{B})^{}ˆ = κ_{B}λ′_{B} and p̂_{B}κ_{B} = λ̂ˆĵ_{B}κ_{B} = λ̂κ_{A}j_{B} = λj_{B}. From this we obtain that e_{B}ε̂_{B}λ = N̂_{B}ĵ_{B}λ = N̂_{B}p_{B} = p̂_{B}(λ′_{B})^{}ˆ p_{B} = p̂_{B}κ_{B}λ′_{B}p_{B} = λj_{B}N_{B} = e_{B}λε_{B}, i.e. that ε_{B}^{∗} = ε_{B}, as claimed.

In view of Theorem 1.1 (or Theorem 3.3) it is therefore enough to show:

Each right 𝔼-ideal 𝔞 is generated by a unique symmetric idempotent.

To prove this, note first that it is enough to verify the uniqueness assertion because by Theorem 1.1 and Remark 3.4(c) we know that 𝔞 = ε_{B,λ}𝔼 for some B ∈ Sub(A/K). To prove uniqueness, let ε_{1}, ε_{2} ∈ SymId(𝔼) be such that ε_{1}𝔼 = ε_{2}𝔼. Then (1 − ε_{2})ε_{1}𝔼 = (1 − ε_{2})ε_{2}𝔼 = 0, so ε_{2}ε_{1} = ε_{1}, and similarly ε_{1}ε_{2} = ε_{2}. Since ε_{1} and ε_{2} are symmetric, we have from the second equation that ε_{2}ε_{1} = ε_{2}^{∗}ε_{1}^{∗} = (ε_{1}ε_{2})^{∗} = ε_{2}^{∗} = ε_{2}, and hence ε_{1} = ε_{2}ε_{1} = ε_{2}.

Corollary 3.6 Let (C, p) be an abelian quotient of A and let 𝔞 = I′_{A/K}(C, p) be its associated left 𝔼-ideal. Let B ⊂ A be the subvariety corresponding to the right 𝔼-ideal 𝔞^{∗} which is the image of 𝔞 with respect to the Rosati involution ^{∗} of A (associated to a polarization λ : A → Â). Then p_{|B} : B → C is an isogeny.

Proof. By Corollary 3.5 we have 𝔞^{∗} = ε_{B,λ}𝔼, so 𝔞 = 𝔞^{∗∗} = 𝔼ε_{B,λ}^{∗} = 𝔼ε_{B,λ} since ε_{B,λ} is symmetric. With the notation of Remark 3.4(c), write N_{B,λ} = hp′ where (C′, p′) ∈ Quot(A/K) and h : C′ → B is an isogeny. By Remark 3.4(b) we have I′(C′, p′) = 𝔼ε_{B,λ} = I′(C, p), and hence by Theorem 3.3 there exists an isomorphism f : C →^{∼} C′ such that p′ = fp. Thus hfpj_{B} = hp′j_{B} = N_{B,λ}j_{B} = [n_{B}] is an isogeny, and hence so is p_{|B} = pj_{B}.

4 Algebraic Subspaces of 𝔼-Modules

As before, let A/K be an abelian variety and 𝔼 = End^{0}(A). If V is any faithful left 𝔼-module, then the results of the previous sections show that we have a canonical bijection between the set Sub(A) of abelian subvarieties of A/K and the set Alg_{𝔼}(V) = {aV : a ∈ 𝔼} of algebraic subspaces of V. More precisely:

Theorem 4.1 If V is a faithful left 𝔼-module, then the map B ↦ I_{A/K}(B)V induces an order-preserving bijection

S_{A/K,V} : Sub(A/K) →^{∼} Alg_{𝔼}(V).

Moreover, if V is also finitely generated, then Alg_{𝔼}(V) = End𝔼̃(V), where 𝔼̃ = End_{𝔼}(V), and hence for each left 𝔼̃-submodule W ⊂ V there is a unique abelian subvariety B_{W} ⊂ A such that S_{A/K,V} (B_{W}) = W, and we have a ring isomorphism

θ_{W} : End𝔼̃(W) →^{∼} End^{0}(B_{W})

such that j_{B}θ_{W} ((λ_{f})_{|W}) = f_{|B_{W}}, for all f ∈ 𝔼 with f(W) ⊂ W.

Proof. This follows immediately from Theorem 2.12 and Proposition 3.1 by putting S_{A/K,V} = S_{A,V} ∘ i_{A,K}.

Most of the interesting examples of faithful 𝔼-modules arise from faithful (covariant) functors in the following way.

Corollary 4.2 If F : \underline{Ab}_{K} → \underline{Vec}_{k} is a faithful functor, where \underline{Vec}_{k} is the category of (finite dimensional) vector spaces over some field k, then for every abelian variety A/K the map B ↦ F (B) induces an order preserving bijection

S_{A/K,F} : Sub(A/K) →^{∼} Alg(F (A)) = {Im(F (a)) : a ∈ End(A)}.

Moreover, if k is a finite extension of ℚ, then Alg(F (A)) = End𝔼̃(F (A)), where 𝔼̃ = End_{𝔼}(F (A)).

Proof. Since F is faithful, it follows that char(k) = 0 and so F extends uniquely to a (faithful) functor F : \underline{Ab}_{K}^{0} → \underline{Vec}_{k}. Thus, the first assertion follows immediately from Corollary 2.14. (Note that if a ∈ 𝔼, then na ∈ End(A) for some n > 0 and then Im(F (a)) = Im(F (na)) and hence Alg(F (A)) = Alg_{F} (F (A)).) Moreover, if k is a finite extension of ℚ, then F (A) is a finitely generated 𝔼-module, and hence Alg(F (A)) = End𝔼̃(F (A)) by Theorem 2.12.

Some examples of functors satisfying the above hypotheses are the following.

Example 4.3 (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. Note that dim_{ℚ} H_{1}(A) = 2 dim(A).

(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.

Proof of Corollary 1.3. By hypothesis, Hom_{𝕋}(T_{0}^{∗}(A), T_{0}(A)) = 𝕋φ because T_{0}(A)^{∗} ≃ T_{0}(A) ≃ 𝕋 as 𝕋-modules. Thus, if φ′ is another isomorphism, then φ′ = tφ with t ∈ 𝕋^{×} and so φ′(W) = φ(tW) = φ(W). This proves the first assertion. Moreover, by Example 4.3(b), the tangent space functor T_{0} satisfies the hypotheses of Corollaries 4.2 and Corollary 2.13, and so the second assertion follows. Finally, we note that dim_{ℚ} W = dim_{ℚ} T_{0}(B_{W}) = dim B_{W}, as asserted.

For contravariant functors, we have the following dual version of Corollary 4.2.

Corollary 4.4 If G : \underline{Ab}_{K}^{op} → \underline{Vec}_{k} is a faithful, contravariant functor, then for each abelian variety A/K the map (C, p) ↦ G(A)I′(C, p) = Im(G(p)) ⊂ G(A) defines an order preserving bijection

Q_{G} = Q_{A/K,G} : Quot(A/K) →^{∼} Alg(G(A)) = {Im(G(a)) : a ∈ End(A)}.

Moreover, if k is a finite extension of ℚ, then Alg(F (A)) = End𝔼̃(F (A)), where 𝔼̃ = End_{𝔼}(G(A)) = {f ∈ End(G(A)) : fG(a) = G(a)f, ∀a ∈ 𝔼}. Thus, for each (left) 𝔼̃-module W ⊂ G(A), there is a unique (C_{W} , p_{W}) ∈ Quot(A/K) such that Im(G(p_{W})) = W, and we have Ker(p_{W}) = r_{𝔼}(W)A.

Proof. Using the same reasoning as in the proof of Corollary 4.2, we see that these assertions follow immediately from Corollary 2.15.

The following are examples of such functors.

Example 4.5 (a) (Duals of covariant functors) Let D_{k} : \underline{Vec}_{k} →^{∼} \underline{Vec}_{k}^{op} denote the (contravariant) duality functor defined by D_{k}(V) = V^{∗} = Hom(V, k). Clearly, if F : \underline{Ab}_{K} → \underline{Vec}_{k} is any faithful covariant functor, then its “dual” F^{∗} = D_{k} ∘ F : \underline{Ab}_{K} → \underline{Vec}_{k}^{op} is a faithful contravariant functor. In particular, the duals of the functors H_{1}, T_{0} and T_{ℓ}^{0} considered in Example 4.3 are faithful and 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}^{op} → \underline{Vec}_{K} denote the functor of holomorphic differentials defined by Ω(A) = H^{0}(A, Ω_{A/K}^{1}). Since the (restriction) map ω ↦ ω_{0} ∈ T_{0}^{∗}(A) defines an isomorphism of functors Ω ≃ T_{0}^{∗}, this functor is again faithful.

Proof of Theorem 1.4. The first assertion follows from Corollary 4.4 because the functor Ω satisfies the required hypotheses by Example 4.5(b). The second assertion follows from Corollary 2.14. Finally, since p_{W}^{∗} : Ω(C_{W}) → Ω(A) is injective (cf. Remark 4.6), we have dim C_{W} = dim_{K} Ω(C_{W}) = dim_{K} p_{W}^{∗} Ω(C_{W}) = dim W, as claimed.

Remark 4.6 Let G : \underline{Ab}_{K}^{op} → \underline{Vec}_{k} be as in Corollary 4.4, and let f : A → A′ be a homomorphism of abelian varieties. If f is surjective, then f is a split epi in \underline{Ab}_{K}^{0} (cf. Proposition 3.1) and hence G(f) : G(A′) → G(A) is injective because G maps split epis in \underline{Ab}_{K}^{0} to monics in \underline{Vec}_{k}. (Similarly, if f has finite kernel, then G(f) is surjective, and if f is an isogeny, then G(f) is bijective.) Thus, G(f) induces an injection G(f) : Sub_{k}(G(A′)) ↪ Sub_{k}(G(A)) which maps algebraic subspaces to algebraic subspaces because we have

(18) G(f)(Q_{G}(C′, p′)) = Q_{G}((C′, p′f)^{0}), for all (C′, p′) ∈ Quot(A′/K).

Indeed, if we write (C, p) := (C′, p′f)^{0} ∈ Quot(A/K) (cf. Remark 3.2(a)), then there is an isogeny g : C → C′ such that gp = p′f, and we have G(f)(Q_{G}(C′, p′)) = G(f)(Im(G(p′))) = Im(G(p′f)) = Im(G(p)G(g)) = Im(G(p)) = Q_{G}((C′, p′f)^{0}) because G(g) is bijective.

The following corollary is frequently useful in applications.

Corollary 4.7 If (C, p), (C_{1}, p_{1}), …, (C_{n}, p_{n}) ∈ Quot(A/K) are abelian quotients such that Q_{G}(C, p) = ∑ Q_{G}(C_{i}, p_{i}), where G is as in Corollary 4.4, then there is a unique homomorphism ν : C → C_{1}×…×C_{n} with finite kernel such that (p_{1}, …, p_{n}) = ν ∘ p. Furthermore, ν is an isogeny if and only if the sum of the Q_{G}(C_{i}, p_{i})’s is a direct sum.

Proof. Since Q_{G} is a lattice isomorphism by Corollary 4.4, it follows from the hypothesis that (C, p) is the maximum of the (C_{i}, p_{i})’s, and so the first assertion follows from Remark 2.10(d). Moreover, since Q_{G}(C_{i}, p_{i}) = Q_{G}(A)I′(C_{i}, p_{i}) and since G(A) is a faithfully flat 𝔼-module (cf. proof of Theorem 2.12), we see that the sum of the Q_{G}(C_{i}, p_{i})’s is a direct sum if and only if the I′(C_{i}, p_{i})’s are a direct sum, and so the assertion follows immediately from (the dual version of) (12).

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 [Sh1], Theorem 3.51 or by Atkin-Lehner Theory (cf. [DDT], Lemma 1.35), and so there is a 𝕋-module isomorphism φ : Ω(J) →^{∼} T_{0}(J). We then have 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 quotient (A_{W} , p_{W}) ∈ Quot(J_{Γ}) such that p_{W}^{∗} Ω(A_{W}) = W. Furthermore, dim A_{W} = dim_{ℚ} W and we have an injective ring homomorphism θ_{W} : End_{𝕋}(W) ↪ End_{ℚ}^{0}(A_{W}) such that θ_{W} (t_{|W}) ∘ p_{W} = p_{W} ∘ t, for all t ∈ 𝕋.

(b) Fix a 𝕋-module isomorphism φ : S_{2}(Γ, ℚ) →^{∼} T_{0}(J). Then φ(W) does not depend on the choice of φ, and there exists a unique abelian subvariety A′_{W} ∈ Sub(J_{Γ}) such that φ_{Γ}(W) = T_{0}(A′_{W}) ⊂ T_{0}(J). Furthermore, dim A′_{W} = dim_{ℚ} W and there exists a ring injection θ′_{W} : End_{𝕋}(W) ↪ End^{0}(A′_{W}) such that θ′_{W} (t_{|W}) = t_{|A′_{W}}, for all t ∈ 𝕋.

(c) The restriction p′_{W} = (p_{W})_{|A′_{W}} : A′_{W} → A_{W} of p_{W} to A′_{W} is an isogeny provided that (W ⊗ ℂ)|w_{n} = W ⊗ ℂ, where w_{N} = \binom{0 −1}{N 0}.

Proof. (a) By the above discussion, this follows immediately from Theorem 1.4.

(b) Via the identification S_{2}(Γ, ℚ) = T_{0}(J)^{∗}, all assertions except the last follow from Corollary 1.3. The last assertion follows from Corollary 2.14.

(c) Since W is an 𝔼̃-module, we have by Corollary 4.4 that W = Ω(J)𝔞 for some left ideal 𝔞 of 𝔼. Since w_{N} induces an automorphism of Ω(J ⊗ ℂ), the hypothesis implies that Ω(J ⊗ ℂ)w_{N}^{−1}𝔞w_{N} = Ω(J ⊗ ℂ)𝔞. Thus, since w_{N}^{−1}𝔞w_{N} is a left ideal of 𝔼 (cf. Lemma 5.2 below), it follows that w_{N}^{−1}𝔞w_{N} = 𝔞. Thus, using the map φ of Lemma 5.2, we have by (19) that φ(W) = φ(Ω(J)w_{N}𝔞w_{N}^{−1}) = 𝔞^{∗}T_{0}(J) (because w_{N}^{∗} = w_{N}^{−1}). This shows that φ(W) = 𝔞^{∗}T_{0}(J), and so the assertion follows from Corollary 3.6.

Lemma 5.2 The map α ↦ w_{N}^{−1}αw_{N} defines an automorphism of 𝔼 = End_{ℚ}^{0}(J). Thus, if α ↦ α^{∗} denotes the Rosati involution on 𝔼 induced by the canonical polarization of J, then there exists a 𝕋-module isomorphism φ : S_{2}(Γ, ℚ) →^{∼} T_{0}(J) such that φ(fα) = (w_{N}^{−1}α^{∗}w_{n})φ(f), for all f ∈ S_{2}(Γ, ℚ), α ∈ 𝔼. (19)

Proof. We first verify that w_{N}^{−1}αw_{n} ∈ 𝔼, if α ∈ 𝔼. For this we observe that the automorphism w_{N} is defined over K := ℚ(ζ_{N}) and satisfies w_{N}^{τ_{a}} = w_{N}σ_{a}^{−1}, where τ_{a} ∈ Gal(K/ℚ) is given by ζ_{N}^{τ_{a}} = ζ_{N}^{a} and σ_{a} ≡ \binom{a^{−1} 0}{0 a} (mod N). Since σ_{a} ∈ 𝕋′ is in the centre of 𝔼 (cf. [Ka], Corollary 2), it follows that w_{N}^{−1}αw_{n} ∈ End_{K}^{0}(J) is τ_{a}-invariant and hence lies in 𝔼.

It thus follows that T_{0}(J) becomes a right 𝔼-module via the rule vα = (w_{N}^{−1}α^{∗}w_{N})v (for v ∈ T_{0}(J)). Moreover, T_{0}(J) is a faithful right 𝔼-module under this twisted action because T_{0}(J) is a faithful left 𝔼-module under the usual action. Thus, since 𝔼 is semisimple, it follows that every irreducible 𝔼-module apears at least once in the decomposition of T_{0}(J) into irreducible submodules. Now since Ω(J) = S_{2}(Γ, ℚ) is also faithful right 𝔼-module, the same is true for Ω(J). Moreover, since End_{𝔼}(Ω(J)) ⊂ End_{𝕋}(Ω(J)) = 𝕋 is commutative, it follows that every irreducible 𝔼-module appears only once in the 𝔼-module decomposition of Ω(J), and so we see that we have an 𝔼module embedding φ : Ω(J) → T_{0}(J), which is an isomorphism because dim Ω(J) = dim T_{0}(J). Moreover, since w_{N}^{−1}tw_{N} = t^{∗}, for all t ∈ 𝕋, it follows that twisted action of 𝕋 on T_{0}(J) is the same as the usual action of 𝕋 on T_{0}(J). In particular, φ is a 𝕋-module isomorphism satisfying (19).

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

Example 5.3 (Shimura) Let f ∈ S_{2}(Γ) be a (normalized) 𝕋-eigenfunction, and put W_{f,ℂ} = ∑ ℂf^{σ}, where the sum is over all Aut(ℂ)-conjugates of f. Clearly, W_{f,ℂ} is Aut(ℂ)-invariant and hence is of the form W_{f,ℂ} = W_{f} ⊗ ℂ for a unique subspace W_{f} ⊂ S_{2}(Γ, ℚ). Moreover, dim_{ℚ} W_{f} = dim_{ℂ} W_{f,ℂ} = [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 A′_{f} = A′_{W_{f}} ≤ J = J_{Γ} and an abelian quotient p = p_{f} : J → A_{f} := A_{W_{f}} together with maps θ′_{f} : K_{f} ↪ End_{ℚ}^{0}(A′_{f}) and θ_{f} : K_{f} ↪ End_{ℚ}^{0}(A_{f}) such that (A′_{f}, θ′_{f}) and (A_{f}, p_{f}, θ_{f}) satisfy the following conditions (which are in fact identical to those of Theorems 1 and 2 of [Sh2]):

(i) A′_{f} ∈ Sub(J/ℚ) and (A_{f}, p_{f}) ∈ Quot(J/ℚ).

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

(iii) dim A′_{f} = dim A_{f} = [K_{f} : ℚ].

(iv) T_{0}(A′_{f}) = φ_{Γ}(W_{f}) and p_{f}^{∗}Ω(A_{f}) = W_{f}.

There are other natural 𝕋-submodules to which the (generalized) Shimura construction can be applied, as the following example shows.

Example 5.4 Let f ∈ S_{2}(Γ) be a newform of some level N_{f}|N and let S_{f} = ∑_{d|(N/N_{f})} ℂf|β_{d} denote the Atkin-Lehner eigenspace of f, where β_{d} = \binom{d 0}{0 1}. Moreover, let S_{[f]} = (∑_{σ} S_{f^{σ}}) ∩ S_{2}(Γ, ℚ), where σ ∈ G_{ℚ} runs over all elements of the absolute Galois group G_{ℚ} = Gal(\overline{ℚ}/ℚ). Then by Atkin-Lehner theory (cf. [DDT], p.

36) each S_{[f]} is a 𝕋-module of dimension σ_{0}(N/N_{f})[K_{f} : ℚ] and we have the following 𝕋-module decomposition in which the sum extends over the set N (Γ) of all newforms of all levels modulo the action of G_{ℚ}:

S_{2}(Γ, ℚ) = ⨁_{f∈N (Γ)/G_{ℚ}} S_{[f]}

Thus, by Theorem 5.1 there exists a unique abelian quotient (Ã_{[f]}, p_{[f]}) (respectively, abelian subvariety Ã′_{[f]}) of J_{Γ} such that p_{[f]}^{∗}Ω(Ã_{[f]}) = S_{[f]} (respectively, such that φ_{Γ}(S_{[f]}) = T_{0}(Ã′_{[f]}) ⊂ T_{0}(J)), and dim Ã_{[f]} = Ã′_{[f]} = dim_{ℚ} S_{[f]}. Moreover, by Corollary 4.7 it follows from the above decomposition that we have isogenies

J_{Γ} ∼ ∏ Ã_{[f]} ∼ ∏ Ã′_{[f]}. (20)

Since in general 𝕋 ≠ 𝔼̃ (because 𝔼̃ is semi-simple whereas in general 𝕋 is not), we cannot expect that every abelian subvariety and/or quotient can be obtained by the Shimura construction (Theorem 5.1). For example, the following varieties A_{[f],d} cannot be obtained from the Shimura construction.

Example 5.5 As before, let f ∈ N (Γ) be a newform of level N_{f}|N and assume for simplicity that Γ = Γ_{1}(N). For a divisor d|\frac{N}{N_{f}} put S_{f,d} = ℂf|β_{d} and S_{[f],d} = ∑ σ(S_{f^{σ},d}) ∩ S_{2}(Γ, ℚ); thus S_{f} = ⊕_{d}S_{f,d} and S_{[f]} = ⊕_{d}S_{[f],d}. In general, S_{[f],d} is not a 𝕋-module because S_{[f],d}|T_{p} ⊂ S_{[f],d/p}, if p|d. However, S_{[f],d} is an algebraic subspace of S_{2}(Γ, ℚ) = Ω(J_{Γ}) (as we shall see below) and hence corresponds to an abelian quotient p_{[f],d} : J_{Γ} → A_{[f],d}.

To see that S_{[f],d} is algebraic, we shall use the “degeneracy map” π_{N_{f} ,d} : X_{1}(N) → X_{1}(N_{f}) which is induced by the map g ↦ g|β_{d} on the function fields. Thus, if π̃_{N_{f} ,d} = (π_{N_{f} ,d})_{∗} : J_{1}(N) → J_{1}(N_{f}) denotes the Albanese map induced by autoduality of the Jacobian, then we have π̃_{N_{f} ,d}^{∗}W_{f} = S_{[f],d}. Since W_{f} is algebraic by Example 5.3 (because f is a 𝕋-eigenform in S_{2}(Γ_{1}(N_{f}))) with corresponding quotient (A_{f}, p_{f}) ∈ Quot(J_{1}(N_{f})), we see from Remark 4.6 that S_{[f],d} is also algebraic with corresponding abelian quotient (A_{[f],d}, p_{[f],d}) := (A_{f}, p_{f} π̃_{N_{f} ,d})^{0}. In particular, there is an isogeny ν_{[f],d} : A_{[f],d} → A_{f} such that p_{f} ∘ π̃_{N_{f} ,d} = ν_{[f],d} ∘ p_{[f],d}. Thus, since S_{[f]} = ⊕_{d}S_{[f],d}, it follows from Corollary 4.7 that

Ã_{[f]} ∼ A_{f}^{n_{f}} , where n_{f} = σ_{0}(N/N_{f}). (21)

Combining this with the relation (20), we thus obtain the isogeny relation

J_{1}(N) ∼ ∏_{f∈N (Γ_{1}(N))/G_{ℚ}} A_{f}^{n_{f}} . (22)

Remark 5.6 (a) It follows from the work of Ribet that the above isogeny relation (22) is the isogeny decomposition of J_{1}(N), for by Ribet we have that each A_{f} is a ℚ-simple abelian variety and that A_{f} ∼ A′_{f} ⇔ [f] = [f′].

(b) In [Ka] it is shown that in fact 𝔼̃ := End_{𝔼}(Ω(J_{Γ})) = 𝕋′, where 𝕋′ = ℚ[{T_{n} : (n, N) = 1}] ⊂ 𝕋. Thus, by the dictionary of Corollary 4.4 we have that the map (C, p) ↦ p^{∗}Ω(C) ⊂ Ω(J_{Γ}) = S_{2}(Γ, ℚ) induces a bijection

Quot(J_{Γ}) →^{∼} Sub_{𝕋′}(S_{2}(Γ, ℚ))

between the set of abelian quotients of J_{Γ}/ℚ and the set of 𝕋′-submodules of S_{2}(Γ, ℚ). Furthermore, if (C_{i}, p_{i}) i = 1, 2 are two such quotients, then we have a canonical isomorphism

Hom_{ℚ}^{0}(C_{1}, C_{2}) ≃ Hom_{𝕋′}(p^{∗}Ω(C_{2}), p^{∗}Ω(C_{1})).

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