On the Image of l-Adic Galois Representations

for Abelian Varieties of Type I and II

Dedicated to John Coates on the

occasion of his 60-th birthday

G. Banaszak, W. Gajda, P. Krasoń

Received: August 9, 2005 Revised: August 10, 2006

Abstract. In this paper we investigate the image of the l-adic representation attached to the Tate module of an abelian variety over a number field with endomorphism algebra of type I or II in the Albert classification. We compute the image explicitly and verify the classical conjectures of Mumford-Tate, Hodge, Lang and Tate for a large family of abelian varieties of type I and II. In addition, for this family, we prove an analogue of the open image theorem of Serre.

2000 Mathematics Subject Classification: 11F80, 11G10

Keywords and Phrases: abelian varieties, l-adic representations

1. Introduction. Let A be an abelian variety defined over a number field F. Let l be an odd prime number. In this paper we study the images of the l-adic representation ρ_{l} : G_{F} ⟶ GL(T_{l}(A)) and the mod l representation \overline{ρ}_{l} : G_{F} ⟶ GL(A[l]) of the absolute Galois group G_{F} = G(F̄/F) of the field F, associated with the Tate module, for A of type I or II in the Albert classification list cf. [M]. In our previous paper on the subject cf. [BGK], we computed the images of the Galois representations for some abelian varieties with real (type I) and complex multiplications (type IV) by the field E=End_{F} (A) ⊗ ℚ and for l which splits completely in the field E loc. cit., Theorem 2.1 and Theorem 5.3.

In the present paper we extend results proven in [BGK] to a larger class (cf. Definition of class A below) of abelian varieties which includes some varieties with non-commutative algebras of endomorphisms, and to almost all prime numbers l. In order to get these results, we had to implement the Weil restriction functor R_{L/K} for a finite extension of fields L/K. In section 2 of the paper we give an explicit description of the Weil restriction functor for affine group schemes which we use in the following sections. In a very short section 3 we prove two general lemmas about bilinear forms which we apply to Weil pairing in the following section. Further in section 4, we collect some auxiliary facts about abelian varieties. In section 5 we obtain the integral versions of the results of Chi cf. [C2], for abelian varieties of type II and compute Lie algebras and endomorphism algebras corresponding to the λ-adic representations related to the Tate module of A. In section 6 we prove the main results of the paper which concern images of Galois representations ρ_{l}, ρ_{l} ⊗ ℚ_{l} : G_{F} → GL(V_{l}(A)), the mod l-representation \overline{ρ}_{l} and the associated group schemes G_{l}^{alg}, G_{l}^{alg} and G(l)^{alg}, respectively.

The main results proven in this paper concern the following class of abelian varieties:

Definition of class A. We say that an abelian variety A/F, defined over a number field F is of class A, if the following conditions hold:

(i) A is a simple, principally polarized abelian variety of dimension g (ii) R = EndF̄(A) = End_{F} (A) and the endomorphism algebra D = R⊗_{ℤ}ℚ,

is of type I or II in the Albert list of division algebras with involution (cf. [M], p. 201).

(iii) the field F is such that for every l the Zariski closure G_{l}^{alg} of ρ_{l}(G_{F}) in

GL_{2g}/ℚ_{l} is a connected algebraic group (iv) g = hed, where h is an odd integer, e = [E : ℚ] is the degree of the

center E of D and d^{2} = [D : E].

Let us recall the definition of abelian varieties of type I and II in the Albert’s classification list of division algebras with involution [M], p. 201. Let E ⊂ D = EndF̄(A) ⊗_{Z} ℚ be the center of D and E be a totally real extension of ℚ of degree e. Abelian varieties of type I are such that D = E. Abelian varieties of type II are those for which D is an indefinite quaternion algebra with the center E, such that D ⊗_{ℚ} ℝ ≅ ∏_{i=1}^{e} M_{2,2}(ℝ).

We have chosen to work with principal polarizations, however the main results of this paper have their analogs for any simple abelian variety A with a fixed polarization, provided A satisfies the above conditions (ii), (iii) and (iv). The most restrictive of the conditions in the definition of class A is condition (iv) on the dimension of the variety A. We need this condition to perform computations with Lie algebras in the proof of Lemma 5.33, which are based on an application of the minuscule conjecture cf. [P]. Note that due to results of Serre, the assumption (iii) is not very restrictive. It follows by [Se1] and [Se4] that for an abelian variety A defined over a number field K, there exists a finite extension K^{conn}/K for which the Zariski closure of the group ρ_{l}(G_{K^{conn}}) in GL is a connected variety for any prime l. Hence, to make A meet the condition (iii), it is enough to enlarge the base field, if necessary. Note that the field K^{conn} can be determined in purely algebraic terms, as the intersection of a family of fields of division points on the abelian variety A cf. [LP2], Theorem 0.1.

Main results

Theorem A. [Theorem 6.9] If A is an abelian variety of class A, then for l ≫ 0, we have equalities of group schemes:

(G_{l}^{alg})′ = ∏_{λ|l} R_{E_{λ}/ℚ_{l}}(Sp_{2h})

(G(l)^{alg})′ = ∏_{λ|l} R_{k_{λ}/𝔽_{l}}(Sp_{2h}),

where G′ stands for the commutator subgroup of an algebraic group G, and R_{L/K}(−) denotes the Weil restriction functor.

Theorem B. [Theorem 6.16] If A is an abelian variety of class A, then for l ≫ 0, we have:

\overline{ρ_{l}}(G′_{F}) = ∏_{λ|l} Sp_{2h}(k_{λ}) = Sp_{2h}(O_{E}/lO_{E})

ρ_{l}( \overline{G′_{F}} ) = ∏_{λ|l} Sp_{2h}(O_{λ}) = Sp_{2h}(O_{E} ⊗_{ℤ} ℤ_{l}),

where \overline{G′_{F}} is the closure of G′_{F} in the profinite topology in G_{F} .

As an application of Theorem A we obtain:

Theorem C. [Theorem 7.12] If A is an abelian variety of class A, then

G_{l}^{alg} = MT (A) ⊗ ℚ_{l},

for every prime number l, where MT (A) denotes the Mumford-Tate group of A, i.e., the Mumford -Tate conjecture holds true for A.

Using the approach initiated by Tankeev [Ta5] and Ribet [R2], futher developed by V.K. Murty [Mu] combined with some extra work on the Hodge groups in section 7, we obtain:

Theorem D. [Theorems 7.34, 7.35] If A is an abelian variety of class A, then the Hodge conjecture and the Tate conjecture on the algebraic cycle maps hold true for the abelian variety A.

In the past there has been an extensive work on the Mumford-Tate, Tate and Hodge conjectures for abelian varieties. Special cases of the conjectures were verified for some classes of abelian varieties, see for example papers: [Ab], [C2], [Mu], [P], [Po], [R2], [Se1], [Se5], [Ta1], [Ta2], [Ta3]. For an abelian variety A of type I or II the above mentioned papers consider the cases where A is such that End(A) ⊗ ℚ is either ℚ or has center ℚ. The papers [Ta4], [C1] and [BGK] considered some cases with the center larger than ℚ. For more complete list of results concerning the Hodge conjecture see [G]. In the current work we prove the conjectures in the case when the center of End(A)⊗ℚ is an arbitrary totally real extension of ℚ. To prove the conjectures for such abelian varieties we needed to do careful computations using the Weil restriction functor.

Moreover, using a result of Wintenberger (cf. [Wi], Cor. 1, p.5), we were able to verify that for A of class A, the group ρ_{l}(G_{F}) contains the group of all the homotheties in GL_{T_{l}(A)}(ℤ_{l}) for l ≫ 0, i.e., the Lang conjecture holds true for A cf. Theorem 7.38.

As a final application of the method developed in this paper, we prove an analogue of the open image theorem of Serre cf. [Se1] for the class of abelian varieties we work with.

Theorem E. [Theorem 7.42] If A is an abelian variety of class A, then for every prime number l, the image ρ_{l}(G_{F}) is open in the group C_{R}(GSp_{(Λ, ψ)})(ℤ_{l}) of ℤ_{l}-points of the commutant of R=End A in the group GSp_{(Λ, ψ)} of symplectic similitudes of the bilinear form ψ : Λ × Λ ⟶ ℤ associated with the polarization of A. In addition, for l ≫ 0 we have:

ρ_{l}(\overline{G′_{F}}) = C_{R}(Sp_{(Λ, ψ)})(ℤ_{l}).

As an immediate corollary of Theorem E we obtain that for any A of class A and for every l, the group ρ_{l}(G_{F}) is open in G_{l}^{alg}(ℤ_{l}) (in the l-adic topology), where G_{l}^{alg} is the Zariski closure of ρ_{l}(G_{F}) in GL_{2g}/ℤ_{l}. cf. Theorem 7.48. Recently, the images of Galois representations coming from abelian varieties have also been considered by A.Vasiu (cf. [Va1],[Va2]).

2. Weil restriction functor R_{E/K} for affine schemes and Lie algebras. In this section we describe the Weil restriction functor and its basic properties which will be used in the paper c.f. [BLR], [V1], [V2, pp. 37-40], [W1] and [W2, pp. 4-9]. For the completeness of the exposition and convenience of the reader we decided to include the results although some of them might be known to specialists. Let E/K be a separable field extension of degree n. Let \{σ_{1}, σ_{2}, …, σ_{n}\} denote the set of all imbeddings E → E^{σ_{i}} ⊂ \overline{K} fixing K. Define M to be the composite of the fields E^{σ_{i}}

M = E^{σ_{1}} … E^{σ_{n}}.

Let X = [x_{1}, x_{2}, … x_{r}] denote a multivariable. For polynomials f_{k} = f_{k}(X) ∈ E[X], 1 ≤ k ≤ s, we denote by I = (f_{1}, f_{2}, …, f_{s}) the ideal generated by the f_{k}’s and put I^{σ_{i}} = (f_{1}^{σ_{i}}(X), f_{2}^{σ_{i}}(X), …, f_{s}^{σ_{i}}(X)) for any 1 ≤ i ≤ n. Let A = E[X]/I. Define E-algebras A^{σ_{i}} and \overline{A} as follows:

A^{σ_{i}} = A ⊗_{E,σ_{i}} M ≅ M[X]/ I^{σ_{i}}M[X],
\overline{A} = A^{σ_{1}} ⊗_{M} ⋯ ⊗_{M} A^{σ_{n}}.

Let X^{σ_{1}}, …, X^{σ_{n}} denote the multivariables

X^{σ_{i}} = [x_{i,1}, x_{i,2}, …, x_{i,r}]

on which the Galois group G = G(M/K) acts naturally on the right. Indeed for any imbedding σ_{i} and any σ ∈ G the composition σ_{i} ∘ σ, applied to E on the right, gives uniquely determined imbedding σ_{j} of E into \overline{K}, for some 1 ≤ j ≤ n. Hence we define the action of G(M/K) on the elements X^{σ_{i}} in the following way:

(X^{σ_{i}})^{σ} = X^{σ_{j}}.

We see that

\overline{A} ≅ M[X^{σ_{1}}, …, X^{σ_{n}}]/ (I_{1} + ⋯ + I_{n}),

where I_{k} = M[X^{σ_{1}}, …, X^{σ_{n}}]I_{(k)} and I_{(k)} = (f_{1}^{σ_{k}}(X^{σ_{k}}), …, f_{s}^{σ_{k}}(X^{σ_{k}})), for any 1 ≤ k ≤ n.

Lemma 2.1.

\overline{A}^{G} ⊗_{K} M ≅ \overline{A}.

Proof. Let α_{1}, …, α_{n} be a basis of E over K. It is clear that

∑_{i=1}^{n} α_{j}^{σ_{i}}X^{σ_{i}} ∈ \overline{A}^{G}.

Since [α_{j}^{σ_{i}}]_{i,j} is an invertible matrix with coefficients in M, we observe that

X^{σ_{1}}, …, X^{σ_{n}} are in the subalgebra of \overline{A} generated by M and \overline{A}^{G}. But X^{σ_{1}}, …, X^{σ_{n}} and M generate \overline{A} as an algebra. □

Remark 2.2. Notice that the elements ∑_{i=1}^{n} α_{j}^{σ_{i}}X^{σ_{i}} for j = 1, …, n generate

\overline{A}^{G} as a K-algebra. Indeed if C denotes the K-subalgebra of \overline{A}^{G} generated by these elements and if C were smaller than \overline{A}^{G}, then C ⊗_{K} M would be smaller than \overline{A}^{G} ⊗_{K} M, contrary to Lemma 2.1.

Definition 2.3. Put V = spec A, and W = spec \overline{A}^{G}. Weil’s restriction functor R_{E/K} is defined by the following formula:

R_{E/K}(V) = W.

Note that we have the following isomorphisms:

W ⊗_{K} M = spec (\overline{A}^{G} ⊗_{K} M) ≅ spec \overline{A} ≅
spec (A^{σ_{1}} ⊗_{M} ⋯ ⊗_{M} A^{σ_{n}}) ≅ (V ⊗_{E,σ_{1}} M) ⊗_{M} ⋯ ⊗_{M} (V ⊗_{E,σ_{n}} M),

hence

R_{E/K}(V) ⊗_{K} M ≅ (V ⊗_{E,σ_{1}} M) ⊗_{M} ⋯ ⊗_{M} (V ⊗_{E,σ_{n}} M).

Lemma 2.4. Let V′ ⊂ V be a closed imbedding of affine schemes over E. Then R_{E/K}(V′) ⊂ R_{E/K}(V) is a closed imbedding of affine schemes over K.

Proof. We can assume that V = spec (E[X]/I) and V′ = spec (E[X]/J) for two ideals I ⊂ J of E[X]. Put A = E[X]/I and B = E[X]/J and let ϕ : A → B be the natural surjective ring homomorphism. The homomorphism ϕ induces the surjective E-algebra homomorphism

\overline{ϕ} : \overline{A} → \overline{B}

which upon taking fix points induces the K-algebra homomorphism

(2.5)  \overline{ϕ}^{G} : \overline{A}^{G} → \overline{B}^{G}.

By Remark 2.2 we see that \overline{B}^{G} is generated as a K-algebra by elements ∑_{i=1}^{n} α_{j}^{σ_{i}}X^{σ_{i}} (more precisely their images in \overline{B}^{G}). Similarly \overline{A}^{G} is generated as

a K-algebra by elements ∑_{i=1}^{n} α_{j}^{σ_{i}}X^{σ_{i}} (more precisely their images in \overline{A}^{G}). It

is clear that \overline{ϕ}^{G} sends the element ∑_{i=1}^{n} α_{j}^{σ_{i}}X^{σ_{i}} ∈ \overline{A}^{G} into ∑_{i=1}^{n} α_{j}^{σ_{i}}X^{σ_{i}} ∈ \overline{B}^{G}.

Hence \overline{ϕ}^{G} is onto. □

Let α_{1}, …, α_{n} be a basis of E over K and let β_{1}, …, β_{n} be the corresponding dual basis with respect to T r_{E/K}. Define block matrices:

𝔸 = ( \begin{matrix}α_{1}^{σ_{1}}I_{r} & α_{1}^{σ_{2}}I_{r} & … & α_{1}^{σ_{n}}I_{r} \\ α_{2}^{σ_{1}}I_{r} & α_{2}^{σ_{2}}I_{r} & … & α_{2}^{σ_{n}}I_{r} \\ ⋮ & ⋮ & … & ⋮ \\ α_{n}^{σ_{1}}I_{r} & α_{n}^{σ_{2}}I_{r} & … & α_{n}^{σ_{n}}I_{r} \end{matrix} ) , 𝔹 = ( \begin{matrix}β_{1}^{σ_{1}}I_{r} & β_{2}^{σ_{1}}I_{r} & … & β_{n}^{σ_{1}}I_{r} \\ β_{1}^{σ_{2}}I_{r} & β_{2}^{σ_{2}}I_{r} & … & β_{n}^{σ_{2}}I_{r} \\ ⋮ & ⋮ & … & ⋮ \\ β_{1}^{σ_{n}}I_{r} & β_{2}^{σ_{n}}I_{r} & … & β_{n}^{σ_{n}}I_{r} \end{matrix} )

Notice that by definition of the dual basis 𝔸𝔹 = 𝔹𝔸 = I_{rn}. Define block diagonal matrices:

𝕏 = ( \begin{matrix}X^{σ_{1}} & 0I_{r} & … & 0I_{r} \\ 0I_{r} & X^{σ_{2}} & … & 0I_{r} \\ ⋮ & ⋮ & … & ⋮ \\ 0I_{r} & 0I_{r} & … & X^{σ_{n}} \end{matrix} ) , 𝕐 = ( \begin{matrix}Y^{σ_{1}} & 0I_{r} & … & 0I_{r} \\ 0I_{r} & Y^{σ_{2}} & … & 0I_{r} \\ ⋮ & ⋮ & … & ⋮ \\ 0I_{r} & 0I_{r} & … & Y^{σ_{n}} \end{matrix} ) ,

where Y^{σ_{1}}, …, Y^{σ_{n}} and X^{σ_{1}}, …, X^{σ_{n}}, are multivariables written now in a form of r × r matrices indexed by σ_{1}, …, σ_{n}. Let T_{ij} and S_{ij}, for all 1 ≤ i ≤ n, 1 ≤ j ≤ n, be r × r multivariable matrices. Define block matrices of multivariables:

𝕋 = ( \begin{matrix}T_{11} & T_{12} & … & T_{1n} \\ T_{21} & T_{22} & … & T_{2n} \\ ⋮ & ⋮ & … & ⋮ \\ T_{n1} & T_{n2} & … & T_{nn} \end{matrix} ) , 𝕊 = ( \begin{matrix}S_{11} & S_{12} & … & S_{1n} \\ S_{21} & S_{22} & … & S_{2n} \\ ⋮ & ⋮ & … & ⋮ \\ S_{n1} & S_{n2} & … & S_{nn} \end{matrix} )

Notice that:

𝔸𝕏𝔹 =( \begin{matrix}∑_{j=1}^{n}(α_{1}β_{1})^{σ_{j}}X^{σ_{j}} & ∑_{j=1}^{n}(α_{1}β_{2})^{σ_{j}}X^{σ_{j}} & … & ∑_{j=1}^{n}(α_{1}β_{n})^{σ_{j}}X^{σ_{j}} \\ ∑_{j=1}^{n}(α_{2}β_{1})^{σ_{j}}X^{σ_{j}} & ∑_{j=1}^{n}(α_{2}β_{2})^{σ_{j}}X^{σ_{j}} & … & ∑_{j=1}^{n}(α_{2}β_{n})^{σ_{j}}X^{σ_{j}} \\ ⋮ & ⋮ & … & ⋮ \\ ∑_{j=1}^{n}(α_{n}β_{1})^{σ_{j}}X^{σ_{j}} & ∑_{j=1}^{n}(α_{n}β_{2})^{σ_{j}}X^{σ_{j}} & … & ∑_{j=1}^{n}(α_{n}β_{n})^{σ_{j}}X^{σ_{j}} \end{matrix} )

𝔸𝕐𝔹 =( \begin{matrix}∑_{j=1}^{n}(α_{1}β_{1})^{σ_{j}}Y^{σ_{j}} & ∑_{j=1}^{n}(α_{1}β_{2})^{σ_{j}}Y^{σ_{j}} & … & ∑_{j=1}^{n}(α_{1}β_{n})^{σ_{j}}Y^{σ_{j}} \\ ∑_{j=1}^{n}(α_{2}β_{1})^{σ_{j}}Y^{σ_{j}} & ∑_{j=1}^{n}(α_{2}β_{2})^{σ_{j}}Y^{σ_{j}} & … & ∑_{j=1}^{n}(α_{2}β_{n})^{σ_{j}}Y^{σ_{j}} \\ ⋮ & ⋮ & … & ⋮ \\ ∑_{j=1}^{n}(α_{n}β_{1})^{σ_{j}}Y^{σ_{j}} & ∑_{j=1}^{n}(α_{n}β_{2})^{σ_{j}}Y^{σ_{j}} & … & ∑_{j=1}^{n}(α_{n}β_{n})^{σ_{j}}Y^{σ_{j}} \end{matrix} ).

Observe that the entries of 𝔸𝕏𝔹 and 𝔸𝕐𝔹 are G-equivariant. Hence, there is a well defined homomorphism of K-algebras

(2.6) Φ : K[𝕋, 𝕊]/(𝕋𝕊−I_{rn}, 𝕊𝕋−I_{rn}) → (M[𝕏, 𝕐]/(𝕏𝕐−I_{rn}, 𝕐𝕏−I_{rn}))^{G}
𝕋 → 𝔸𝕏𝔹
𝕊 → 𝔸𝕐𝔹

The definition of Φ and the form of the entries of matrices 𝔸𝕏𝔹 and 𝔸𝕐𝔹 show (by the same argument as in Lemma 2.4) that the map Φ is surjective. Observe that

GL_{rn}/K = spec K[𝕋, 𝕊]/(𝕋𝕊 − I_{rn}, 𝕊𝕋 − I_{rn}),
GL_{r}/E = spec E[X, Y]/(XY − I_{r}, Y X − I_{r}),

where X and Y are r × r multivariable matrices.

Lemma 2.7. Consider the group scheme GL_{r}/E. The map Φ induces a natural isomorphism R_{E/K}(GL_{r}) ≅ C_{E}(GL_{rn}/K) of closed group subschemes of GL_{rn}/K, where C_{E}(GL_{rn}/K) is the commutant of E in GL_{rn}/K.

Proof. Observe that there is a natural M-algebra isomorphism

M[𝕏, 𝕐]/(𝕏𝕐 − I_{rn}, 𝕐𝕏 − I_{rn}) ≅ A^{σ_{1}} ⊗_{M} ⋯ ⊗_{M} A^{σ_{n}},

where in this case

A^{σ_{j}} =M[X, Y]/(XY −I_{r}, Y X−I_{r}) ≅ M[X^{σ_{j}}, Y^{σ_{j}}]/(X^{σ_{j}}Y^{σ_{j}} −I_{r}, Y^{σ_{j}}X^{σ_{j}} −I_{r}).

Hence, by Definition 2.3 we get a natural isomorphism of schemes over K :

R_{E/K}(GL_{r}) ≅ spec (M[𝕏, 𝕐]/(𝕏𝕐 − I_{rn}, 𝕐𝕏 − I_{rn}))^{G}

and it follows that Φ induces a closed imbedding of schemes R_{E/K}(GL_{r}) → GL_{rn} over K. Moreover we easily check that Ker Φ is generated by elements α ∘ 𝕋 − 𝕋 ∘ α and α ∘ 𝕊 − 𝕊 ∘ α for all α ∈ E, where ∘ denotes the multiplication in GL_{rn}/K. Note that C_{E}(GL_{rn}/K) is equal to

spec K[𝕋, 𝕊]/(𝕋𝕊 − I_{rn}, 𝕊𝕋 − I_{rn}, α ∘ 𝕋 − 𝕋 ∘ α, α ∘ 𝕊 − 𝕊 ∘ α, ∀_{α∈E}). □

Remark 2.8. Let E/K be an unramified extension of two local fields. Hence the extension of rings of integers O_{E}/O_{K} has an integral basis α_{1}, …, α_{n} of O_{E} over O_{K} such that the corresponding dual basis β_{1}, …, β_{n} with respect to T r_{E/K} is also a basis of O_{E} over O_{K} see [A], Chapter 7. Let R_{O_{E}/O_{K}} be the Weil restriction functor defined analogously to the Weil restriction functor for the extension E/K. Under these assumptions the following Lemmas 2.9 and 2.10 are proven in precisely the same way as Lemmas 2.4 and 2.6.

Lemma 2.9. Let V′ ⊂ V be a closed imbedding of affine schemes over O_{E}. Under the assumptions of Remark 2.8 R_{O_{E}/O_{K}} (V′) ⊂ R_{O_{E}/O_{K}} (V) is a closed imbedding of affine schemes over O_{K}.

Lemma 2.10. Consider the group scheme GL_{r}/O_{E}. Under the assumptions of Remark 2.8 there is a natural isomorphism R_{O_{E}/O_{K}} (GL_{r}) ≅ C_{O_{E}}(GL_{rn}/O_{K}) of closed group subschemes of GL_{rn}/O_{K}, where C_{O_{E}}(GL_{rn}/O_{K}) is the commutant of O_{E} in GL_{rn}/O_{K}.

We return to the case of the arbitrary separable field extension E/K of degree n. Every point of X_{0} ∈ GL_{r}(E) is uniquely determined by the ring homomorphism

h_{X_{0}} : E[X, Y]/(XY − I_{r}, Y X − I_{r}) → E

X ↦ X_{0}, Y ↦ Y_{0},

where Y_{0} is the inverse of X_{0}. This gives immediately the homomorphism

h_{𝕋_{0}} : K[𝕋, 𝕊]/(𝕋𝕊 − I_{rn}, 𝕊𝕋 − I_{rn}) → K
𝕋 ↦ 𝕋_{0} = 𝔸𝕏_{0}𝔹,
𝕊 ↦ 𝕊_{0} = 𝔸𝕐_{0}𝔹

where

𝕏_{0} = ( \begin{matrix}X_{0}^{σ_{1}} & 0I_{r} & … & 0I_{r} \\ 0I_{r} & X_{0}^{σ_{2}} & … & 0I_{r} \\ ⋮ & ⋮ & … & ⋮ \\ 0I_{r} & 0I_{r} & … & X_{0}^{σ_{n}} \end{matrix} ) , 𝕐_{0} = ( \begin{matrix}Y_{0}^{σ_{1}} & 0I_{r} & … & 0I_{r} \\ 0I_{r} & Y_{0}^{σ_{2}} & … & 0I_{r} \\ ⋮ & ⋮ & … & ⋮ \\ 0I_{r} & 0I_{r} & … & Y_{0}^{σ_{n}} \end{matrix} ) ,

and the action of σ_{i} on X_{0} and Y_{0} is the genuine action on the entries of X_{0} and Y_{0}. Obviously h_{𝕋_{0}} determines uniquely the point 𝕋_{0} ∈ GL_{rn}(K) with the inverse 𝕊_{0}.

Definition 2.11. Assume that Z = {X_{t}; t ∈ T } ⊂ GL_{r}(E) is a set of points. We define the corresponding set of points:

Z_{Φ} = {𝕋_{t} = 𝔸𝕏_{t}𝔹; t ∈ T } ⊂ GL_{rn}(K),

where

𝕏_{t} = ( \begin{matrix}X_{t}^{σ_{1}} & 0I_{r} & … & 0I_{r} \\ 0I_{r} & X_{t}^{σ_{2}} & … & 0I_{r} \\ ⋮ & ⋮ & … & ⋮ \\ 0I_{r} & 0I_{r} & … & X_{t}^{σ_{n}} \end{matrix} ) .

We denote by Z^{alg} the Zariski closure of Z in GL_{r}/E and by Z_{Φ}^{alg} the Zariski closure of Z_{Φ} in GL_{rn}/K.

Proposition 2.12. We have a natural isomorphism of schemes over K :

R_{E/K}(Z^{alg}) ≅ Z_{Φ}^{alg}.

Proof. Let

J_{t} = (XY − I_{r}, Y X − I_{r}, X − X_{t}, Y − Y_{t})

be the prime ideal of E[X, Y] corresponding to the point X_{t} ∈ GL_{r}(E). Let

J = ⋂_{t∈T} J_{t}.

By definition Z^{alg} = spec (E[X, Y]/J). Let

𝕁_{t} = (𝕋𝕊 − I_{rn}, 𝕊𝕋 − I_{rn}, 𝕋 − 𝔸𝕏_{t}𝔹, 𝕊 − 𝔸𝕐_{t}𝔹)

be the prime ideal in K[𝕋, 𝕊]/(𝕋𝕊 − I_{rn}, 𝕊𝕋 − I_{rn}) corresponding to the point 𝔸𝕏_{t}𝔹 ∈ GL_{rn}(K). Define

𝕁 = ⋂_{t∈T} 𝕁_{t}.

By definition Z_{Φ}^{alg} = spec (K[𝕋, 𝕊]/𝕁). Put A = E[X, Y]/(XY − I_{r}, Y X − I_{r}). Observe that the ring \overline{A}^{G} is generated as a K-algebra by 𝔸𝕏𝔹 and 𝔸𝕐𝔹, since \overline{A} is generated by 𝕏 and 𝕐 as an M-algebra. Define

𝕁′_{t} = (𝔸𝕏𝔹 − 𝔸𝕏_{t}𝔹, 𝔸𝕐𝔹 − 𝔸𝕐_{t}𝔹)

which is an ideal of \overline{A}^{G}. Put

𝕁′ = ⋂_{t∈T} 𝕁′_{t}.

We have the following isomorphism induced by Φ.

(2.13)  K[𝕋, 𝕊]/𝕁_{t} ≅ \overline{A}^{G}/ 𝕁′_{t} ≅ K.

Hence, Φ^{−1}(𝕁′_{t}) = 𝕁_{t} and Φ^{−1}(𝕁′) = 𝕁. This gives the isomorphism

(2.14)  K[𝕋, 𝕊]/𝕁 ≅ \overline{A}^{G}/ 𝕁′.

Let B = E[X, Y]/J. There is a natural surjective homomorphism of K-algebras coming from the construction in the proof of Lemma 2.4 (see (2.5)):

(2.15)  \overline{A}^{G}/ 𝕁′ → \overline{B}^{G}

induced by the quotient map A → B. We want to prove that (2.15) is an isomorphism. Observe that there is natural isomorphism of K-algebras:

(2.16)  \overline{A}^{G}/ 𝕁′_{t} ≅ \overline{A/J_{t}}^{G} ≅ K.

Consider the following commutative diagram of homomorphisms of K-algebras:

(2.17)  \begin{matrix}\overline{A}^{G}/ 𝕁′ & ⟶ & \overline{B}^{G} \\ ↓ & & ↓ \\ ∏_{t∈T} \overline{A}^{G}/ 𝕁′_{t} & ⟶^{≅} & ∏_{t∈T} \overline{A/J_{t}}^{G} \end{matrix}

The left vertical arrow is an imbedding by definition of 𝕁′ and the bottom horizontal arrow is an isomorphism by (2.16). Hence the top horizontal arrow is an imbedding, i.e., the map (2.15) is an isomorphism. The composition of maps (2.14) and (2.15) gives a natural isomorphism of K-algebras

(2.18)  K[𝕋, 𝕊]/𝕁 ≅ \overline{B}^{G}.

But Z_{Φ}^{alg} = spec (K[𝕋, 𝕊]/𝕁). In addition, Z^{alg} = spec B, hence R_{E/K}(Z^{alg}) = spec \overline{B}^{G} and Proposition 2.12 follows by (2.18). □

Remark 2.19. If Z is a subgroup of GL_{r}(E), then Z_{Φ} is a subgroup of GL_{rn}(K). In this case Z^{alg} is a closed algebraic subgroup of GL_{r}/E and Z_{Φ}^{alg} is a closed algebraic subgroup of GL_{rn}/K.

Definition 2.20. Let H = spec A be an affine algebraic group scheme defined over E and 𝔥 its Lie algebra. We define 𝔤 = R_{E/K}𝔥 to be the Lie algebra obtained from 𝔥 by considering it over K with the same bracket.

Lemma 2.21. There is the following equality of Lie algebras

Lie(R_{E/K}H) = R_{E/K}𝔥.

Proof. Let n = [E : K] and G = Gal(E/K). Since H is an algebraic group 𝔥 = Der(A) is the Lie algebra of derivations of the algebra A of functions on H [ H1]. Let ϕ : Der(A) → Der(Ā) be the homomorphism of Lie algebras (considered over E) given by the following formula:

ϕ(δ) = Σ_{i=1}^{n}id ⊗ ⋯ ⊗ id ⊗ δ_{i} ⊗ id ⊗ ⋯ ⊗ id,

where δ_{i} = δ⊗1 as an element of Der(A^{σ_{i}}). Recall that A^{σ_{i}} = A⊗_{E,σ_{i}}M. If σ ∈ G and σ(a_{1}⊗⋯⊗a_{n}) = σ(a_{k_{1}})⊗⋯⊗σ(a_{k_{n}}) one readily sees that δ_{j}(σ(a_{k_{j}})) = σ(δ_{k_{j}}(a_{k_{j}})) and therefore ϕ(δ) is G-equivariant i.e., ϕ(δ) ∈ Der(Ā^{G}). It is easy to see that ϕ(δ) as an element of Der(Ā) is nontrivial if δ is nontrivial. Since ϕ(δ) is M-linear and Ā^{G}⊗_{K}M = Ā, we see that ϕ(δ) is a nontrivial element of Der(Ā^{G}) = Lie(R_{E/K}H). On the other hand, observe that

Lie(R_{E/K}H)⊗_{K} K̄ = Lie(R_{E/K}H⊗_{K} K̄) =
= Lie(H̄ ×_{K} ⋯ ×_{K} H̄) = (⊕𝔥) ⊗_{E} K̄ = 𝔤 ⊗_{K} K̄.

This shows that Lie(R_{E/K}H) and R_{E/K}𝔥 have the same dimensions and therefore are equal. □

Lemma 2.22. Let 𝔤 be a Lie algebra over E and let 𝔤′ be its derived algebra. Then

R_{E/K}(𝔤′) = (R_{E/K}(𝔤))′

Proof. This follows immediately from the fact that R_{E/K}(𝔤) and 𝔤 have the same Lie bracket (cf. Definition 2.20) □

Lemma 2.23. If G is a connected, algebraic group over E of characteristic 0, then

R_{E/K}(G′) = (R_{E/K}G)′

Proof. We have the following identities:

Lie((R_{E/K}(G))′) = (Lie(R_{E/K}(G)))′ = (R_{E/K}(Lie(G)))′ =
= R_{E/K}((Lie(G))′) = R_{E/K}(Lie(G′)) = Lie(R_{E/K}(G′))

The first and the fourth equality follow from Corollary on p.75 of [H1]. The second and fifth equality follow from Lemma 2.21. The third equality follows from Lemma 2.22. The Lemma follows by Theorem on p. 87 of [H1] and Proposition on p. 110 of [H1]. □

3. Some remarks on bilinear forms.

Let E be a finite extension of ℚ of degree e. Let E_{l} = E⊗ℚ_{l} and O_{E_{l}} = O_{E}⊗ℤ_{l}. Hence E_{l} = ∏_{λ|l} E_{λ} and O_{E_{l}} = ∏_{λ|l} O_{λ}. Let O′_{λ} be the dual to O_{λ} with respect to the trace T r_{E_{λ}/ℚ_{l}}. For l ≫ 0 we have O′_{λ} = O_{λ} see [A], Chapter 7. From now on we take l big enough to ensure that O′_{λ} = O_{λ} for all primes λ in O_{E} over l and that an abelian variety A we consider, has good reduction at all primes in O_{F} over l. The following lemma is the integral version of the sublemma 4.7 of [D].

Lemma 3.1. Let T_{1} and T_{2} be finitely generated, free O_{E_{l}}-modules. For any ℤ_{l}-bilinear (resp. nondegenerate ℤ_{l}-bilinear ) map

ψ_{l} : T_{1} × T_{2} → ℤ_{l}

such that ψ_{l}(ev_{1}, v_{2}) = ψ_{l}(v_{1}, ev_{2}) for all e ∈ O_{E_{l}}, v_{1} ∈ T_{1}, v_{2} ∈ T_{2}, there is a unique O_{E_{l}}-bilinear (resp. nondegenerate O_{E_{l}}-bilinear ) map

ϕ_{l} : T_{1} × T_{2} → O_{E_{l}}

such that T r_{E_{l}/ℚ_{l}}(ϕ_{l}(v_{1}, v_{2})) = ψ_{l}(v_{1}, v_{2}) for all v_{1} ∈ T_{1} and v_{2} ∈ T_{2}.

Proof. Similary to Sublemma 4.7, [D] we observe that the map

T r_{E_{l}/ℚ_{l}} : Hom_{O_{E_{l}}} (T_{1} ⊗_{O_{E_{l}}} T_{2} ; O_{E_{l}}) → Hom_{ℤ_{l}}(T_{1} ⊗_{O_{E_{l}}} T_{2} ; ℤ_{l})

is an isomorphism since it is a surjective map of torsion free ℤ_{l}-modules of the same ℤ_{l}-rank. The surjectivity of T r_{E_{l}/ℚ_{l}} can be seen as follows. The ℤ_{l}-basis of the module T_{1}⊗_{O_{E_{l}}} T_{2} is given by

B = ((0, …, 0, α_{k}^{λ}, 0, …, 0)e_{i} ⊗ e′_{j})

where (0, …, 0, α_{k}^{λ}, 0, …, 0) ∈ ∏_{λ|l} O_{λ} and α_{k}^{λ} is an element of a basis of O_{λ} over ℤ_{l} and e_{i} (resp. e′_{j}) is an element of the standard basis of T_{1} (resp. T_{2}) over O_{E_{l}}. Let ψ_{k,i,j}^{λ} ∈ Hom_{ℤ_{l}}(T_{1} ⊗_{O_{E_{l}}} T_{2} ; ℤ_{l}) be the homomorphism which takes value 1 on the element (0, …, 0, α_{k}^{λ}, 0, …, 0)e_{i} ⊗ e′_{j} of the basis B and takes value 0 on the remaining elements of the basis B. Let us take ϕ_{i,j} ∈ Hom_{O_{E_{l}}} (T_{1} ⊗_{O_{E_{l}}} T_{2} ; O_{E_{l}}) such that

ϕ_{i,j}(e_{r} ⊗ e′_{s}) = \{ \begin{matrix}1 & if i = r and j = s \\ 0 & if i ≠ r or j ≠ s \end{matrix}

Then for each k there exist elements (the dual basis) β_{k}^{λ} ∈ O_{λ} such that T r_{E_{λ}/ℚ_{l}}(β_{k}^{λ}α_{n}^{λ}) = δ_{k,n}. If we put ϕ_{i,j,k}^{λ} = (0, …, 0, β_{k}^{λ}, 0, …, 0)ϕ_{i,j} then clearly T r_{E_{l}/ℚ_{l}}(ϕ_{i,j,k}^{λ}(t_{1}, t_{2})) = ψ_{i,j,k}^{λ}(t_{1}, t_{2}). Hence the proof is finished since the elements ψ_{i,j,k}^{λ}(t_{1}, t_{2}) form a basis of Hom_{ℤ_{l}}(T_{1} ⊗_{O_{E_{l}}} T_{2} ; ℤ_{l}) over ℤ_{l}. □

Consider the case T_{1} = T_{2} and put T_{l} = T_{1} = T_{2}. Assume in addition that ψ_{l} is nondegenerate. Let

\overline{ψ}_{l} : T_{l}/l T_{l} × T_{l}/l T_{l} → ℤ/l

be the ℤ/l-bilinear pairing obtained by reducing the form ψ_{l} modulo l. Define

T_{λ} = e_{λ}T_{l} ≅ T_{l} ⊗_{O_{E_{l}}} O_{λ}
V_{λ} = T_{λ} ⊗_{O_{λ}} E_{λ}

where e_{λ} is the standard idempotent corresponding to the decomposition O_{E_{l}} = ∏_{λ} O_{λ}. Let π_{λ} : O_{E_{l}} → O_{λ} be the natural projection. We can define an O_{λ}- nondegenerate bilinear form as follows:

ψ_{λ} : T_{λ} × T_{λ} → O_{λ}
ψ_{λ}(e_{λ}v_{1}, e_{λ}v_{2}) = π_{λ}(ϕ_{l}(v_{1}, v_{2}))

for any v_{1}, v_{2} ∈ T_{l}. Put k_{λ} = O_{λ}/λ O_{λ}. This gives the k_{λ}-bilinear form \overline{ψ}_{λ} = ψ_{λ} ⊗_{O_{λ}} k_{λ}

\overline{ψ}_{λ} : T_{λ}/λ T_{λ} × T_{λ}/λ T_{λ} → k_{λ}.

We also have the E_{λ}-bilinear form ψ_{λ}^{0} := ψ_{λ} ⊗_{O_{λ}} E_{λ}

ψ_{λ}^{0} : V_{λ} × V_{λ} → E_{λ}.

Lemma 3.2. Assume that the form \overline{ψ}_{l} is nondegenerate. Then the forms \overline{ψ}_{λ}, ψ_{λ} and ψ_{λ}^{0} are nondegenerate for each λ|l.

Proof. First we prove that \overline{ψ}_{λ} is nondegenerate for all λ|l. Assume that \overline{ψ}_{λ} is degenerate for some λ. Without loss of generality we can assume that the left radical of \overline{ψ}_{λ} is nonzero. So there is a nonzero vector e_{λ}v_{0} ∈ T_{λ} (for some v_{0} ∈ T_{l}) which maps to a nonzero vector in T_{λ}/λ T_{λ} such that ψ_{λ}(e_{λ}v_{0}, e_{λ}w) ∈ λ O_{λ} for all w ∈ T_{l}. Now use the decomposition T_{l} = ⊕_{λ}T_{λ}, Lemma 3.1 and the O_{E_{l}}-linearity of ϕ_{l} to observe that for each w ∈ T_{l}

ψ_{l}(e_{λ}v_{0}, w) = T r_{E_{l}/ℚ_{l}}(ϕ_{l}(e_{λ}v_{0}, ∑_{λ′} e_{λ′}w)) = T r_{E_{λ}/ℚ_{l}}ψ_{λ}(e_{λ}v_{0}, e_{λ}w) ∈ lℤ_{l}.

This contradicts the assumption that \overline{ψ}_{l} is nondegenerate. Similarly, but in an easier way, we prove that ψ_{λ} is nondegenerate. From this it immediately follows that ψ_{λ}^{0} is nondegenerate. □

4. Auxiliary facts about abelian varieties. Let A/F be a principally polarized, simple abelian variety of dimension g with the polarization defined over F. Put R = EndF̄(A) We assume that EndF̄(A) = End_{F} (A), hence the actions of R and G_{F} on A(\overline{F}) commute. Put D = EndF̄(A) ⊗_{ℤ} ℚ. The ring R is an order in D. Let E_{1} be the center of D and let

E := {a ∈ E_{1}; a′ = a},

where ′ is the Rosati involution. Let R_{D} be a maximal order in D containing R. Put O_{E}^{0} := R∩E. The ring O_{E}^{0} is an order in E. Take l that does not divide the index [R_{D} : R]. Then R_{D} ⊗_{ℤ} ℤ_{l} = R ⊗_{ℤ} ℤ_{l} and O_{E} ⊗_{ℤ} ℤ_{l} = O_{E}^{0} ⊗_{ℤ} ℤ_{l}

The polarization of A gives a ℤ_{l}-bilinear, nondegenerate, alternating pairing

(4.1)  ψ_{l} : T_{l}(A) × T_{l}(A) → ℤ_{l}.

Because A has the principal polarization, for any endomorphism α ∈ R we get α′ ∈ R, (see [Mi] chapter 13 and 17) where α′ is the image of α by the Rosati involution. Hence for any v, w ∈ T_{l}(A) we have ψ_{l}(αv, w) = ψ_{l}(v, α′w) (see loc. cit.).

Remark 4.2. Notice that if an abelian variety were not principally polarized, one would have to assume that l does not divide the degree of the polarization of A, to get α′ ⊗ 1 ∈ R ⊗ ℤ_{l} for α ∈ R.

By Lemma 3.1 there is a unique nondegenerate, O_{E_{l}}-bilinear pairing

(4.3)  ϕ_{l} : T_{l}(A) × T_{l}(A) → O_{E_{l}}

such that T r_{E_{l}/ℚ_{l}}(ϕ_{l}(v_{1}, v_{2})) = ψ_{l}(v_{1}, v_{2}). As in the general case define

T_{λ}(A) = e_{λ}T_{l}(A) ≅ T_{l}(A) ⊗_{O_{E_{l}}} O_{λ}

V_{λ}(A) = T_{λ}(A) ⊗_{O_{λ}} E_{λ}.

Note that T_{λ}(A)/λT_{λ}(A) ≅ A[λ] as k_{λ}[G_{F}]-modules. Again as in the general case define nondegenerate, O_{λ}-bilinear form

(4.4)  ψ_{λ} : T_{λ}(A) × T_{λ}(A) → O_{λ}
ψ_{λ}(e_{λ}v_{1}, e_{λ}v_{2}) = π_{λ}(ϕ_{l}(v_{1}, v_{2}))

for any v_{1}, v_{2} ∈ T_{l}(A), where π_{λ} : O_{E_{l}} → O_{λ} is the natural projection. The form ψ_{λ} gives the forms:

(4.5)  \overline{ψ}_{λ} : A[λ] × A[λ] → k_{λ}.

(4.6)  ψ_{λ}^{0} : V_{λ}(A) × V_{λ}(A) → E_{λ}.

Notice that all the bilinear forms ψ_{λ}, \overline{ψ}_{λ} and ψ_{λ}^{0} are alternating forms. For l relatively prime to the degree of polarization the form ψ_{l} is nondegenerate. Hence by lemma 3.2 the forms ψ_{λ}, \overline{ψ}_{λ} and ψ_{λ}^{0} are nondegenerate.

Lemma 4.7. Let χ_{λ} : G_{F} → ℤ_{l} ⊂ O_{λ} be the composition of the cyclotomic character with the natural imbedding ℤ_{l} ⊂ O_{λ}.

(i) For any σ ∈ G_{F} and all v_{1}, v_{2} ∈ T_{λ}(A)

ψ_{λ}(σv_{1}, σv_{2}) = χ_{λ}(σ)ψ_{λ}(v_{1}, v_{2}).

(ii) For any α ∈ R and all v_{1}, v_{2} ∈ T_{λ}(A)

ψ_{λ}(αv_{1}, v_{2}) = ψ_{λ}(v_{1}, α′v_{2}).

Proof. The proof is the same as the proof of Lemma 2.3 in [C2]. □

Remark 4.8. After tensoring appropriate objects with ℚ_{l} in lemmas 3.1 and 4.6 we obtain Lemmas 2.2 and 2.3 of [C2].

Let A/F be an abelian variety defined over a number field F such that EndF̄(A) = End_{F} (A). We introduce some notation. Let G_{l^{∞}}, G_{l}, G_{l^{∞}}^{0} denote the images of the corresponding representations:

ρ_{l} : G_{F} → GL(T_{l}(A)) ≅ GL_{2g}(ℤ_{l}),
\overline{ρ_{l}} : G_{F} → GL(A[l]) ≅ GL_{2g}(𝔽_{l}),
ρ_{l} ⊗ ℚ_{l} : G_{F} → GL(V_{l}(A)) ≅ GL_{2g}(ℚ_{l}).

Let G_{l}^{alg}, (G_{l}^{alg} resp.) denote the Zariski closure of the image of the representation ρ_{l}, (ρ_{l} ⊗ ℚ_{l}, resp.) in GL_{2g}/ℤ_{l}, (GL_{2g}/ℚ_{l}, resp). We define G(l)^{alg} to be the special fiber of the ℤ_{l}−scheme G_{l}^{alg}.

Due to our assumptions on the G_{F} -action and the properties of the forms ψ_{λ}, \overline{ψ}_{λ} and ψ_{λ}^{0} we get:

(4.9)  G_{l^{∞}} ⊂ G_{l}^{alg}(ℤ_{l}) ⊂ ∏_{λ|l} GSp_{T_{λ}(A)}(O_{λ}) ⊂ GL_{T_{l}(A)}(ℤ_{l})

(4.10)  G_{l} ⊂ G(l)^{alg}(𝔽_{l}) ⊂ ∏_{λ|l} GSp_{A[λ]}(k_{λ}) ⊂ GL_{A[l]}(𝔽_{l})

(4.11)  G_{l^{∞}}^{0} ⊂ G_{l}^{alg}(ℚ_{l}) ⊂ ∏_{λ|l} GSp_{V_{λ}(A)}(E_{λ}) ⊂ GL_{V_{l}(A)}(ℚ_{l}).

Before we proceed further let us state and prove some general lemmas concerning l-adic representations. Let K/ℚ_{l} be a local field extension and O_{K} the ring of integers in K. Let T be a finitely generated, free O_{K}-module and let V = T ⊗_{O_{K}} K. Consider a continuous representation ρ : G_{F} → GL(T) and the induced representation ρ^{0} = ρ ⊗ K : G_{F} → GL(V). Since G_{F} is compact and ρ^{0} is continuous, the subgroup ρ^{0}(G_{F}) of GL(V) is closed. By [Se7], LG. 4.5, ρ^{0}(G_{F}) is an analytic subgroup of GL(V).

Lemma 4.12. Let 𝔤 be the Lie algebra of the group ρ^{0}(G_{F})

(i) There is an open subgroup U_{0} ⊂ ρ^{0}(G_{F}) such that

End_{U_{0}} (V) = End_{𝔤} (V).

(ii) For all open subgroups U ⊂ ρ^{0}(G_{F}) we have

End_{U} (V) ⊂ End_{𝔤} (V).

(iii) Taking union over all open subgroups U ⊂ ρ^{0}(G_{F}) we get

⋃_{U} End_{U} (V) = End_{𝔤} (V).

Proof. (i) Note that for any open subgroup Ũ of 𝔤 we have

(4.13)  EndŨ (V) = End_{𝔤} (V)

because K Ũ = 𝔤. By [B], Prop. 3, III.7.2, for some open Ũ ⊂ 𝔤, there is an exponential map

exp : Ũ → ρ^{0}(G_{F})

which is an analytic isomorphism and such that exp (Ũ) is an open subgroup of ρ^{0}(G_{F}). The exponential map can be expressed by the classical power series for exp (t). On the other hand by [B], Prop. 10, III.7.6, for some open U ⊂ ρ^{0}(G_{F}), there is a logarithmic map

log : U → 𝔤

which is an analytic isomorphism and the inverse of exp. The logarithmic map can be expressed by the classical power series for ln t. Hence, we can choose Ũ_{0} such that U_{0} = exp (Ũ_{0}) and log (U_{0}) = Ũ_{0}. This gives

(4.14)  End_{U_{0}} (V) = EndŨ_{0} (V).

and (i) follows by (4.13) and (4.14).

(ii) Observe that for any open U ⊂ ρ^{0}(G_{F}) we have

End_{U} (V) ⊂ End_{U_{0}∩U} (V).

Hence (ii) follows by (i).

(iii) Follows by (i) and (ii). □

Lemma 4.15. Let A/F be an abelian variety over F such that End_{F} (A) = End\overline{F} (A). Then

End_{G_{F}} (V_{l}(A)) = End_{𝔤_{l}} (V_{l}(A)).

Proof. By the result of Faltings [Fa], Satz 4,

End_{L} (A) ⊗ ℚ_{l} = End_{G_{L}} (V_{l}(A))

for any finite extension L/F. By the assumption End_{F} (A) = End_{L} (A). Hence

End_{G_{F}} (V_{l}(A)) = End_{U′} (V_{l}(A))

for any open subgroup U′ of G_{F} . So the claim follows by Lemma 4.12 (iii). □

Let A be a simple abelian variety defined over F and E be the center of the algebra D = End_{F} (A) ⊗ ℚ. Let λ|l be a prime of O_{E} over l. Consider the following representations.

ρ_{λ} : G_{F} → GL(T_{λ}(A)),
\overline{ρ_{λ}} : G_{F} → GL(A[λ]),
ρ_{λ} ⊗_{O_{λ}} E_{λ} : G_{F} → GL(V_{λ}(A)),

where λ|l. Let G_{λ}^{alg}, ( G_{λ}^{alg} resp.) denote the Zariski closure of the image of the representation ρ_{λ}, (ρ_{λ} ⊗ E_{λ} resp.) in GL_{T_{λ}(A)}/O_{λ}, ( GL_{V_{λ}(A)}/E_{λ} resp.) We define G(λ)^{alg} to be the special fiber of the O_{λ}-scheme G_{λ}^{alg}.

Theorem 4.16. Let A be a simple abelian variety with the property that R = EndF̄(A) = End_{F} (A). Let R_{λ} = R ⊗_{O_{E}^{0}} O_{λ} and let D_{λ} = D ⊗_{E} E_{λ}. Then

(i) End_{O_{λ}[G_{F}]} (T_{λ}(A)) ≅ R_{λ} (ii) End_{R_{λ}[G_{F}]} (V_{λ}(A)) ≅ D_{λ} (iii) End_{k_{λ}[G_{F}]} (A[λ]) ≅ R_{λ} ⊗_{O_{λ}} k_{λ} for l ≫ 0.

Proof. It follows by [Fa], Satz 4 and [Za], Cor. 5.4.5. □

Lemma 4.17. Let K be a field and let R be a unital K-algebra. Put D = End_{R}(M) and let L be a subfield of the center of D. Assume that L/K is a finite separable extension. If M is a semisimple R-module then M is also a semisimple R ⊗_{K} L-module with the obvious action of R ⊗_{K} L on M.

Proof. Take α ∈ L such that L = K(α). Let [L : K] = n. Let us write M = ⊕_{i}M_{i} where all M_{i} are simple R modules. For any i we put M̃_{i} = ∑_{k=0}^{n−1} α^{k} M_{i}. Then M̃_{i} is an R ⊗_{K} L-module. Because M_{i} is a simple R-module we can write

M̃_{i} = ⨁_{k=0}^{m−1} α^{k} M_{i},

for some m. Observe that if m = 1, then M̃_{i} is obviously a simple R ⊗_{K} Lmodule. If m>1, we pick any simple R-submodule N_{i} ⊂ M̃_{i}, N_{i} ≠ M_{i}. There is an R- isomorphism ϕ : M_{i} → N_{i} by semisimplicity of M̃_{i}. We can write M = M_{i}⊕N_{i}⊕M′, where M′ is an R-submodule of M. Define Ψ ∈ Aut_{R}(M) ⊂ End_{R}(M) by Ψ|_{M_{i}} = ϕ, Ψ|_{N_{i}} = ϕ^{−1} and Ψ|_{M′} = Id_{M′}. Note that

(4.18)  Ψ(⨁_{k=0}^{m−1} α^{k} M_{i}) = ⨁_{k=0}^{m−1} α^{k} N_{i}

since α is in the center of D. Hence M̃_{i} = ⨁_{k=0}^{m−1} α^{k} N_{i} by the classification of semisimple R-modules. We conclude that M̃_{i} is a simple R ⊗_{K} L-module. Indeed, if N ⊂ M̃_{i} were a nonzero R ⊗_{K} L-submodule of M̃_{i} then we could pick any simple R-submodule N_{i} ⊂ N. If N_{i} = M_{i} then N = M̃_{i}. If N_{i} ≠ M_{i} then by (4.18) M̃_{i} = ⨁_{k=0}^{m−1} α^{k} N_{i} ⊂ N. Since M = ∑_{i} M̃_{i}, we see that M is a semisimple R ⊗_{K} L-module. □

Theorem 4.19. Let A be a simple abelian variety with the property that R = EndF̄(A) = End_{F} (A). Let R_{λ} = R⊗_{O_{E}^{0}} O_{λ} and let D_{λ} = D ⊗_{E} E_{λ}. Then G_{F} acts on V_{λ}(A) and A[λ] semisimply and G_{λ}^{alg} and G(λ)^{alg} are reductive algebraic groups. The scheme G_{λ}^{alg} is a reductive group scheme over O_{λ} for l big enough.

Proof. It follows by [Fa], Theorem 3 and our Lemma 4.17. The last statement follows by [LP1], Proposition 1.3, see also [Wi], Theoreme 1. □

5. Abelian varieties of type I and II. In this section we work with abelian varieties of type I and II in the Albert’s classification list of division algebras with involution [M], p. 201, i.e. E ⊂ D = EndF̄(A) ⊗_{Z} ℚ is the center of D and E is a totally real extension of ℚ of degree e. To be more precise D is either E (type I) or an indefinite quaternion algebra with the center E, such that D ⊗_{ℚ} ℝ ≅ ∏_{i=1}^{e} M_{2,2}(ℝ) (type II). In the first part of this section we prove integral versions of the results of Chi [C2] for abelian varieties of type II. Let l be a sufficiently large prime number that does not divide the index [R_{D} : R] and such that D ⊗_{E} E_{λ} splits over E_{λ} for any prime λ in O_{E} over l. Hence, D_{λ} = M_{2,2}(E_{λ}). Then by [R, Corollary 11.2 p. 132 and Theorem 11.5 p. 133] the ring R_{λ} is a maximal order in D_{λ}. So by [R] Theorem 8.7 p. 110 we get R_{λ} = M_{2,2}(O_{λ}), hence R_{λ}⊗_{O_{λ}}k_{λ} = M_{2,2}(k_{λ}). Similarly to [C2] we put

t = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) , u = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) .

Let e = \frac{1}{2}(1+t), f = \frac{1}{2}(1+u), X = e T_{λ}(A), Y = (1−e) T_{λ}(A), X′ = f T_{λ}(A), Y′ = (1 − f) T_{λ}(A). Put X = X ⊗_{O_{λ}}E_{λ}, X′ = X′⊗_{O_{λ}}E_{λ}, Y = Y⊗_{O_{λ}}E_{λ}, Y′ = Y′⊗_{O_{λ}}E_{λ}, \overline{X} = X ⊗_{O_{λ}}k_{λ}, \overline{X}′ = X′⊗_{O_{λ}}k_{λ}, \overline{Y} = Y⊗_{O_{λ}}k_{λ}, \overline{Y}′ = Y′⊗_{O_{λ}}k_{λ}. Because ueu = 1 − e, the matrix u gives an O_{λ}[G_{F}]-isomorphism between X and Y, hence it yields an E_{λ}[G_{F}]-isomorphism between X and Y and a k_{λ}[G_{F}]-isomorphism between \overline{X} and \overline{Y}. Multiplication by t gives an O_{λ}[G_{F}]- isomorphism between X′ and Y′, hence it yields an E_{λ}[G_{F}]-isomorphism between X′ and Y′ and a k_{λ}[G_{F}]-isomorphism between \overline{X}′ and \overline{Y}′. Observe that

(5.1)  End_{O_{λ}[G_{F}]} (X) ≅ End_{O_{λ}[G_{F}]}(X′) ≅ O_{λ}
(5.2)  End_{E_{λ}[G_{F}]} (X) ≅ End_{E_{λ}[G_{F}]}(X′) ≅ E_{λ}
(5.3)  End_{k_{λ}[G_{F}]} (\overline{X}) ≅ End_{k_{λ}[G_{F}]}(\overline{X}′) ≅ k_{λ}.

So the representations of G_{F} on the spaces X, Y, X′, Y′ (resp. \overline{X} , \overline{Y}, \overline{X}′, \overline{Y}′) are absolutely irreducible over E_{λ} (resp. over k_{λ}). Hence, the bilinear form ψ_{λ}^{0} cf. (4.4) (resp. \overline{ψ}_{λ} cf. (4.5)) when restricted to any of the spaces X, X′, Y, Y′, (resp. spaces \overline{X} , \overline{X}′, \overline{Y}, \overline{Y}′) is either nondegenerate or isotropic.

We obtain the integral version of Theorem A of [C2].

Theorem 5.4. If A is of type II, then there is a free O_{λ}-module W_{λ}(A) of rank 2h such that

(i) we have an isomorphism of O_{λ}[G_{F}]- modules T_{λ}(A) ≅ W_{λ}(A) ⊕ W_{λ}(A) (ii) there is an alternating pairing ψ_{λ} : W_{λ}(A) × W_{λ}(A) → O_{λ} (ii’) the induced alternating pairing ψ_{λ}^{0} : W_{λ}(A) × W_{λ}(A) → E_{λ} is nonde-

generate, where W_{λ}(A) = W_{λ}(A) ⊗_{O_{λ}} E_{λ} (ii”) the induced alternating pairing \overline{ψ}_{λ} : \overline{W}_{λ}(A) × \overline{W}_{λ}(A) → k_{λ} is nonde-

generate, where \overline{W}_{λ}(A) = W_{λ}(A) ⊗_{O_{λ}} k_{λ}.

The pairings in (ii), (ii’) and (ii”) are compatible with the G_{F} -action in the same way as the pairing in Lemma 4.7 (i).

Proof. (ii’) is proven in [C2], while (i) and (ii) are straightforward generalizations of the arguments in loc. cit. The bilinear pairing ϕ_{l} is nondegenerate, hence the bilinear pairing \overline{ϕ}_{l} is nondegenerate, since the abelian variety A is principally polarized by assumption. (Actually \overline{ϕ}_{l} is nondegenerate for any abelian variety with polarization degree prime to l). So, by Lemma 3.2 the form \overline{ψ}_{λ} is nondegenerate for all λ hence simultaneously the forms ψ_{λ}^{0} and \overline{ψ}_{λ} are nondegenerate. Now we finish the proof of (ii”) arguing for A[λ] similarly as it is done for V_{λ} in [C2], Lemma 3.3. □

From now on we work with the abelian varieties of type either I or II. We assume that the field F of definition of A is such that G_{l}^{alg} is a connected algebraic group. Let us put

(5.5)  T_{λ} = \{ \begin{matrix}T_{λ}(A) & if A is of type I \\ W_{λ}(A) , & if A is of type II \end{matrix}

Let V_{λ} = T_{λ} ⊗_{O_{λ}} E_{λ} and A_{λ} = V_{λ}/T_{λ}. With this notation we have:

(5.6)  V_{l}(A) = \{ \begin{matrix}⨁_{λ|l} V_{λ} & if A is of type I \\ ⨁_{λ|l}(V_{λ} ⊕ V_{λ}) , & if A is of type II \end{matrix}

We put

(5.7)  V_{l} = ⨁_{λ|l} V_{λ}

Let V_{Φ_{λ}} be the space V_{λ} considered over ℚ_{l}. We define ρ_{Φ_{λ}}(g) = 𝕋_{λ} = 𝔸_{λ}𝕏_{λ}𝔹_{λ}, where X_{λ} ∈ GL(V_{λ}) is such that ρ_{λ}(g) = X_{λ}. ( cf. the definition of the map Φ in (2.6) for the choice of 𝔸_{λ} and 𝔹_{λ}). Proposition 2.12 motivates the definition of ρ_{Φ_{λ}}. We have the following equality of ℚ_{l}-vector spaces:

(5.8)  V_{l} = ⨁_{λ|l} V_{Φ_{λ}}

The l-adic representation

(5.9)  ρ_{l} : G_{F} ⟶ GL(V_{l}(A))

induces the following representations (note that we use the notation ρ_{l} for both representations (5.9) and (5.10) cf. Remark 5.13 ):

(5.10)  ρ_{l}: G_{F} ⟶ GL(V_{l})
(5.11)  ∏ ρ_{λ}: G_{F} ⟶ ∏_{λ} GL(V_{λ})

(5.12)  ∏ ρ_{Φ_{λ}} : G_{F} ⟶ ∏_{Φ_{λ}}GL(V_{Φ_{λ}}).

Remark 5.13. In the case of abelian variety of type II we have V_{l}(A) = V_{l} ⊕V_{l} and the action of G_{F} on the direct sum is the diagonal one as follows from Theorem 5.4. Hence, the images of the Galois group via the representations (5.9), (5.10) and (5.12) are isomorphic. Also the Zariski closures of the images of these three representations are isomorphic as algebraic varieties over ℚ_{l} in the corresponding GL-groups. Similarly, V_{λ}(A) = V_{λ} ⊕ V_{λ} with the diagonal action of G_{F} on the direct sum by Theorem 5.4. Hence, the images of the representations given by G_{F} -actions on V_{λ} and V_{λ}(A) are isomorphic and so are their Zariski closures in corresponding GL-groups. For this reason, in the sequel, we will identify the representation of G_{F} on V_{l}(A) (respectively on V_{λ}(A)) with its representation on V_{l} (resp. V_{λ}).

By Remark 5.13 we can consider G_{l}^{alg} (resp. G_{λ}^{alg}) to be the Zariski closure in GL_{V_{l}} (resp. GL_{V_{λ}}) of the image of the representation ρ_{l} of (5.10) (resp. ρ_{λ} of (5.11)). Let G_{Φ_{λ}}^{alg} denote the Zariski closure in GL_{V_{Φ_{λ}}} of the image of the representation ρ_{Φ_{λ}} of (5.12). Let 𝔤_{l} be the Lie algebra of G_{l}^{alg}, 𝔤_{λ} be the Lie algebra of G_{λ}^{alg} and let 𝔤_{Φ_{λ}} be the Lie algebra of G_{Φ_{λ}}^{alg}. By definition, we have the following inclusions:

(5.14)  G_{l}^{alg} ⊂ ∏_{λ|l}G_{Φ_{λ}}^{alg}

(5.15)  (G_{l}^{alg})′ ⊂ ∏_{λ|l}(G_{Φ_{λ}}^{alg})′

(5.16)  𝔤_{l} ⊂ ⨁_{λ|l}𝔤_{Φ_{λ}}

(5.17)  𝔤_{l}^{ss} ⊂ ⨁_{λ|l}𝔤_{Φ_{λ}}^{ss}.

The map (5.14) gives a map

(5.18)  G_{l}^{alg} → G_{Φ_{λ}}^{alg},

which induces the natural map of Lie algebras:

(5.19)  𝔤_{l} → 𝔤_{Φ_{λ}}.

Lemma 5.20. The map (5.19) of Lie algebras is surjective for any prime λ|l. Hence the following map of Lie algebras:

(5.21)  𝔤_{l}^{ss} → 𝔤_{Φ_{λ}}^{ss}

is surjective.

Proof. We know by the result of Tate, [T2] that the ℚ_{l}[G_{F}]-module V_{l}(A) is of Hodge-Tate type for any prime v of O_{F} dividing l. Hence by the theorem of Bogomolov cf. [Bo] we have

𝔤_{l} = Lie (ρ_{l}(G_{F})).

Since each ℚ_{l}[G_{F}]-module V_{Φ_{λ}} is a direct summand of the ℚ_{l}[G_{F}]-module V_{l}, then the ℚ_{l}[G_{F}]-module V_{Φ_{λ}} is also of Hodge-Tate type for any prime v of O_{F} dividing l. It follows by the theorem of Bogomolov, [Bo] that

𝔤_{Φ_{λ}} = Lie (ρ_{Φ_{λ}}(G_{F})).

But the surjective map of l-adic Lie groups ρ_{l}(G_{F}) → ρ_{Φ_{λ}}(G_{F}) induces the surjective map of l-adic Lie algebras Lie (ρ_{l}(G_{F})) → Lie (ρ_{Φ_{λ}}(G_{F})). □

Lemma 5.22. Let A/F be an abelian variety over F of type I or II such that End_{F} (A) = End\overline{F} (A). Then

(5.23)  End_{𝔤_{λ}} (V_{λ}) ≅ End_{E_{λ}[G_{F}]} (V_{λ}) ≅ E_{λ}

(5.24)  End_{𝔤_{Φ_{λ}}} (V_{Φ_{λ}}) ≅ End_{ℚ_{l}[G_{F}]} (V_{Φ_{λ}}) ≅ E_{λ}.

Proof. By [F], Theorem 4, the assumption End_{F} (A) = End_{L} (A) for any finite extension L/F, Theorem 4.16 (ii), the equality (5.2) and Theorem 5.4 we get

(5.25)  E_{λ} ≅ End_{E_{λ}[G_{F}]} (V_{λ}) ≅ End_{E_{λ}[G_{L}]} (V_{λ}).

This implies the equality

End_{G_{F}} (V_{λ}) = End_{U} (V_{λ})

for any open subgroup U of G_{F} . Hence, the equality (5.23) follows by Lemma 4.12 (iii). For any F ⊂ L ⊂ \overline{F} we have M_{2,2}(End_{ℚ_{l}[G_{L}]}(V_{l})) = End_{ℚ_{l}[G_{L}]}(V_{l}^{2}) = End_{ℚ_{l}[G_{L}]}(V_{l}(A)) and

(5.26)  End_{ℚ_{l}[G_{L}]}(V_{l}(A)) ≅ ∏_{λ|l} D_{λ} ≅ ∏_{λ|l} M_{2,2}(E_{λ}).

On the other hand

(5.27)  ∏_{λ|l} E_{λ} ≅ ∏_{λ|l} End_{E_{λ}[G_{L}]}(V_{λ}) ⊂ End_{ℚ_{l}[G_{L}]}(V_{l}).

Hence, comparing the dimensions over ℚ_{l} in (5.26) and (5.27) we get

(5.28)  ∏_{λ|l} End_{E_{λ}[G_{L}]}(V_{λ}) ≅ End_{ℚ_{l}[G_{L}]}(V_{l}).

By (5.28) we clearly have

(5.29)  ∏_{λ|l} End_{ℚ_{l}[G_{L}]}(V_{Φ_{λ}}) ⊂ End_{ℚ_{l}[G_{L}]}(V_{l}) ≅ ∏_{λ|l} E_{λ},

and

(5.30)  End_{E_{λ}[G_{L}]}(V_{λ}) ⊂ End_{ℚ_{l}[G_{L}]}(V_{Φ_{λ}}).

It follows by (5.25), (5.29) and by (5.30) that for any finite field extension F ⊂ L contained in \overline{F} we have

(5.31)  End_{ℚ_{l}[G_{L}]}(V_{Φ_{λ}}) ≅ End_{E_{λ}[G_{L}]}(V_{λ}) ≅ E_{λ}.

The isomorphisms (5.31) imply that

(5.32)  End_{G_{F}} (V_{Φ_{λ}}) ≅ End_{U}(V_{Φ_{λ}})

for any open subgroup U of G_{F} . The isomorphism (5.24) follows by (5.32) and Lemma 4.12 (iii). □

Lemma 5.33. 𝔤_{λ}^{ss} = sp_{2h}(E_{λ}).

Proof. In the proof we adapt to the current situation the argument from [BGK], Lemma 3.2. The only thing to check is the minuscule conjecture for the λ-adic representations ρ_{F} : G_{F} → GL(V_{λ}). By the work of Pink cf. [P], Corollary 5.11, we know that 𝔤_{l}^{ss} ⊗ ℚ̄_{l} may only have simple factors of types A, B, C or D. By the semisimplicity of 𝔤_{l}^{ss} and Lemma 5.20 the simple factors of 𝔤_{Φ_{λ}}^{ss} ⊗ ℚ̄_{l} are of the same types. By Proposition 2.12 and Lemmas 2.21, 2.22, 2.23 we get

(5.34)  𝔤_{Φ_{λ}}^{ss} ≅ R_{E_{λ}/ℚ_{l}}𝔤_{λ}^{ss}.

Since

𝔤_{Φ_{λ}}^{ss} ⊗_{ℚ_{l}} \overline{ℚ}_{l} ≅ 𝔤_{λ}^{ss}⊗_{E_{λ}}E_{λ}⊗_{ℚ_{l}} \overline{ℚ} ≅ ⨁_{E_{λ}↪\overline{ℚ}_{l}} 𝔤_{λ}^{ss}⊗_{E_{λ}} \overline{ℚ}

we see that the simple factors of 𝔤_{λ}^{ss}⊗_{E_{λ}} \overline{ℚ} are of types A, B, C or D. The rest of the argument is the same as in the proof of Lemma 3.2 of [BGK]. □

Lemma 5.35. There are natural isomorphisms of ℚ_{l}-algebras.

(5.36)  End_{𝔤_{Φ_{λ}}^{ss}} (V_{Φ_{λ}}) ≅ End_{𝔤_{λ}^{ss}} (V_{λ}) ≅ E_{λ}

Proof. Since 𝔤_{λ} is reductive and it acts irreducibly on the module V_{λ} (cf. Lemma 5.33) by [H2], Prop. p. 102 we have:

(5.37)  𝔤_{λ} = Z(𝔤_{λ}) ⊕ 𝔤_{λ}^{ss}

and Z(𝔤_{λ}) = 0 or Z(𝔤_{λ}) = E_{λ}. This gives

(5.38)  End_{𝔤_{λ}^{ss}} (V_{λ}) = End_{𝔤_{λ}} (V_{λ}).

The Weil restriction functor commutes with the operation of taking the center of a Lie algebra, hence we get Z(𝔤_{Φ_{λ}}) = 0 or E_{λ} and by (5.34):

𝔤_{Φ_{λ}} = Z(𝔤_{Φ_{λ}}) ⊕ 𝔤_{Φ_{λ}}^{ss}.

Since 𝔤_{Φ_{λ}} ≅ R_{E_{λ}/ℚ_{l}}𝔤_{λ}, it is clear that

End_{𝔤_{Φ_{λ}}^{ss}} (V_{Φ_{λ}}) = End_{𝔤_{Φ_{λ}}} (V_{Φ_{λ}}).

The lemma follows now from Lemma 5.22. □

Proposition 5.39. There is an equality of Lie algebras:

(5.40)  𝔤_{l}^{ss} = ⨁_{λ|l} 𝔤_{Φ_{λ}}^{ss}

Proof. Put \overline{V}_{l} = V_{l} ⊗_{ℚ_{l}} \overline{ℚ}_{l}, \overline{V}_{λ} = V_{λ} ⊗_{E_{λ}} \overline{ℚ}_{l}, \overline{𝔤}_{l}^{ss} = 𝔤_{l}^{ss} ⊗_{ℚ_{l}} \overline{ℚ}_{l}, \overline{𝔤}_{Φ_{λ}}^{ss} = 𝔤_{Φ_{λ}}^{ss} ⊗_{ℚ_{l}} \overline{ℚ}_{l}. By (5.34) we get

(5.41) \overline{𝔤}_{Φ_{λ}}^{ss} ≅ 𝔤_{λ}^{ss} ⊗_{E_{λ}} E_{λ} ⊗_{ℚ_{l}} \overline{ℚ}_{l} ≅ ∏_{E_{λ}↪\overline{ℚ}_{l}} 𝔤_{λ}^{ss} ⊗_{E_{λ}} \overline{ℚ}_{l} ≅ ∏_{E_{λ}↪\overline{ℚ}_{l}} sp (\overline{V}_{λ})

By Corollary 1.2.2 of [C1] we have 𝔤_{l} = ℚ_{l} ⊕ 𝔤_{l}^{ss}, hence

End_{𝔤_{l}^{ss}} (V_{l}(A)) = End_{𝔤_{l}} (V_{l}(A)).

By Lemmas 5.20 and 5.35

(5.42)  ∏_{λ|l} E_{λ} ≅ ∏_{λ|l} End_{𝔤_{Φ_{λ}}^{ss}} (V_{Φ_{λ}}) ≅ ∏_{λ|l} End_{𝔤_{l}^{ss}}(V_{Φ_{λ}}) ⊂ End_{𝔤_{l}^{ss}}(V_{l}).

But by assumption on l and (5.42)

∏_{λ|l} D_{λ} ≅ ∏_{λ|l} M_{2,2}(E_{λ}) ≅ M_{2,2}(∏_{λ|l} E_{λ}) ⊂ M_{2,2}(End_{𝔤_{l}^{ss}}(V_{l})) =

(5.43)  = End_{𝔤_{l}^{ss}}(V_{l}(A)) = End_{𝔤_{l}}(V_{l}(A)) ≅ ∏_{λ|l} D_{λ}.

Comparing dimensions in (5.43) we get

(5.44)  End_{𝔤_{l}^{ss}}(V_{l}) ≅ ∏_{λ|l} E_{λ}.

Hence we get

(5.45) End\overline{𝔤}_{l}^{ss} (\overline{V}_{l}) ≅ End_{𝔤_{l}^{ss}} (V_{l})⊗_{ℚ_{l}} \overline{ℚ}_{l} ≅ ∏_{λ|l} E_{λ}⊗_{ℚ_{l}} \overline{ℚ}_{l} ≅ ∏_{λ|l} ∏_{E_{λ}↪\overline{ℚ}_{l}} \overline{ℚ}_{l}.

(5.46) End_{\overline{ℚ}_{l}[G_{F}]} (\overline{V}_{λ}) ≅ End_{E_{λ}[G_{F}]} (V_{λ}) ⊗_{E_{λ}} \overline{ℚ}_{l} ≅ E_{λ} ⊗_{E_{λ}} \overline{ℚ}_{l} ≅ \overline{ℚ}_{l}.

(5.47)  \overline{V}_{l} ≅ ⨁_{λ|l} V_{λ} ⊗_{ℚ_{l}} \overline{ℚ}_{l} ≅ ⨁_{λ|l} ⨁_{E_{λ}↪\overline{ℚ}_{l}} \overline{V}_{λ}.

By (5.21) the map of Lie algebras \overline{𝔤}_{l}^{ss} → \overline{𝔤}_{Φ_{λ}}^{ss} is surjective. Isomorphisms (5.45), (5.46) and (5.47) show that the simple \overline{𝔤}_{l}^{ss} modules 𝔤_{λ}^{ss} ⊗_{E_{λ}} \overline{ℚ}_{l}, for all λ|l and all E_{λ} ↪ \overline{ℚ}_{l}, are pairwise nonisomorphic submodules of \overline{𝔤}_{l}^{ss}. Hence by [H2], Theorem on page 23

(5.48)  ⨁_{λ|l} ⨁_{E_{λ}↪\overline{ℚ}_{l}} 𝔤_{λ}^{ss} ⊗_{E_{λ}} \overline{ℚ}_{l} ⊂ \overline{𝔤}_{l}^{ss}.

Tensoring (5.17) with \overline{ℚ}_{l} and comparing with (5.48) we get

(5.49)  ⨁_{λ|l} ⨁_{E_{λ}↪\overline{ℚ}_{l}} 𝔤_{λ}^{ss} ⊗_{E_{λ}} \overline{ℚ}_{l} ≅ \overline{𝔤}_{l}^{ss}.

Hence for dimensional reasons (5.17), (5.41) and (5.49) imply (5.40). □

Corollary 5.50. The representations ρ_{Φ_{λ}}, for λ|l are pairwise nonisomorphic. The representations of the Lie algebra 𝔤_{l}^{ss} on V_{Φ_{λ}} are pairwise nonisomorphic over ℚ_{l}.

Proof. It follows by Lemmas 5.20 and 5.22 and equalities (5.8), (5.36), (5.44). □

Corollary 5.51. There is an equality of ranks of group schemes over ℚ_{l}:

(5.52)  rank (G_{l}^{alg})′ = rank ∏_{λ|l} R_{E_{λ}/ℚ_{l}}(Sp_{2h}/E_{λ}).

Proof. The Corollary follows by Lemma 5.33, equality (5.40), the isomorphism (5.34) and Lemma 2.21. □

Taking into account (4.10), (4.11) and Remark 5.13 we get:

(5.53)  G(l)^{alg} ⊂ ∏_{λ|l} R_{k_{λ}/𝔽_{l}}(GSp_{A_{λ}[λ]}) ≅ ∏_{λ|l} R_{k_{λ}/𝔽_{l}}(GSp_{2h})

(5.54)  G_{l}^{alg} ⊂ ∏_{λ|l} R_{E_{λ}/ℚ_{l}}(GSp_{V_{λ}}) ≅ ∏_{λ|l} R_{E_{λ}/ℚ_{l}}(GSp_{2h}).

6. Computation of the images of the Galois representations ρ_{l} and \overline{ρ}_{l}.

In this section we explicitly compute the images of the l-adic representations induced by the action of the absolute Galois group on the Tate module of a large class of abelian varieties of types I and II described in the definition below.

Definition of class A. We say that an abelian variety A/F, defined over a number field F, is of class A, if the following conditions hold:

(i) A is a simple, principally polarized abelian variety of dimension g (ii) R = EndF̄(A) = End_{F} (A) and the endomorphism algebra D = R⊗_{ℤ}ℚ,

is of type I or II in the Albert list of the division algebras with involution cf. [Mu], p. 201

(iii) the field F is such that for every l the Zariski closure G_{l}^{alg} of ρ_{l}(G_{F}) in

GL_{2g}/ℚ_{l} is a connected algebraic group (iv) g = hed, where h is an odd integer, e = [E : Q] is the degree of the

center E of D and d^{2} = [D : E].

Let L be a local field with the ring of integers O_{L} with maximal ideal 𝔪_{L} = 𝔪 and the residue field k = O_{L}/𝔪.

Lemma 6.1. Let

(6.2)  \xymatrix{G_{1} \ar@{^{(}->}[r] & G_{2}}

be a closed immersion of two smooth, reductive group schemes over O_{L}. Let

(6.3)  \xymatrix{G_{1} \ar@{^{(}->}[r] & G_{2}}

be the base change to L of the arrow (6.2) and let

(6.4)  \xymatrix{G_{1}(𝔪) \ar@{^{(}->}[r] & G_{2}(𝔪)}

be the base change to k of the arrow (6.2). If rank G_{1} = rank G_{2} then rank G_{1}(𝔪) = rank G_{2}(𝔪).

Proof. By [SGA3, Th. 2.5 p. 12] applied to the special point of the scheme spec O_{L} there exists an étale neighborhood S′ → spec O_{L} of the geometric point over the special point such that the group schemes G_{1,S′} = G_{1} ×_{spec O_{L}} S′ and G_{2,S′} = G_{2} ×_{spec O_{L}} S′ have maximal tori T_{1,S′} and T_{1,S′} respectively. By [SGA3] XXII, Th. 6.2.8 p. 260 we observe (we do not need it here but in the Theorem 6.6 below) that (G_{i,S′})′ ∩ T_{i,S′} is a maximal torus of (G_{i,S′})′. By the definition of a maximal torus and by [SGA3] XIX, Th. 2.5, p. 12 applied to the special point of spec O_{L}, we obtain that the special and generic fibers of each scheme G_{i,S′} have the same rank. But clearly the generic (resp. special) fibers of schemes G_{i,S′} and G_{i} have the same rank for i = 1, 2. Hence going around the diagram

(6.5)  \xymatrix{G_{1} \ar@{^{(}->}[d] \ar@{^{(}->}[r] & G_{2} \ar@{^{(}->}[d] \\ G_{1} \ar@{^{(}->}[r] & G_{2} \\ G_{1}(𝔪) \ar@{^{(}->}[r] \ar@{^{(}->}[u] & G_{2}(𝔪) \ar@{^{(}->}[u]}

and taking into account the assumptions that the ranks of the upper corners are the same we get rank G_{1}(𝔪) = rank G_{2}(𝔪). □

Theorem 6.6. Let A/F be an abelian variety of class A. Then for all l ≫ 0, we have equalitiy of ranks of group schemes over 𝔽_{l}:

(6.7)  rank (G(l)^{alg})′ = rank ∏_{λ|l} R_{k_{λ}/𝔽_{l}}(Sp_{2h})

Proof. By [LP1] Prop.1.3 and by [Wi], Th.1 and 2.1, for l ≫ 0 the group scheme G_{l}^{alg} over spec ℤ_{l} is smooth and reductive. For such an l the structure morphism (G_{l}^{alg})′ → spec ℤ_{l} is the base change of the smooth morphism G_{l}^{alg} → D_{ℤ_{l}}(D_{ℤ_{l}}(G_{l}^{alg})) via the unit section of D_{ℤ_{l}}(D_{ℤ_{l}}(G_{l}^{alg})), see [SGA3] XXII, Th. 6.2.1, p. 256 where D_{S}(G) = \underline{Hom}_{S−gr}(G, 𝔾_{m,S}) for a scheme S. Hence, the group scheme (G_{l}^{alg})′ is also smooth over ℤ_{l}. By [SGA3] loc. cit, the group scheme (G_{l}^{alg})′ is semisimple. We finish the proof by taking L = ℚ_{l}, G_{1} = (G_{l}^{alg})′, G_{2} = ∏_{λ|l} R_{O_{λ}/ℤ_{l}}(Sp_{2h}) in Lemma 6.1 and applying Corollary 5.51. □

Remark 6.8. If G is a group scheme over S_{0} then the derived subgroup G′ is defined as the kernel of the natural map

G → D_{S_{0}}(D_{S_{0}}(G))

[V], [SGA3]. Since this map is consistent with the base change, we see that for any scheme S over S_{0} we get

G′ ×_{S_{0}} S = (G ×_{S_{0}} S)′.

Theorem 6.9. Let A/F be an abelian variety of class A. Then for all l ≫ 0, we have equalities of group schemes:

(6.10)  (G_{l}^{alg})′ = ∏_{λ|l} R_{E_{λ}/ℚ_{l}}(Sp_{2h})

(6.11)  (G(l)^{alg})′ = ∏_{λ|l} R_{k_{λ}/𝔽_{l}}(Sp_{2h})

Proof. The proof is similar to the proof of Lemma 3.4 of [BGK]. We prove the equality (6.11). The proof of the equality (6.10) is analogous. Let

\underline{ρ}_{l} : G(l)^{alg} → GL_{2g}

denote the representation induced by the inclusion G(l)^{alg} ⊂ GL_{2g}. By the result of Faltings cf. [Fa], the representation \underline{ρ}_{l} is semisimple and the commutant of \underline{ρ}_{l}(G(l)^{alg}) in the matrix ring M_{2g,2g} is EndF̄(A) ⊗_{ℤ} 𝔽_{l}. The representation \underline{ρ}_{l} factors through the imbedding (5.53). Projecting onto the λ component in (5.53) we obtain the representation

(6.12)  \underline{ρ}_{Φ_{λ}} : G(l)^{alg} → R_{k_{λ}/𝔽_{l}}(GSp_{A[λ]}) ≅ R_{k_{λ}/𝔽_{l}}(GSp_{2h}).

This map corresponds to the map

(6.13)  G(l)^{alg} ⊗_{𝔽_{l}} k_{λ} → GSp_{2h}.

By Remark 6.8 restriction of the the map (6.13) to the derived subgroups gives the following map:

(6.14)  (G(l)^{alg})′ ⊗_{𝔽_{l}} k_{λ} → Sp_{2h}

which in turn gives the representation

\underline{ρ}_{Φ_{λ}} : (G(l)^{alg})′ → R_{k_{λ}/𝔽_{l}}(Sp_{2h}).

Now by (5.3) we have the natural isomorphisms:

∏_{k_{λ}↪\overline{𝔽}_{l}} \overline{𝔽}_{l} ≅ k_{λ} ⊗_{𝔽_{l}} \overline{𝔽}_{l} ≅ End_{k_{λ}⊗_{𝔽_{l}} \overline{𝔽}_{l}[G_{F}]}(A_{λ}[λ] ⊗_{𝔽_{l}} \overline{𝔽}_{l}) ≅

≅ End_{k_{λ}⊗_{𝔽_{l}} \overline{𝔽}_{l}[G_{F}]}(A_{λ}[λ] ⊗_{k_{λ}} k_{λ} ⊗_{𝔽_{l}} \overline{𝔽}_{l}) ≅

(6.15)  ≅ ∏_{k_{λ}↪\overline{𝔽}_{l}} End_{\overline{𝔽}_{l}[G_{F}]}(A_{λ}[λ] ⊗_{k_{λ}} \overline{𝔽}_{l}).

Note that Z(Sp_{2h}) ≅ μ_{2} and this isomorphism holds over any field of definition. The isomorphisms (6.15) imply by the Schur’s Lemma:

\underline{ρ}_{Φ_{λ}}(Z((G(l)^{alg})′)) ⊂ R_{k_{λ}/𝔽_{l}}(μ_{2}).

Hence

Z((G(l)^{alg})′) ⊂ ∏_{λ|l} R_{k_{λ}/𝔽_{l}}(μ_{2}) = Z(∏_{λ|l} R_{k_{λ}/𝔽_{l}}(Sp_{2h})).

Observe that both groups (G(l)^{alg})′ and ∏_{λ|l} R_{k_{λ}/𝔽_{l}}(Sp_{2h}) are reductive. Now the proof is finished in the same way as the proof of Lemma 3.4 in [BGK]. □

Theorem 6.16. Let A/F be an abelian variety of class A. Then for l ≫ 0, we have:

(6.17)  \overline{ρ_{l}}(G′_{F}) = ∏_{λ|l} Sp_{2h}(k_{λ}) = Sp_{2h}(O_{E}/lO_{E}),

(6.18)  ρ_{l}(\overline{G′_{F}}) = ∏_{λ|l} Sp_{2h}(O_{λ}) = Sp_{2h}(O_{E} ⊗_{ℤ} ℤ_{l}),

where \overline{ρ_{l}} is the representation ρ_{l} mod l and \overline{G′_{F}} is the closure of the commutator subgroup G′_{F} ⊂ G_{F} computed with respect to the natural profinite topology of G_{F} .

Proof. To prove the equality (6.17), note that the group scheme ∏_{λ|l} R_{k_{λ}/𝔽_{l}}(Sp_{2h}) is simply connected, since its base change to \overline{𝔽}_{l} is ∏_{λ|l} ∏_{k_{λ}↪\overline{𝔽}_{l}} Sp_{2h}/\overline{𝔽}_{l}, which is clearly simply connected. From now on the argument is the same as in the proof of Theorem 3.5 in [BGK]. Namely: it follows by (6.11) that (G(l)^{alg})′ is simply connected. So (G(l)^{alg})′(𝔽_{l}) = (G(l)^{alg})′(𝔽_{l})_{u}. Hence, by a theorem of Serre (cf. [Wi], Th.4) we get

(G(l)^{alg})′(𝔽_{l}) ⊂ (\overline{ρ_{l}}(G_{F}))′ = \overline{ρ_{l}}(G′_{F}).

On the other hand, by definition of the group G(l)^{alg}, it is clear that

\overline{ρ_{l}}(G′_{F}) = (\overline{ρ_{l}}(G_{F}))′ ⊂ (G(l)^{alg})′(𝔽_{l}).

As for the second equality in (6.18) we have

(6.19)  ρ_{l}( \overline{G′_{F}} ) = \overline{(ρ_{l}(G_{F}))′} ⊂ ∏_{λ|l} Sp_{2h}(O_{λ}),

where \overline{(ρ_{l}(G_{F}))′} denotes the closure of (ρ_{l}(G_{F}))′ in the natural (λ-adic in each factor) topology of the group ∏_{λ|l} Sp_{2h}(O_{λ}). Using equality (6.17) and Lemma 6.20 stated below, applied to X = \overline{(ρ_{l}(G_{F}))′}, we finish the proof. □

Lemma 6.20. Let X be a closed subgroup in ∏_{λ|l} Sp_{2h}(O_{λ}) such that its image via the reduction map

∏_{λ|l} Sp_{2h}(O_{λ}) → ∏_{λ|l} Sp_{2h}(k_{λ})

is all of ∏_{λ|l} Sp_{2h}(k_{λ}). Then X = ∏_{λ|l} Sp_{2h}(O_{λ}).

Proof. The proof is similar to the proof of Lemma 3 in [Se] chapter IV, 3.4. □

7. Applications to classical conjectures.

Choose an imbedding of F into the field of complex numbers ℂ. Let V = H^{1}(A(ℂ), ℚ) be the singular cohomology group with rational coefficients. Consider the Hodge decomposition

V ⊗_{ℚ} ℂ = H^{1,0} ⊕ H^{0,1},

where H^{p,q} = H^{p}(A; Ω_{A/ℂ}^{q}) and \overline{H^{p,q}} = H^{q,p}. Observe that H^{p,q} are invariant subspaces with respect to D = End\overline{F} (A) ⊗ ℚ action on V ⊗_{ℚ} ℂ. Hence, in particular H^{p,q} are E-vector spaces. Let

ψ : V × V → ℚ

be the ℚ-bilinear, nondegenerate, alternating form coming from the Riemann form of A. Since A has a principal polarization by assumption, the form ψ is given by the standard matrix

J = ( \begin{matrix}0 & I_{g} \\ −I_{g} & 0 \end{matrix} ) .

Define the cocharacter

μ_{∞} : 𝔾_{m}(ℂ) → GL(V ⊗_{ℚ} ℂ) = GL_{2g}(ℂ)

such that, for any z ∈ ℂ^{×}, the automorphism μ_{∞}(z) is the multiplication by z on H^{1,0} and the identity on H^{0,1}.

Definition 7.1. The Mumford-Tate group of the abelian variety A/F is the smallest algebraic subgroup MT (A) ⊂ GL_{2g}, defined over ℚ, such that MT (A)(ℂ) contains the image of μ_{∞}. The Hodge group H(A) is by definition the connected component of the identity in MT (A) ∩ SL_{V} ≅ MT (A) ∩ SL_{2g}.

We refer the reader to [D] for an excellent exposition on the Mumford-Tate group. In particular, MT (A) is a reductive group loc. cit. Since, by definition

μ_{∞}(ℂ^{×}) ⊂ GSp_{(V, ψ)}(ℂ) ≅ GSp_{2g}(ℂ),

it follows that the group MT (A) is a reductive subgroup of the group of symplectic similitudes GSp_{(V, ψ)} ≅ GSp_{2g} and that

(7.2)  H(A) ⊂ Sp_{(V, ψ)} ≅ Sp_{2g}.

Remark 7.3. Let V be a finite dimensional vector space over a field K such that it is also an R-module for a K-algebra R. Let G be a K-group subscheme of GL_{V} . Then by the symbol C_{R}(G) we will denote the commutant of R in G. The symbol C_{R}^{∘}(G) will denote the connected component of identity in C_{R}(G). Let β : V × V → K be a bilinear form and let G_{(V,β)} ⊂ GL_{V} be the subscheme of GL_{V} of all isometries with respect to the bilinear form β. It is easy to check that C_{R}(G_{(V,β)}) ⊗_{K} L ≅ C_{R⊗_{K}L}(G_{(V ⊗_{K}L, β⊗_{K}L)}). Note that MT (A) ⊂ C_{D}(GSp_{(V, ψ)}) by definitions.

Definition 7.4. The algebraic group L(A) = C_{D}^{∘}(Sp_{(V, ψ)}) is called the Lefschetz group of a principally polarized abelian variety A. Note that the group L(A) does not depend on the form ψ cf. [R2].

By [D], Sublemma 4.7, there is a unique E-bilinear, nondegenerate, alternating pairing

ϕ : V × V → E

such that T r_{E/ℚ}(ϕ) = ψ. Taking into account that the actions of H(A) and L(A) on V commute with the E-structure, we get

(7.5)  H(A) ⊂ L(A) ⊂ R_{E/ℚ}Sp_{(V, ϕ)} ⊂ Sp_{(V, ψ)}.

But R_{E/ℚ}(Sp_{(V, ϕ)}) = C_{E}(Sp_{(V, ψ)}) hence C_{D}(R_{E/ℚ}(Sp_{(V, ϕ)})) = C_{D}(Sp_{(V, ψ)}) so

(7.6)  H(A) ⊂ L(A) = C_{D}^{∘}(R_{E/ℚ}(Sp_{(V, ϕ)})) ⊂ C_{D}(R_{E/ℚ}(Sp_{(V, ϕ)})).

Definition 7.7. If L/ℚ is a field extension of ℚ we put

MT (A)_{L} := MT (A) ⊗_{ℚ} L, H(A)_{L} := H(A) ⊗_{ℚ} L, L(A)_{L} := L(A) ⊗_{ℚ} L.

Conjecture 7.8 (Mumford-Tate cf. [Se5], C.3.1). If A/F is an abelian variety over a number field F, then for any prime number l

(7.9)  (G_{l}^{alg})^{∘} = MT (A)_{ℚ_{l}},

where (G_{l}^{alg})^{∘} denotes the connected component of the identity.

Theorem 7.10 (Deligne [D], I, Prop. 6.2). If A/F is an abelian variety over a number field F and l is a prime number, then

(7.11)  (G_{l}^{alg})^{∘} ⊂ MT (A)_{ℚ_{l}}.

Theorem 7.12. The Mumford-Tate conjecture holds true for abelian varieties of class A defined in the beginning of Section 6.

Proof. By [LP1], Theorem 4.3, it is enough to verify (7.9) for a single prime l only. We use the equality (6.10) for a big enough prime l. The proof goes similarly to the proof of Theorem 3.6 in [BGK]. In the proof we will make some additional computations, which provide an extra information on the Hodge group H(A). The Hodge group H(A) is semisimple (cf. [G], Prop. B.63) and the center of MT (A) is 𝔾_{m} (cf. [G], Cor. B.59). Since MT (A) = 𝔾_{m}H(A), we get

(7.13)  (MT (A)_{ℚ_{l}})′ = (H(A)_{ℚ_{l}})′ = H(A)_{ℚ_{l}}.

By (7.11), (7.13) and (6.10)

(7.14)  ∏_{λ|l} R_{E_{λ}/ℚ_{l}}(Sp_{(V_{λ},ψ_{λ}^{0})}) ≅ ∏_{λ|l} R_{E_{λ}/ℚ_{l}}(Sp_{2h}) ⊂ H(A)_{ℚ_{l}}.

On the other hand by (7.6)

(7.15)  H(A)_{ℚ_{l}} ⊂ L(A)_{ℚ_{l}} ⊂ C_{D}(R_{E/ℚ}(Sp_{(V, ϕ)})) ⊗_{ℚ} ℚ_{l}.

Since R_{E/ℚ}(Sp_{(V, ϕ)}) = C_{E}(Sp_{(V, ψ)}), by Remark 7.3, formulae (7.14) and (7.15) we get:

(7.16)  ∏_{λ|l} R_{E_{λ}/ℚ_{l}}(Sp_{(V_{λ},ψ_{λ}^{0})}) ⊂ ∏_{λ|l} C_{D_{λ}}(R_{E_{λ}/ℚ_{l}}(Sp_{(V_{λ}(A), ψ_{λ}^{0})})).

For A of type I, D_{λ} = E_{λ} and V_{λ}(A) = V_{λ} hence, trivially, the inclusion (7.16) is an equality. Assume that A is of type II. Since V_{λ}(A) = V_{λ} ⊕ V_{λ} and D_{λ} = M_{2,2}(E_{λ}), evaluating both sides of the inclusion (7.16) on the \overline{ℚ}_{l}-points, we get equality with both sides equal to

∏_{λ|l} ∏_{E_{λ}↪\overline{ℚ}_{l}}(Sp_{(V_{λ},ϕ_{λ}|_{V_{λ}})})(\overline{ℚ}_{l})

which is an irreducible algebraic variety over \overline{ℚ}_{l}. Then we use Prop. II, 2.6 and Prop. II, 4.10 of [H] in order to conclude that the groups H(A)_{\overline{ℚ}_{l}}, L(A)_{\overline{ℚ}_{l}} and C_{D}(R_{E/ℚ}(Sp_{(V, ϕ)})) ⊗_{ℚ} \overline{ℚ}_{l} are connected. Hence all the groups H(A), L(A) and C_{D}(R_{E/ℚ}(Sp_{(V, ϕ)})) are connected, and we have

(7.17)  ∏_{λ|l} R_{E_{λ}/ℚ_{l}}(Sp_{(V_{λ},ϕ_{λ}|_{V_{λ}})}) ≅ ∏_{λ|l} R_{E_{λ}/ℚ_{l}}(Sp_{2h}) =
= H(A)_{ℚ_{l}} = L(A)_{ℚ_{l}} = C_{D}(R_{E/ℚ}(Sp_{(V, ϕ)})) ⊗_{ℚ} ℚ_{l}.

By (6.10), (7.17) and [Bo], Corollary 1. p. 702 we get

(7.18)  MT (A)_{ℚ_{l}} = 𝔾_{m}H(A)_{ℚ_{l}} = 𝔾_{m}(G_{l}^{alg})′ ⊂ G_{l}^{alg}.

The Theorem follows by (7.11) and (7.18). □

Corollary 7.19. If A is an abelian variety of class A, then

(7.20)  H(A)_{ℚ} = L(A)_{ℚ} = C_{D}(R_{E/ℚ}(Sp_{(V, ϕ)})) = C_{D}(Sp_{(V, ψ)}).

Proof. Taking Lie algebras of groups in (7.17) we deduce by a simple dimension argument that

(7.21)  Lie H(A) = Lie L(A) = Lie C_{D}(R_{E/ℚ}(Sp_{(V, ϕ)})).

In the proof of Theorem 7.12 we have showed that the groups H(A), L(A) and C_{D}(R_{E/ℚ}(Sp_{(V, ϕ)})) are connected. Hence, by Theorem p. 87 of [H1] we conclude that

(7.22)  H(A) = L(A) = C_{D}(R_{E/ℚ}(Sp_{(V, ϕ)})).  □

Corollary 7.23. If A is an abelian variety of class A, then for all l:

(7.24)  H(A)_{ℚ_{l}} = ∏_{λ|l} C_{D_{λ}}(R_{E_{λ}/ℚ_{l}}(Sp_{(V_{λ}(A), ϕ⊗_{ℚ}E_{λ})})).

In particular, for l ≫ 0 we get

(7.25)  H(A)_{ℚ_{l}} = ∏_{λ|l} R_{E_{λ}/ℚ_{l}}(Sp_{(V_{λ}, ϕ⊗_{ℚ}E_{λ})}).

Proof. Equality (7.24) follows immediately from Corollary 7.19. Equality (7.25) follows then from (7.17). □

We have:

H^{1}(A(ℂ); ℝ) ≅ V ⊗_{ℚ} ℝ ≅ ⨁_{σ:E↪ℝ} V ⊗_{E,σ} ℝ.

Put V_{σ}(A) = V ⊗_{E,σ} ℝ and let ϕ_{σ} be the form ϕ ⊗_{E,σ} ℝ : V_{σ}(A) ⊗_{ℝ} V_{σ}(A) → ℝ.

Lemma 7.26. If A is simple, principally polarized abelian variety of type II, then for each σ : E ↪ ℝ there is an ℝ-vector space W_{σ}(A) of dimension \frac{g}{e} = \frac{4 dimA}{[D: ℚ]} such that:

(i) V_{σ}(A) ≅ W_{σ}(A) ⊕ W_{σ}(A), (ii) the restriction of ϕ ⊗_{ℚ} ℝ to W_{σ}(A) gives a nondegenerate, alternating

pairing

ψ_{σ} : W_{σ}(A) × W_{σ}(A) → ℝ.

Proof. Using the assumption that D ⊗_{ℚ} ℝ ≅ M_{2,2}(ℝ) the proof is similar to the proof of Theorem 5.4. □

We put

W_{∞,σ} = \{ \begin{matrix}V_{σ}(A) & if A is of type I \\ W_{σ}(A) , & if A is of type II \end{matrix}

and

ψ_{σ} = \{ \begin{matrix}ϕ_{σ} & if A is of type I \\ ϕ_{σ}|_{W_{σ}(A)} , & if A is of type II. \end{matrix}

Observe that

dim_{ℝ} W_{∞,σ} = \{ \begin{matrix}\frac{2g}{e} = \frac{2 dimA}{[D: ℚ]} & if A is of type I \\ \frac{g}{e} = \frac{4 dimA}{[D: ℚ]} , & if A is of type II. \end{matrix}

Corollary 7.27. If A is an abelian variety of class A, then

(7.28)  H(A)_{ℝ} = L(A)_{ℝ} = ∏_{σ:E↪ℝ} Sp_{(W_{∞,σ}, ψ_{σ})}

(7.29)  H(A)_{ℂ} = L(A)_{ℂ} = ∏_{σE↪ℝ} Sp_{(W_{∞,σ}⊗_{ℂ}ℂ, ψ_{σ}⊗_{ℝ}ℂ)}.

Proof. It follows from Lemma 7.26 and Corollary 7.19. □

We recall the conjectures of Tate and Hodge in the case of abelian varieties. See [G], [K] and [T1] for more details.

Conjecture 7.30 (Hodge). If A/F is a simple abelian variety over a number field F, then for every 0 ≤ p ≤ g the natural cycle map induces an isomorphism

(7.31)  A^{p}(A) ≅ H^{2p}(A(ℂ); ℚ) ∩ H^{p,p},

where A^{p}(A) is the ℚ-vector space of codimension p algebraic cycles on A modulo the homological equivalence.

Conjecture 7.32 (Tate). If A/F is a simple abelian variety over a number field F and l is a prime number, then for every 0 ≤ p ≤ g the cycle map induces an isomorphism:

(7.33)  A^{p}(A) ⊗_{ℚ} ℚ_{l} ≅ H_{et}^{2p}(\overline{A}; ℚ_{l}(p))^{G_{F}}

where \overline{A} = A ⊗_{F} \overline{F} .

Theorem 7.34. The Hodge conjecture holds true for abelian varieties of class A.

Proof. By [Mu], Theorem 3.1 the Hodge conjecture follows from the equality (7.20) of Corollary 7.19. □

Theorem 7.35. The Tate conjecture holds true for abelian varieties of class A.

Proof. It is known (see Proposition 8.7 of [C1]) that Mumford-Tate conjecture implies the equivalence of Tate and Hodge conjectures. Hence the Tate conjecture follows by Theorems 7.12 and 7.34. □

Conjecture 7.36 (Lang). Let A be an abelian variety over a number field F. Then for l ≫ 0 the group ρ_{l}(G_{F}) contains the group of all homotheties in GL_{T_{l}(A)}(ℤ_{l}).

Theorem 7.37 (Wintenberger [Wi], Cor. 1, p. 5). Let A be an abelian variety over a number field F. The Lang conjecture holds for such abelian varieties A for which the Mumford-Tate conjecture holds or if dim A < 5.

Theorem 7.38. The Lang’s conjecture holds true for abelian varieties of class A.

Proof. It follows by Theorem 7.12 and Theorem 7.37. □

We are going to use Theorem 7.12 and Corollary 7.19 to prove an analogue of the open image Theorem of Serre cf. [Se8]. We start with the following remark which is a plain generalization of remark 7.3.

Remark 7.39. Let B_{1} ⊂ B_{2} be two commutative rings with identity. Let Λ be a free, finitely generated B_{1}-module such that it is also an R-module for a B_{1}- algebra R. Let G be a B_{1}-group subscheme of GL_{Λ}. Then C_{R}(G) will denote the commutant of R in G. The symbol C_{R}^{∘}(G) will denote the connected component of identity in C_{R}(G). Let β : Λ × Λ → B_{1} be a bilinear form and let G_{(Λ,β)} ⊂ GL_{Λ} be the subscheme of GL_{Λ} of the isometries with respect to the form β. Then we check that C_{R}(G_{(Λ,β)}) ⊗_{B_{1}} B_{2} ≅ C_{R⊗_{B_{1}} B_{2}}(G_{(Λ⊗_{B_{1}} B_{2}, β⊗_{B_{1}} B_{2})}).

Consider the bilinear form:

(7.40)  ψ : Λ × Λ → ℤ

associated with the variety A. Abusing notation sligthly, we will denote by ψ the Riemann form ψ ⊗_{ℤ} ℚ, i.e., we put:

ψ : V × V → ℚ.

Consider the group scheme C_{R}(Sp_{(Λ, ψ)}) over Spec ℤ. Since C_{R}(Sp_{(Λ, ψ)}) ⊗_{ℤ} ℚ = C_{D}(Sp_{(V, ψ)}) (see Remark 7.39), there is an open imbedding in the l-adic topology:

(7.41)  C_{R}(Sp_{(Λ, ψ)})(ℤ_{l}) ⊂ C_{D}(Sp_{(V, ψ)})(ℚ_{l}).

Note that the form ψ_{l} of (4.1) is obtained by tensoring (7.40) with ℤ_{l}.

Theorem 7.42. If A is an abelian variety of class A, then for every prime number l, ρ_{l}(G_{F}) is open in the group

C_{R}(GSp_{(Λ, ψ)})(ℤ_{l}) = C_{R⊗_{ℤ}ℤ_{l}}(GSp_{(T_{l}(A), ψ_{l})})(ℤ_{l}).

In addition, for l ≫ 0 we have:

(7.43)  ρ_{l}(\overline{G′_{F}}) = C_{R}(Sp_{(Λ, ψ)})(ℤ_{l}).

Proof. For any ring with identity R the group GSp_{2g}(R) is generated by subgroups Sp_{2g}(R) and

\{( \begin{matrix}aI_{g} & 0 \\ 0 & I_{g} \end{matrix} ); a ∈ R^{×}\}.

One checks easily that the group ℤ_{l}^{×}Sp_{2g}(ℤ_{l}) has index 2 (index 4 resp.) in GSp_{2g}(ℤ_{l}), for l > 2 (for l = 2 resp.). Here the symbol ℤ_{l}^{×} stands for the subgroup of homotheties in GL_{2g}(ℤ_{l}). Since by assumption A has a principal polarization, Sp_{2g}(ℤ_{l}) ≅ Sp_{(Λ, ψ)})(ℤ_{l}). By [Bo], Cor. 1. on p. 702, there is an open subgroup U ⊂ ℤ_{l}^{×} such that U ⊂ ρ_{l}(G_{F}). Hence U C_{R}(Sp_{(Λ, ψ)})(ℤ_{l}) = C_{R}(U Sp_{(Λ, ψ)}(ℤ_{l})) is an open subgroup of C_{R}(GSp_{(Λ, ψ)})(ℤ_{l}) = C_{R}(GSp_{(Λ, ψ)}(ℤ_{l})). By [Bo], Th. 1, p. 701, the group ρ_{l}(G_{F}) is open in G_{l}^{alg}(ℚ_{l}). By Theorem 7.12, Corollary 7.19 and Remark 7.3

U C_{R}(Sp_{(Λ, ψ)})(ℤ_{l}) ⊂ ℚ_{l}^{×} C_{D}(Sp_{(V, ψ)})(ℚ_{l}) =

(7.44)  = 𝔾_{m}(ℚ_{l})H(A)(ℚ_{l}) ⊂ MT (A)(ℚ_{l}) = G_{l}^{alg}(ℚ_{l}).

Hence, U C_{R}(Sp_{(Λ, ψ)})(ℤ_{l}) ∩ ρ_{l}(G_{F}) is open in U C_{R}(Sp_{(Λ, ψ)})(ℤ_{l}) and we get that ρ_{l}(G_{F}) is open in C_{R}(GSp_{(Λ, ψ)})(ℤ_{l}). Using Remark 7.39 and the universality of the fiber product, we observe that

(7.45)  C_{R}(Sp_{(Λ, ψ)})(ℤ_{l}) = C_{R⊗_{ℤ}ℤ_{l}}(Sp_{(T_{l}(A), ψ_{l})})(ℤ_{l}).

For l ≫ 0 we get

C_{R⊗_{ℤ}ℤ_{l}}(Sp_{(T_{l}(A), ψ_{l})}) ≅ C_{R⊗_{ℤ}ℤ_{l}}(C_{O_{E}⊗_{ℤ}ℤ_{l}}(Sp_{(T_{l}(A), ψ_{l})})) ≅

(7.46)  ≅ C_{R⊗_{ℤ}ℤ_{l}}(∏_{λ|l} R_{O_{λ}/ℤ_{l}}(Sp_{(T_{λ}(A), ψ_{λ})})).

Evaluating the group schemes in (7.46) on Spec ℤ_{l} we get

C_{R⊗_{ℤ}ℤ_{l}}(Sp_{(T_{l}(A), ψ_{l})})(ℤ_{l}) ≅ C_{R⊗_{ℤ}ℤ_{l}}(∏_{λ|l} R_{O_{λ}/ℤ_{l}}(Sp_{(T_{λ}(A), ψ_{λ})}))(ℤ_{l}) ≅

(7.47) ≅ ∏_{λ|l} C_{R_{λ}}Sp_{(T_{λ}(A), ψ_{λ})}(O_{λ}) ≅ ∏_{λ|l} Sp_{(T_{λ}, ψ_{λ})}(O_{λ}) ≅ ∏_{λ|l} Sp_{2h}(O_{λ}).

Hence by (7.45), (7.46), (7.47), (6.18) and Theorem 7.38, we conclude that for l ≫ 0 the equality (7.43) holds. □

Theorem 7.48. If A is an abelian variety of class A, then for every prime number l, the group ρ_{l}(G_{F}) is open in the group G_{l}^{alg}(ℤ_{l}) in the l-adic topology.

Proof. By Theorem 7.42 the group ρ_{l}(G_{F}) is open in C_{R⊗_{ℤ}ℤ_{l}}(GSp_{(T_{l}(A), ψ_{l})})(ℤ_{l}) in the l-adic topology, so ρ_{l}(G_{F}) has a finite index in the group C_{R⊗_{ℤ}ℤ_{l}}(GSp_{(T_{l}(A), ψ_{l})})(ℤ_{l}). By the definition of G_{l}^{alg}, we have:

ρ_{l}(G_{F}) ⊂ G_{l}^{alg}(ℤ_{l}) ⊂ C_{R⊗_{ℤ}ℤ_{l}}(GSp_{(T_{l}(A), ψ_{l})})(ℤ_{l}).

Hence, ρ_{l}(G_{F}) has a finite index in G_{l}^{alg}(ℤ_{l}), and the claim follows since C_{R⊗_{ℤ}ℤ_{l}}(GSp_{(T_{l}(A), ψ_{l})})(ℤ_{l}) is a profinite group. □

Acknowledgements. The first author would like to thank G. Faltings for conversations concerning l-adic representations associated with abelian varieties. The first and the second authors would like to thank Mathematics Department of Ohio State University and Max Planck Institut in Bonn for financial support during visits in 2003 and 2004. The second author thanks the CRM in Barcelona for the hospitality during the visit in June and July 2004. The research has been partially sponsored by a KBN grant 1 P03A 027 27 and by Marie Curie Research Training Network ”Arithmetic Algebraic Geometry” MRTN-CT-2003-504917.

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G. Banaszak Department of Mathematics Adam Mickiewicz University Poznań Poland banaszak@amu.edu.pl W. Gajda Department of Mathematics Adam Mickiewicz University Poznań Poland gajda@amu.edu.pl

P. Krasoń

Department of Mathematics Szczecin University Szczecin Poland krason@sus.univ.szczecin.pl 76

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