ON THE MUMFORD-TATE CONJECTURE FOR ABELIAN VARIETIES WITH REDUCTION CONDITIONS

Alex Lesin

Abstract. We study monodromy action on abelian varieties satisfying certain bad reduction conditions. These conditions allow us to get some control over the Galois image. As a consequence we verify the Mumford-Tate conjecture for such abelian varieties.

Contents

Introduction I. Preliminaries, first applications

0. Basics and notations

1. On the Hodge-Tate decomposition

2. Abelian 4-folds II. Abelian varieties with reduction conditions

3. Bad reduction and monodromy action

4. Minimal reduction

5. Applications of minimal reduction

6. Another type of bad reduction

7. A curious result References

Introduction

There are two long outstanding conjectures due to Hodge and Tate related to the structure of the ring of algebraic cycles modulo homological equivalence. The Mumford-Tate conjecture implies that for abelian varieties the two are equivalent. The focus of this work is on the Mumford-Tate conjecture for special classes of abelian varieties.

Let X be a smooth projective algebraic variety over ℂ. Then its r-th cohomology group admits the Hodge decomposition: H^{r}(X, ℂ) = ⊕_{p+q=r}H^{p,q}(X). The Hodge cycles are those rational cohomology classes, i.e., elements of H˙ (X, ℚ) that sit in the components H^{p,p}(X) via the canonical embedding H˙ (X, ℚ) ↪ H˙ (X, ℂ). Denote H^{p}(X) := H^{p,p}(X) ∩ H^{2p}(X, ℚ) the group of codimension p Hodge cycles. Then that H(X) := ⊕_{p}H^{p}(X) has a ring structure with respect to the cup-product. It is immediate that rational linear combinations of the cohomology classes of algebraic subvarieties in X (=: algebraic cycles) are Hodge. The Hodge conjecture claims that the converse is also true, viz., all the Hodge cycles are algebraic.

1991 Mathematics Subject Classification. 11G10, 14C25, 14C30.

Key words and phrases. abelian variety, algebraic cycles, Galois representations, monodromy action, Mumford-Tate conjecture, Tate conjecture, Hodge conjecture.

Research at MSRI supported in part by N.S.F. grant #DMS 9022140.

The only general result in this direction is the Lefschetz (1,1)-theorem asserting algebraicity of all codimension 1 Hodge cycles (= rational (1,1) cohomology classes, hence the name). Denote D(X) the subring of H(X) generated by H^{1}, D^{p}(X) := D(X) ∩ H^{p}(X) is the group of codimension p cycles which are linear combinations of cup-products of divisors. If D(X) = H(X), then the (1,1)-theorem implies the Hodge conjecture.

On the other hand, for an algebraic variety defined over an algebraic number field, say K ⊂ \overline{ℚ}, one can consider ℓ-adic étale cohomology H_{ ˙ét}(X\overline{ℚ}, ℚ_{ℓ}). The Galois group Gaℓ(\overline{ℚ}/K) acts continuously on H_{ ˙ét}(X\overline{ℚ}, ℚ_{ℓ}). If F is a finite extension of K, then the open subgroup Gaℓ(\overline{ℚ}/F) of Gaℓ(\overline{ℚ}/K) acts by the p^{th} power of the inverse of the cyclotomic character χ_{ℓ} on the cohomology classes of F -rational codimension p algebraic cycles. If for an arbitrary ℓ-adic Galois representation W and an integer n ∈ ℤ W (n) := W ⊗χ_{ℓ}^{n} denotes the n^{th} Tate twist of W, then the these cohomology classes are in H_{ét}^{2p}(X\overline{ℚ}, ℚ_{ℓ})(p)^{Gaℓ(\overline{ℚ}/F)}. By a codimension p Tate cycle we mean a cohomology class in H_{ét}^{2p}(X\overline{ℚ}, ℚ_{ℓ})(p) fixed by an open subgroup of Gaℓ(\overline{ℚ}/K). Since H_{ ˙ét}(X\overline{ℚ}, ℚ_{ℓ}) is a finite dimensional ℓ-adic representation, we can find the largest open subgroup of the Galois group fixing all the Tate cycles (hence these cycles are all defined over a large enough number field). Let 𝔤 be the Lie algebra of the image of the Galois group in End_{ℚ_{l}}(H_{ ˙ét}(X\overline{ℚ}, ℚ_{ℓ})). Then codimension p Tate cycles T_{ℓ}^{p}(X) are, by definition, the 𝔤-invariants H_{ét}^{2p}(X\overline{ℚ}, ℚ_{ℓ})(p)^{𝔤}. Tate [T 0] conjectured that all the Tate cycles are algebraic. We denote T_{ℓ} := ⊕_{p} T_{ℓ}^{p} the ring (under the cup-product) of the Tate cycles.

From now on, we restrict ourselves to the case of abelian varieties defined over number fields. On the one hand, this case is more concrete, and some progress has been made; on the other, it has important arithmetical applications.

Although not known in general, the analog of the (1,1)-theorem for the Tate cycles of codimension 1 for abelian varieties has been proved by Faltings [F]. Hence, as above, we can conclude that the Tate conjecture holds for an abelian variety A satisfying T_{ℓ}(A) = D_{ℓ}(A), where D_{ℓ}(A) is the ring of étale cohomology classes generated by divisors.

It is known that generically, but not always, the Hodge (resp. the Tate) cycles are all generated by divisors [Ma], [Ab 1].

The first (counter)example due to Mumford (cf. [Po]) features a CM abelian 4fold. Weil [W] has shown that the essential feature of Mumford’s example causing H = D to fail is an action in a special way of a quadratic imaginary field 𝕜 on an abelian variety. Namely, consider a family of abelian varieties of even dimension, say 2d, whose endomorphism algebra contains such a field 𝕜 with the signature of the 𝕜-action (d, d). Generically for such a family, the ring of Hodge cycles is generated by divisors together with an exceptional (non-divisorial) cycle of codimension d.

Recently, C. Schoen proved the Hodge conjecture for one family of abelian 4-folds of Weil type (with an action of ℚ(μ_{3})).

Our initial motivation was to answer a question of Tate on whether the Tate conjecture holds for this family (cf. [T 2], p. 82). The affirmative answer was obtained independently by Moonen-Zarhin (cf. [MZ]).

In general, both conjectures seem to be very difficult in codimensions > 1.

The existence of the comparison isomorphisms between the ℓ-adic and singular cohomology theories carrying algebraic cycles in one theory to another suggests that the Hodge and the Tate conjectures describe essentially the same object. So, it is natural to ask if the two conjectures are equivalent in some sense. The precise statement in the case of abelian varieties constitutes the Mumford-Tate conjecture, which we denote by MT. It asserts that the Hodge and the Tate conjectures are equivalent for an abelian variety and all its self-products.

Concretely, for an abelian variety A over a number field K, there exists a connected reductive algebraic subgroup Hg(A) of GL(V) defined over ℚ, V = H^{1}(A_{ℂ}, ℚ) (resp. a connected reductive algebraic subgroup G_{ℓ}(A) of GL(V_{ℓ}) defined over ℚ_{ℓ}, V_{ℓ} = H_{ét}^{1}(A\overline{ℚ}, ℚ_{ℓ}) for some prime number ℓ∈ℤ), such that the Hodge (resp. the Tate) cycles of codimension p are obtained as invariants in H^{2p}(A, ℚ) ≅ ⋀^{2p} V (resp. in H_{ét}^{2p}(A\overline{ℚ}, ℚ_{ℓ}) ≅ ⋀^{2p} V_{ℓ}) of 𝔥 = Lie(Hg(A)) (resp. 𝔤_{ℓ} = Lie(G_{ℓ}(A))). Because of the comparison isomorphism V_{ℓ} ≅ V ⊗_{ℚ} ℚ_{ℓ} between the two cohomology theories, Hg_{ℓ}(A) := Hg(A) ⊗_{ℚ} ℚ_{ℓ} acts on V_{ℓ}. Let 𝔥_{ℓ} := Lie(Hg_{ℓ}) = 𝔥 ⊗ ℚ_{ℓ}. MT asserts that 𝔤_{ℓ} = 𝔥_{ℓ}. (Note that 𝔤_{ℓ}(A) is not the Lie algebra of the image of Gaℓ(\overline{ℚ}/K) in GL(V_{ℓ}), but the intersection of the Lie algebra of this image with 𝔰𝔩(V_{ℓ}). That is why we do not Tate-twist the étale cohomology group.)

Deligne, Piatetskii-Shapiro and Borovoi proved a “half” of MT, viz., 𝔥_{ℓ}(A) ⊇ 𝔤_{ℓ}(A). Hence the Tate conjecture implies the Hodge conjecture.

MT for abelian varieties of CM-type is a consequence of the results of Shimura and Taniyama (cf. [ShT], [Po]). This must have been the motivating factor behind Mumford-Tate.

MT has been proved in a few (non-CM) cases by imposing restrictions on the size of the endomorphism algebra and adding some divisibility conditions on the dimension of abelian varieties, cf. [S 0], [C 0]. In [Z 4], [Z 5], [LZ] MT was verified for abelian varieties satisfying certain conditions on the Galois action at a prime of good reduction. In all these cases the Tate cycles are generated by divisors, hence the Tate conjecture holds. The result now follows from the “known half” of MT.

The main thrust of this work is to show that under suitable bad reduction conditions we can control the image of Galois; in particular, MT holds for a class of abelian varieties, including some Weil-type abelian varieties (for which the Tate conjecture is not known).

Note that if A is an absolutely simple abelian variety, e = (End^{∘}(A) : ℚ) the degree over ℚ of its endomorphism algebra, and A has bad reduction at some prime ℘, then e divides the dimension of the toric part of the reduction.

The following is the main result of this work (see Theorem 6.4)

Main Theorem. Let A be an absolutely simple abelian variety, End^{∘}(A) = 𝕜 : imaginary quadratic field, g = dim(A). Assume A has bad semi-stable reduction at some prime ℘, with the dimension of the toric part of the reduction equal to 2r, and gcd(r, g) = 1, and (r, g) ≠ (15, 56) or (m − 1, \frac{m(m+1)}{2}). Then MT holds.

Roughly speaking, the idea is the following. If A is an abelian variety with bad semi-stable reduction at some prime ℘ (of its field of definition), then the action of the inertia at ℘ on on the ℓ-adic (℘ ∤ ℓ) Tate module of A is unipotent of “rank” equal to the dimension of the toric part of the identity component of the the special fiber of the Nèron model of A at ℘ (=: toric rank). If A satisfies the conditions on the “size” of the endomorphism algebra and the toric rank imposed above, then the rank of the unipotents (in the inertia image) is prime to the dimension of the Galois representation, which is a sufficiently restrictive condition, given our knowledge of the possible Galois representations arising in this situation.

Note that people have looked at the special elements in the monodromy action before (cf. 6.8).

As mentioned above, MT is known to hold for CM abelian varieties. It is also known (e.g., [ST]) that such abelian varieties have good reduction at all primes, after possibly a finite base change. But the set of abelian varieties with (potentially) good reduction everywhere is “small” in a corresponding moduli space, i.e., it is a very rare occasion that an abelian variety has everywhere (potentially) good reduction. Indeed, such abelian varieties correspond to “integral” points of the moduli space (cf. loc. cit., Remark (1), p. 498) and as “sparse” as integers in a number field.

So, “most” of the abelian varieties do have bad reduction “somewhere.” We have reasons to believe that abelian varieties with minimal bad reduction (e.g., the case r = 1 of Theorem A) are the “most typical” (cf. [L]).

Along the way we established various other results. They are:

• If A is an abelian variety with End^{∘}(A) = ℚ and the dimension of the toric part of its reduction is either 2 or prime to 2 dim(A), then MT holds (Theorems 6.5, 6.6).

• MT holds for some abelian 4-folds A with End^{∘}(A) = ℚ (Theorem 5.2).

• For some abelian varieties, either MT or the Hodge conjecture holds (Theorem 7.1, Remark 7.2(4)).

The question of existence of abelian varieties considered here and the “size” of the set of such varieties in the corresponding moduli spaces is addressed in [L].

This work started with 4-dimensional case, part of which was independently obtained by Moonen-Zarhin. Since this has been published ([MZ]), we do not treat this case here (see, however, section 2).

Acknowledgment. I greatly benefited from conversations with many people over the period of time I was working on this paper. I would like to thank my advisor, Dinakar Ramakrishnan, for encouragement and guidance throughout the research and preparation of my PhD thesis (at Caltech) in which most of these results appeared. I am especially grateful to Don Blasius, Haruzo Hida and Michael Aschbacher for numerous helpful remarks and enlightening comments.

This work could not have been possible without generous help of Fedor Bogomolov. I can hardly thank him enough.

I. Preliminaries, first applications

0. Basics and Notation

0.0. Let A be a simple (:=absolutely simple) abelian variety defined over some number field, say, K ↪ \overline{ℚ} with a fixed embedding, D := End^{∘}(A) := End∘ ; \overline{ℚ}(A) ⊗_{ℤ} ℚ, V := H_{1}(A(ℂ), ℚ). Then D ↪ End_{ℚ}(V).

We recall the Albert classification of the possible types of the endomorphism algebras D (cf. [MAV, §20]):

I : D is a totally real field; II : D is a indefinite quaternion algebra over a totally real field F, i.e., D ⊗_{F} ℝ

is a sum of f := (F : ℚ) copies of M_{2}(ℝ); III : D is an definite quaternion algebra over a totally real field F, i.e., D ⊗_{F} ℝ

is a sum of f := (F : ℚ) copies of the Hamiltonian quaternions ℍ; IV : D is a division algebra over a CM-field; By abuse of language we say that A is “of type I (II, ...)” if its D is of this type.

0.1. Recall that V_{ℝ} := V ⊗_{ℚ} ℝ is given a complex structure induced by the natural isomorphism between V_{ℝ} and the universal covering space of A(ℂ) (cf. [MAV]). Therefore we obtain a homomorphism of algebraic groups,

φ : T → GL(V),

defined over ℝ, where T is the compact one-dimensional torus over ℝ, i.e., T_{ℝ} = {z ∈ ℂ | |z| = 1}, by the formula

φ(e^{iθ}) = the element of GL(V), which is multiplication by e^{iθ}

in the complex structure on V_{ℝ}.

Note that there is a non-degenerate skew symmetric (Riemann) form Θ : V ×V → ℚ and that φ satisfies the Riemann conditions (cf. [M 1]):

1. φ(T) ⊆ Sp(V, Θ),

2. Θ(v, φ(i) · v) > 0, ∀v ∈ V, x ≠ 0.

Definition. The Hodge group Hg(A) of A is the smallest algebraic subgroup of Sp(V) := Sp(V, Θ) defined over ℚ which after extension of scalars to ℝ contains the image of φ.

0.1.1. For the purpose of completeness and further reference, we give the following reformulation of the above construction and definition. The reference for what follows is [D 3, § 3].

Since A(ℂ) is a compact smooth Kähler manifold, V_{ℂ} := H_{1}(A(ℂ), ℂ) admits a Hodge decomposition

H_{1}(A(ℂ), ℂ) = H_{−1,0}(A) ⊕ H_{0,−1}(A).

Thus we obtain a homomorphism

μ : 𝔾_{m}_{,ℂ} → GL(V)_{ℂ}

by defining μ(z), ∀z ∈ ℂ^{×}, to be the automorphism of V_{ℂ} which is multiplication by z on H_{−1,0}(A) and by the identity on H_{0,−1}(A).

Definition. The Mumford-Tate group M(A) of A is the smallest algebraic subgroup of GL(V) defined over ℚ which after extension of scalars to ℂ contains the image of μ.

Clearly, over ℂ, M(A) is the subgroup of GL(V)_{ℂ} generated by the conjugates σμ, ∀σ ∈ Aut(ℂ).

Definition. The Hodge group Hg(A) of A (or the special Mumford-Tate group of A) is the connected component of the identity of the intersection M(A) ∩ SL(V) in GL(V).

Remarks. 1. The construction of M(A) furnishes it with a canonical character ν : M(A) → 𝔾_{m} defined over ℚ and characterized by the condition ν ∘ μ = id_{𝔾_{m}}. Then Hg(A) = Ker(ν). This is the reason why we use the Hodge group instead of the Mumford-Tate group (cf. also 0.3).

2. One can easily show that the two definitions of Hg(A) are equivalent.

0.1.2. The following theorem lists the properties of Hg(A) (cf. [M 1, § 2, Theorem (i)], [Ta, Lemma 1.4]).

Theorem.

1. Hg(A) is a connected reductive group.

2. D (= End^{∘}(A)) = End_{Hg(A)}(V) = End_{𝔥}(A), where 𝔥 := Lie(Hg(A)).

3. Hg(A) is semi-simple for an abelian variety A of type I, II or III.

4. Hg(A^{a} × B^{b}) ≅ Hg(A × B) for any abelian varieties A, B and a, b ∈ ℕ.

Remarks. 1. Part 2 of the theorem implies that A is simple if and only if V is 𝔥-simple, if and only if (Schur’s lemma) D is a division algebra.

2. One can refine part 3, cf. 0.7.1.

0.1.3. Recall that the Hodge classes of A are classes of type (p, p) in the Hodge decomposition of homology of A.

The Hodge conjecture states that all the Hodge classes are algebraic.

0.1.4. By the Künneth formula H_{.}(A) = ⋀^{.} H_{1}(A) (cf. [MAV]), hence Hg(A) acts on H_{.}(A). One can show (cf. [M 1]) that the Hodge classes of A are exactly those classes in H_{.}(A) that are fixed by Hg(A). In fact, the Hodge group has the following characteristic property (loc. cit., § 2, Corollary).

Theorem. The Hodge group Hg(A) is the largest (reductive) subgroup of GL(V) fixing all the Hodge classes of A^{s}, s ≥ 1.

By the Künneth formula H_{2}(A^{s}) = ⊕_{i=1}^{s}H_{2}(A). Hence, in the view of the previous theorem, the Lefschetz (1,1)-theorem for abelian varieties takes the following form.

Theorem. Let s ∈ ℕ, sV := V ⊕ ... ⊕ V (s times), then the 𝔥-invariants (⋀^{2} ; ℚ sV)^{𝔥}

is exactly the (ℚ-span of homological classes of) divisors on A^{s} = A × ... × A (s times).

0.1.5. Following Ribet and Murty (cf. [Ri], [Mu]) we make the following

Definition. The Lefschetz group L(A) of A is the connected component of the identity of the centralizer of End^{∘}(A) in Sp(V, Θ) (inside End_{ℚ}(V), Θ is a polarization, cf. 0.1).

The following are the main results about the Lefschetz group (cf. [Mu], also [Ri], [H 1]).

Theorem.

0.(i) L(A) is a connected reductive algebraic group defined over ℚ.

(ii) Hg(A) ⊆ L(A).

(iii) L(A) is semi-simple for A of type I, II or III; moreover, it is symplectic for

A of type I and II, orthogonal for type III. (iv) L(A_{1}^{n_{1}} × ... × A_{s}^{n_{s}}) = L(A_{1}) × ... × L(A_{s}).

1. All the Hodge classes on A^{s} are divisorial if and only if Hg(A) = L(A) and A is not of type III.

2. If A is of type III, then it has a non-divisorial Hodge class.

Let 𝔩 := Lie(L(A)), then 0.(ii)′ 𝔥 ↪ 𝔩 ↪ 𝔰𝔭(V), 𝔥^{ss} ↪ 𝔩^{ss}, C_{𝔥} ↪ C_{𝔩}. Here C_{?} is the center of ?, and ?^{ss} is the semi-simple part of ?.

0.2. Let 𝕜 ↪ D be an imaginary quadratic field, Gaℓ(𝕜/ℚ) = {σ, ρ}, ρ (= σ^{2}) is the fixed (identity) embedding 𝕜 ↪ \overline{ℚ}. In this case V_{ℝ} := V ⊗ ℝ has two complex structures. One is given by the isomorphism V_{ℝ} = Lie(A(ℂ)), (cf. 0.1), and the other by the action of 𝕜 ⊗_{ℚ} ℝ(≃ ℂ). Hence the splitting V_{ℝ} = V^{σ} ⊕ V^{ρ} (𝕜 acts by σ(𝕜) on V^{σ} and by ρ(𝕜) on V^{ρ}). The two complex structures coincide on one of the subspaces, say V^{ρ} , and conjugate on the other, V^{σ}. If m_{σ} = dim_{ℂ}(V^{σ}), m_{ρ} = dim_{ℂ}(V^{ρ}), then (m_{σ}, m_{ρ}) is the signature of the 𝕜-action; m_{σ} + m_{ρ} = g = dim_{ℂ}(V_{ℝ}) = dim(A).

0.2.1. Recall that the Rosati involution is the involution on D = End^{∘}(A) induced by the Riemann form (cf. 0.1). The Rosati involution is positive, consequently, the field it fixes is totally real (cf. [MAV]).

In the case 𝕜 ⊆ D we always assume that the Rosati involution preserves 𝕜. The positivity of the involution implies that it acts on 𝕜 non-trivially. Hence this action coincides with (the complex conjugation) σ.

0.2.2. Since 𝔥 and 𝔩 centralize D (cf. 0.1.2 and 0.1.5)

𝔥 ↪ 𝔩 ↪ 𝔰𝔭(V)^{𝕜} ↪ 𝔰𝔭(V),

where 𝔰𝔭(V)^{𝕜} is the centralizer of 𝕜 in 𝔰𝔭(V). By [D 3, Lemma 4.6]

𝔰𝔭(V)^{𝕜} = 𝔲(V),

the Lie algebra of the unitary group of a 𝕜-Hermitian form on V viewed as the 𝕜-vector space (cf. 0.2). Extending scalars to 𝕜 we get

(0.2.2.1)  𝔥_{𝕜} ↪ 𝔩_{𝕜} ↪ 𝔲(V) × 𝔲(V)^{σ} ↪ 𝔰𝔭(V_{𝕜}),

where 𝔥_{𝕜} := 𝔥 ⊗_{ℚ} 𝕜, 𝔩_{𝕜} := 𝔩 ⊗_{ℚ} 𝕜, V_{𝕜} := V ⊗_{ℚ} 𝕜 = V ⊕ U, U is the same V, but with the conjugate 𝕜-vector space structure.

The 𝔥-invariant 𝕜-Hermitian form referred to above is a non-degenerate element of V̌ ⊗ Ǔ (cf. [D 3, Lemma 4.6]), hence the isomorphism

U ≅ V̌

of 𝔥-modules. Clearly the projection of 𝔥_{𝕜} to 𝔲(V) is 𝔥, thus we can rewrite (0.2.2.1) as

(0.2.2.2)  𝔥 ↪ 𝔩 ↪ 𝔲(V) ↪^{Δ} 𝔰𝔭(V ⊕ V̌), g ↦^{Δ} ( \begin{matrix}g & 0 \\ 0 & ^{t}g^{−1} \end{matrix} ) .

(Note. The embeddings above are considered over 𝕜.)

0.2.3. From this we get:

\overline{𝔥} ↪ \overline{𝔩} ↪ 𝔤𝔩(W) ↪^{Δ} 𝔰𝔭(W ⊕ W̌),

where \overline{𝔥} := 𝔥 ⊗_{ℚ} \overline{ℚ}, \overline{𝔩} := 𝔩 ⊗_{ℚ} \overline{ℚ}, W := V ⊗_{𝕜,ρ} \overline{ℚ}, W̌ : dual of W (= W ⊗_{𝕜,σ} \overline{ℚ}).

0.3. Let V_{ℓ} := T_{ℓ}(A) ⊗_{ℤ_{ℓ}} ℚ_{ℓ} ≅ V ⊗_{ℚ} ℚ_{ℓ}, where T_{ℓ}(A) is the ℓ-adic Tate module (= H_{1}^{ét}(A ×_{K} \overline{ℚ}, ℤ_{ℓ}) = \underleftarrow{lim}_{n}_{ℓ^{n}}A(\overline{ℚ}), where _{ℓ^{n}}A(\overline{ℚ}) = kernel of multiplication by

ℓ^{n} : A(\overline{ℚ}) → A(\overline{ℚ})), V = H_{1}(A(ℂ), ℚ) as above. By abuse of language we call V_{ℓ} Tate module too. Let G_{ℓ} be the image of Gaℓ(\overline{ℚ}/K) in End_{ℚ_{ℓ}}(V_{ℓ}), where K is the base field of A, 𝔊_{ℓ} := Lie(G_{ℓ}). It is known (cf. [Bo]) that 𝔊_{ℓ} is algebraic and ℚ_{ℓ} · 1_{V_{ℓ}} ⊆ 𝔊_{ℓ} ⊂ 𝔤𝔰𝔭(V_{ℓ}). Let 𝔤_{ℓ} := 𝔊_{ℓ} ∩ 𝔰𝔩(V_{ℓ}) ⊂ 𝔰𝔭(V_{ℓ}). Then C_{𝔊_{ℓ}} = C_{𝔤_{ℓ}} ⊕ ℚ_{ℓ}·1_{V_{ℓ}}, 𝔤_{ℓ}^{ss} = 𝔊_{ℓ}^{ss}.

Remarks. 1. 𝔤_{ℓ} does not depend on finite extensions of K (cf. [S 2]).

2. V_{ℓ} ≅ V_{ℓ}(r), ∀r ∈ ℤ, as 𝔤_{ℓ}-modules, but not as 𝔊_{ℓ}-modules.

0.3.0. The Tate conjecture states that the Tate cycles, i.e., the Galois invariants H_{.}^{ét}(A\overline{ℚ}, ℚ_{ℓ})^{𝔤_{ℓ}} = (⋀^{.} H_{1}^{ét}(A\overline{ℚ}, ℚ_{ℓ}))^{𝔤_{ℓ}}, are algebraic.

0.3.1. Faltings (cf. [F]) has proved the analogs of Mumford’s Theorem 0.1.2(1,2) and the (1,1)-theorem (a special case of the Tate conjecture) for abelian varieties.

Theorem.

1. Let s ∈ ℕ, ( ⋀^{2} ; ℚ_{ℓ} sV_{ℓ})^{𝔤_{ℓ}} is exactly the (ℚ_{ℓ}-span of homological classes of)

divisors of A^{s} = A × ... × A (s times).

2. End_{𝔤_{ℓ}}(V_{ℓ}) = D ⊗_{ℚ} ℚ_{ℓ}.

3. 𝔤_{ℓ} is reductive.

0.3.2. 𝔥_{ℓ} := 𝔥⊗_{ℚ} ℚ_{ℓ} ↪ End_{ℚ_{ℓ}}(V_{ℓ}). The known relation between 𝔤_{ℓ} and 𝔥_{ℓ} is given by the following theorem.

Theorem (Deligne [D 3], Piatetskii-Shapiro [P-Sh], Borovoi [Bor]).

𝔤_{ℓ} ⊆ 𝔥_{ℓ} ⊂ End_{ℚ_{ℓ}}(V_{ℓ}).

0.4. The Mumford-Tate conjecture (=: MT) states 𝔤_{ℓ} = 𝔥_{ℓ}. Since 𝔤_{ℓ} and 𝔥_{ℓ} are reductive, it is the same as equivalence of the Hodge and the Tate conjectures for an abelian variety and all its self-products.

0.4.1. In order to prove MT it is enough to establish the conjecture for one ℓ ([LP, Theorem 4.3]).

0.4.2. Moreover, it is enough to show \overline{𝔤}_{ℓ} = \overline{𝔥}_{ℓ}, where \overline{𝔤}_{ℓ} := 𝔤_{ℓ} ⊗_{ℚ_{ℓ}} \overline{ℚ}_{ℓ}, \overline{𝔥}_{ℓ} := 𝔥_{ℓ} ⊗_{ℚ_{ℓ}} \overline{ℚ}_{ℓ} ([Z 2, §5, Key Lemma]).

0.4.3. The (1,1)-theorems imply ( ⋀^{2} ; ℚ_{ℓ}sV_{ℓ})^{𝔤_{ℓ}} = ( ⋀^{2} ; ℚ_{ℓ}sV_{ℓ})^{𝔥_{ℓ}}.

0.4.4. Theorems 0.3.2, 0.1.2(2) and 0.3.1(2) imply 𝔤_{ℓ}^{ss} ⊂ 𝔥_{ℓ}^{ss}, C_{𝔤_{ℓ}} ⊂ C_{𝔥_{ℓ}}.

In fact, C_{𝔤_{ℓ}} = C_{𝔥_{ℓ}}. This can be shown in a way similar to the proof of MT for CM abelian varieties (cf. [ShT], and also [D 1]).

So, in order to prove MT one must show that \overline{𝔤}_{ℓ}^{ss} = \overline{𝔥}_{ℓ}^{ss}.

To simplify notations we write sometimes 𝔤̃ for \overline{𝔤}_{ℓ}^{ss} and 𝔥̃ for \overline{𝔥}_{ℓ}^{ss}.

0.5. Let 𝔞 be a semi-simple Lie algebra over an algebraically closed field of characteristic 0, and 𝔞 = 𝔞_{1} × ... × 𝔞_{n} be the decomposition of 𝔞 into the product of its simple ideals. For any faithful irreducible representation U of 𝔞, U decomposes as a tensor product of irreducible representations U_{i} of 𝔞_{i}. Since U is faithful, none of the U_{i}’s is trivial. Moreover, if the representation U admits a non-degenerate invariant bilinear form, then so does each U_{i}.

We say that the representation is minuscule if the highest weight of each U_{i} is minuscule, see [B, Ch.VIII, §7.3]. The following is the list of minuscule weights, [B, Ch.VIII, §7.3 and Table 2]:

type A_{m} (m ≥ 1) : ϖ_{1}, ϖ_{2}, …, ϖ_{m}; dim(ϖ_{s}) = \binom{m+1}{s};

type B_{m} (m ≥ 2) : ϖ_{1}; dim(ϖ_{1}) = 2m + 1;

type C_{m} (m ≥ 2) : ϖ_{1}; dim(ϖ_{1}) = 2m;

type D_{m} (m ≥ 3) : ϖ_{1}, ϖ_{m−1}, ϖ_{m}; dim(ϖ_{1}) = 2m,

dim(ϖ_{m−1}) = dim(ϖ_{m}) = 2^{m−1};

type E_{6} : ϖ_{1}, ϖ_{6}; dim(ϖ_{1}) = dim(ϖ_{6}) = 27;

type E_{7} : ϖ_{7}; dim(ϖ_{7}) = 56;

there are no minuscule representations for the types E_{8}, F_{4}, G_{2}.

0.5.1. It is known that the representations of 𝔤̃, 𝔥̃ are minuscule (cf. [S ∗], [D 2]). It is also known that 𝔤̃ is not exceptional, see [S ∗, Theorem 7] (for the corresponding result for 𝔥̃ see [D 2, Remarque 1.3.10(i)]).

0.6.1. Let again 𝕜 ↪ D and 𝕜_{ℓ} := 𝕜 ⊗_{ℚ} ℚ_{ℓ}. Then, as in 0.2.2,

𝔤_{ℓ} ⊂ 𝔥_{ℓ} ⊂ 𝔰𝔭(V_{ℓ})^{𝕜_{ℓ}} ⊂ 𝔰𝔭(V_{ℓ}).

If l splits in 𝕜, λ, λ′ being the primes of 𝕜 over l, λ′ = λ^{σ}, then 𝕜_{ℓ} ≅ ℚ_{ℓ} ⊕ ℚ_{ℓ}, V_{ℓ} = V_{λ} ⊕ V_{λ′}, where V_{λ}, V_{λ′} are vector spaces over 𝕜_{λ} ≅ ℚ_{ℓ}, 𝕜_{λ′} ≅ ℚ_{ℓ} respectively, and

𝔤_{ℓ} ⊂ 𝔥_{ℓ} ⊂ 𝔤𝔩(V_{λ}) ⊕ 𝔤𝔩(V_{λ′}) ⊂^{Δ} 𝔰𝔭(V_{λ} ⊕ V_{λ′}).

Since λ′ = λ^{σ}, as in 0.2.2 we conclude V_{λ′} ≅ V̌_{λ} and can rewrite the above sequence as

𝔤_{ℓ} ⊂ 𝔥_{ℓ} ⊂ 𝔤𝔩(V_{λ}) ↪^{Δ} 𝔰𝔭(V_{λ} ⊕ V̌_{λ}).

Remark. If ℓ does not split in 𝕜, then 𝕜_{ℓ} is a field, (𝕜_{ℓ} : ℚ_{ℓ}) = 2, and the rest is identical to 0.2.2.

0.6.2. As in 0.2.2, by extending scalars to \overline{ℚ}_{ℓ} we get

\overline{𝔤}_{ℓ} ⊂ \overline{𝔥}_{ℓ} ⊂ 𝔤𝔩(W_{λ}) ↪^{Δ} 𝔰𝔭(W_{λ} ⊕ W̌_{λ}),

where W_{λ} := V_{ℓ} ⊗_{𝕜_{ℓ},ρ_{λ}} \overline{ℚ}_{ℓ}, ρ_{λ} : 𝕜_{ℓ} → 𝕜_{λ} is the projection.

Remark. For ℓ non-split in 𝕜 , the same holds (cf. 0.2.2, 0.2.3).

0.7. We will need the following simple facts. We assume that D = 𝕜.

0.7.1. Proposition. The representations of \overline{𝔤}_{ℓ} and \overline{𝔥}_{ℓ} are non-self-dual. □

Remarks. 1. This is true for any irreducible subrepresentation of W_{λ} for any type IV abelian variety (e.g., [Mu], [H]).

2. If the abelian variety is of type I (respectively II, respectively III), then the irreducible components are symplectic (respectively symplectic, respectively orthogonal) (loc. cit.).

0.7.2. Proposition. \overline{𝔤}_{ℓ} and \overline{𝔥}_{ℓ} are semi-simple if and only if the signature of the 𝕜-action is (m, m). Further, if this is not the case, the centers C_{\overline{𝔤}_{ℓ}}, C_{\overline{𝔥}_{ℓ}} are 1-dimensional.

Proof. This is essentially proved in [D 3], [W]. Let us, however, briefly explain why this holds and fix notations.

Let μ : 𝔾_{m}_{,ℂ} → GL(V_{ℝ}) be the cocharacter defining the Hodge structure on V, then the map h : 𝕊 = R_{ℂ/ℝ}𝔾_{m} → GL(V_{ℂ}) is given by h(z) = μ(z) on V_{ℝ} (cf. 0.1.1), h(z) = \overline{μ}(z) on U_{ℝ}, where U_{ℝ} is the “same” V_{ℝ} but with the conjugate 𝕜 ⊗ ℝ-action, V_{ℂ} = V_{ℝ} ⊕ U_{ℝ} (cf. 0.2). If the 𝕜-signature is (m_{σ}, m_{ρ}) then V_{ℝ} = V_{ℝ}^{σ} ⊕ V_{ℝ}^{ρ}, dim_{ℂ}(V_{ℝ}^{σ}) = m_{σ}, dim_{ℂ}(V_{ℝ}^{ρ}) = m_{ρ} (𝕜 acts by σ(𝕜) on V_{ℝ}^{σ} and by ρ(𝕜) on V_{ℝ}^{ρ}). But V_{ℝ} = H_{−1,0}, hence the power of z by which μ(z) acts on V_{ℝ}^{σ}, V_{ℝ}^{ρ} is 1. We will call this power the μ-weight. Similarly, V_{ℂ} = H_{1}(A(ℂ), ℂ) = V_{ℂ}^{σ} ⊕ V_{ℂ}^{ρ}. But also V_{ℂ} = H_{−1,0} ⊕ H_{0,−1}(= V_{ℝ} ⊕ U_{ℝ}) and these two decompositions commute, since the former is determined by 𝕜 ⊆ D and the Hodge group centralizes D in End_{ℚ}(V). Hence we can write

V_{ℂ} = V_{ℝ} ⊕ U_{ℝ}
= (V_{ℝ}^{σ} ⊕ V_{ℝ}^{ρ}) ⊕ (U_{ℝ}^{σ} ⊕ U_{ℝ}^{ρ})
= (V_{ℝ}^{σ} ⊕ U_{ℝ}^{σ}) ⊕ (V_{ℝ}^{ρ} ⊕ U_{ℝ}^{ρ})
= V_{ℂ}^{σ} ⊕ V_{ℂ}^{ρ},

where U_{ℝ}^{σ} (respectively U_{ℝ}^{ρ}) is the conjugate of V_{ℝ}^{ρ} (respectively V_{ℝ}^{σ}). Thus dim_{ℂ}(U_{ℝ}^{σ}) = m_{ρ}, dim_{ℂ}(U_{ℝ}^{ρ}) = m_{σ}, so dim_{ℂ}(V_{ℂ}^{σ}) = m_{ρ} + m_{σ} = g = dim_{ℂ}(V_{ℂ}^{ρ}).

The μ-weight of U_{ℝ} is 0, hence the decomposition

V_{ℂ}^{σ} = V_{ℝ}^{σ} ⊕ U_{ℝ}^{σ}

is according to μ-weights 1, 0. (This exactly corresponds to the Hodge-Tate decomposition of the λ-adic representationbelow, cf. (1.0.∗ ∼1.1.∗).) Now the μ-weight (respectively the Hodge type) of

⋀^{g} ; ℂV_{ℂ}^{σ} = ⋀^{m_{σ}} ; ℂV_{ℝ}^{σ} ⊗ ⋀^{m_{ρ}} ; ℂU_{ℝ}^{σ} ⊆ ⋀^{g}V_{ℂ}^{σ}

is m_{σ} (respectively (−m_{σ}, −m_{ρ})). Hence ⋀^{g} ; 𝕜V is a Hodge cycle (i.e., of Hodge type (−\frac{g}{2}, −\frac{g}{2})) if and only if m_{σ} = m_{ρ}. Hence it is fixed by 𝔥 if and only if m_{σ} = m_{ρ}, i.e., 𝔥 ⊂ 𝔰𝔩(V) ∩ 𝔲(V) only in this case (here V is considered as a 𝕜-vector space). In other words, the center C_{𝔥} of 𝔥 kills the determinant ⋀^{g} ; 𝕜V (and hence ≠ {0})

if and only if m_{σ} ≠ m_{ρ}. Now, since \overline{V} := V ⊗ \overline{ℚ} = W ⊕ W̌ , W is irreducible, \overline{𝔥} ⊂ 𝔤𝔩(W) →^{Δ} 𝔰𝔭(\overline{V}). Summarizing,

\overline{𝔥} = \overline{𝔥}^{ss} ⊂ 𝔰𝔩(W) →^{Δ} 𝔰𝔭(W ⊕ W̌) if m_{σ} = m_{ρ},
\overline{𝔥} = \overline{𝔥}^{ss} ⊕ \overline{ℚ} ⊂ 𝔤𝔩(W) →^{Δ} 𝔰𝔭(W ⊕ W̌) if m_{σ} ≠ m_{ρ}.

Using 0.4.4 we conclude

\overline{𝔤}_{ℓ} ⊂ \overline{𝔥}_{ℓ} ⊂ 𝔰𝔩(W_{λ}) →^{Δ} 𝔰𝔭(W_{λ} ⊕ W̌_{λ}),

\overline{𝔤}_{ℓ} = \overline{𝔤}_{ℓ}^{ss}, \overline{𝔥}_{ℓ} = \overline{𝔥}_{ℓ}^{ss}, if m_{σ} = m_{ρ},

\overline{𝔤}_{ℓ} ⊂ \overline{𝔥}_{ℓ} ⊂ 𝔤𝔩(W_{λ}) →^{Δ} 𝔰𝔭(W_{λ} ⊕ W̌_{λ}),

C_{\overline{𝔤}_{ℓ}} = C_{\overline{𝔥}_{ℓ}} = \overline{ℚ}_{ℓ}, if m_{σ} ≠ m_{ρ}. □

0.7.3. As one can see from the proof, if 𝕜 ⊆ End^{∘}(A) (not necessarily equal), and m_{σ} ≠ m_{ρ}, the center C_{𝔥} of 𝔥 must kill the determinant ⋀^{g} ; 𝕜V, hence the center is non-trivial. However, if 𝕜 ≠ End^{∘}(A), then the center can be non-trivial even if m_{σ} = m_{ρ} (e.g., CM case).

But even if 𝕜 ⊊ End^{∘}(A), as follows from the proof, \overline{𝔤}_{ℓ} ⊂ \overline{𝔥}_{ℓ} ⊂ 𝔰𝔩(W_{λ}) if and only if m_{σ} = m_{ρ}. In this form the result (using [D 3, Proposition 4.4]) can be generalized to the case of an arbitrary CM-field E ↪ D (cf. [MZ, Lemma 2.8]).

Note that since 𝔤_{ℓ} and 𝔥_{ℓ} are semi-simple for abelian varieties of types II or III (cf. 0.1.2(3)), if 𝕜 ↪ End^{∘}(A) is Rosati-stable, A of type II or III, then the signature of the 𝕜-action is necessarily (m, m).

Definition. If the signature of the 𝕜-action on an abelian variety is (m, m), we call such an abelian variety a Weil type abelian variety (cf. [W]).

1. On the Hodge-Tate decomposition

1.0. We recall here certain basic facts on the Hodge-Tate decomposition and then give some applications. The classical/standard reference is [T 1].

1.0.1. According to Tate and Raynaud, \overline{V}_{ℓ} := V_{ℓ} ⊗ ℂ_{ℓ} = H_{1}((A ⊗_{K} \overline{K})_{ét}, ℚ_{ℓ}) ⊗_{ℚ_{ℓ}} ℂ_{ℓ}, ℂ_{ℓ} is a completion of \overline{ℚ}_{ℓ}, admits a decomposition

\overline{V}_{ℓ} = \overline{V}_{ℓ}(0) ⊕ \overline{V}_{ℓ}(1),

where \overline{V}_{ℓ}(i) := \overline{V} _{ℓ}^{(i)} ⊗_{ℚ_{ℓ}} ℂ_{ℓ}, i = 1, 2. The ℚ_{ℓ}-subspaces (but not ℂ_{ℓ}-subspaces) \overline{V} _{ℓ}^{(i)}’s of \overline{V}_{ℓ} are defined as follows:

\overline{V} _{ℓ}^{(i)} := \{v ∈ \overline{V}_{ℓ} | v^{σ} = χ_{ℓ}(σ)^{i}·v, ∀σ ∈ I\}, i = 1, 2,

where I is the absolute inertia group at a prime of K (=base field of A) over ℓ (cf. 0.3 for why we take the inertia instead of the whole decomposition group), χ_{ℓ} is the cyclotomic character. Recall that the Galois action is continous and semi-linear on \overline{V}_{ℓ} (see [S 2, 1.2]), and, clearly, the \overline{V} _{ℓ}^{(i)}’s are Galois submodules of \overline{V}_{ℓ}. The Galois action on \overline{V}_{ℓ}(i) is by the formula

(v ⊗ c)^{σ} := v^{σ} ⊗ c^{σ}, ∀v ∈ \overline{V} _{ℓ}^{(i)}, ∀c ∈ ℂ_{ℓ}, ∀σ ∈ I,

extended by linearity.

1.0.2. According to S. Sen ([Se, Section 4, Theorem 1]), to the the Hodge-Tate decomposition on \overline{V}_{ℓ} one can associate a cocharacter

ϕ : 𝔾_{m}_{,ℂ_{ℓ}} → GL(V_{ℓ})_{ℂ_{ℓ}},

by defining ϕ(z), ∀z ∈ ℂℓ^{×}, to be the automorphism of \overline{V}_{ℓ} which is multiplication by z on \overline{V}_{ℓ}(1) and by the identity on \overline{V}_{ℓ}(0). This association is made in such a manner that the algebraic envelope G̃_{ℓ} of the Galois image (cf. 0.3) turns out to be the smallest algebraic group defined over ℚ_{ℓ} which after extension of scalars to ℂ_{ℓ} contains the image of ϕ.

Remark. This cocharacter ϕ is completely analogous to the cocharacter μ associated to the Hodge decomposition on V_{ℂ}, G̃_{ℓ} is the analog of the Mumford-Tate group M(A) and 𝔤_{ℓ} is the analog of 𝔥, see 0.1.1.

1.0.3. Before proceeding, recall that for a Gaℓ(\overline{ℚ}_{ℓ}/ℚ_{ℓ})-module X, the Tate twist X(1) of X is defined to be X ⊗_{ℚ_{ℓ}} ℚ_{ℓ}(1) with the Galois structure of a tensor product of Galois modules (as in 1.0.1). Here ℚ_{ℓ}(1) is the Tate module:

ℚ_{ℓ}(1) := (\underleftarrow{lim}_{n} μ_{ℓ^{n}}) ⊗_{ℤ_{ℓ}} ℚ_{ℓ}, μ_{ℓ^{n}} = \{ζ ∈ \overline{ℚ}_{ℓ} | ζ^{ℓ^{n}} = 1\},

with the natural Gaℓ(\overline{ℚ}_{ℓ}/ℚ_{ℓ})-action by χ_{ℓ}:

ζ^{σ} = ζ^{χ_{ℓ}(σ)}, ∀σ ∈ Gaℓ(\overline{ℚ}_{ℓ}/ℚ_{ℓ}), ζ ∈ μ_{ℓ^{n}}, for some n.

1.0.4. The Hodge-Tate decomposition of \overline{V}_{ℓ} can be rewritten in the following explicit form ([T 1, § 4, Corollary 2], see also the Remark following that Corollary):

\overline{V}_{ℓ} = Lie(A_{ℂ_{ℓ}}^{∨})^{∨} ⊕ Lie(A_{ℂ_{ℓ}})(1),

where Lie(A_{ℂ_{ℓ}}^{∨})^{∨} is the cotangent space of the dual abelian variety A_{ℂ_{ℓ}}^{∨} at its origin and Lie(A_{ℂ_{ℓ}})(1) is the tangent space of A_{ℂ_{ℓ}} at its origin Tate-twisted by χ_{ℓ}.

1.0.5. On the other hand, we have the Hodge decomposition on V_{ℂ} = H_{1}(A(ℂ), ℂ):

H_{1}(A(ℂ), ℂ) = H_{1}(A_{ℂ}, O_{A_{ℂ}}) ⊕ H_{0}(A_{ℂ}, Ω_{A_{ℂ}}^{1})
= Lie(A_{ℂ}^{∨})^{∨} ⊕ Lie(A_{ℂ}),

or, in our notation,

V_{ℂ} = U_{ℝ} ⊕ V_{ℝ},

see the proof of 0.7.2.

1.0.6. Fix an isomorphism ℂ_{ℓ} ≅ ℂ. Then the comparison isomorphism

c : H_{1}^{ét}(A\overline{K}, ℚ_{ℓ}) ⊗_{ℚ_{ℓ}} ℂ_{ℓ} ≅ H_{1}(A(ℂ), ℂ)

or, in our notation,

c : V_{ℓ} ⊗_{ℚ_{ℓ}} ℂ_{ℓ} ≅ V ⊗_{ℚ} ℂ,

provides isomorphisms

c : V_{ℝ} → \overline{V}_{ℓ}(1), c : U_{ℝ} → \overline{V}_{ℓ}(0).

Note, that dim_{ℂ}(V_{ℝ}) = g = dim_{ℂ_{ℓ}}(\overline{V}_{ℓ}(1)), dim_{ℂ}(U_{ℝ}) = g = dim_{ℂ_{ℓ}}(\overline{V}_{ℓ}(0)).

1.0.7. On the other hand, both homology groups admit decompositions according to the action of 𝕜 ⊆ End^{∘}(A) :

\overline{V}_{ℓ} := \overline{V}_{λ} ⊕ \overline{V}_{λ′}

where \overline{V}_{λ} := W_{λ} ⊗_{\overline{ℚ}_{ℓ}} ℂ_{ℓ} = V_{ℓ} ⊗_{𝕜_{ℓ},ρ_{λ}} ℂ_{ℓ}, \overline{V}_{λ′} := W_{λ′} ⊗_{\overline{ℚ}_{ℓ}} ℂ_{ℓ} = V_{ℓ} ⊗_{𝕜_{ℓ},ρ_{λ′}} ℂ_{ℓ}, (cf. 0.6.2) and

V_{ℂ} := V_{ℂ}^{σ} ⊕ V_{ℂ}^{ρ}.

These two types of splittings commute, and consequently V_{ℂ}^{σ}, V_{ℂ}^{ρ} admit the Hodge decomposition, and \overline{V}_{λ}, \overline{V}_{λ′} admit the Hodge-Tate decomposition. The map c respects these splittings, hence maps either V_{ℂ}^{σ} to \overline{V}_{λ}, or V_{ℂ}^{ρ} to \overline{V}_{λ}.

1.0.8. Let’s assume that

c : V_{ℂ}^{σ} →^{∼} \overline{V}_{λ},

hence dim_{ℂ}(V_{ℂ}^{σ}) = g = dim_{ℂ_{ℓ}}(\overline{V}_{λ}).

As a result we conclude

c : V_{ℝ}^{σ} →^{∼} \overline{V}_{λ}(1),

hence dim_{ℂ_{ℓ}}(\overline{V}_{λ}(1)) = m_{σ}, dim_{ℂ_{ℓ}}(\overline{V}_{λ}(0)) = m_{ρ}.

Remark. If c : V_{ℂ}^{ρ} → \overline{V}_{λ}, then m_{σ} and m_{ρ} exchange roles. This does not affect our results.

1.1. For abelian varieties of types I, II and III 𝔤_{ℓ}, 𝔥_{ℓ} are semi-simple and have the same invariants on V_{ℓ} ⊗_{ℚ_{ℓ}} V_{ℓ} (cf. 0.4.3 and Remark 0.7.1(2)). In the case of type IV, the Lie algebras can be non-semi-simple. Since 𝔤_{ℓ}^{ss} ⊆ 𝔥_{ℓ}^{ss}, a priori 𝔤_{ℓ}^{ss} can have more invariants than 𝔥_{ℓ}^{ss} in V_{ℓ}^{⊗2}. However, in our special case the following is true.

Theorem. If D = 𝕜, then 𝔤_{ℓ}^{ss} and 𝔥_{ℓ}^{ss} are non-self-dual.

Remark. If 𝔤_{ℓ} = 𝔤_{ℓ}^{ss}, 𝔥_{ℓ} = 𝔥_{ℓ}^{ss} we get nothing new (cf. 0.7.1).

Proof. (0). If \overline{𝔤}_{ℓ}^{ss} is symplectic or orthogonal, then so is \overline{𝔤}_{ℓ}^{ss} ⊗_{\overline{ℚ}_{ℓ}} ℂ_{ℓ}. If \overline{𝔥}_{ℓ}^{ss} ⊗_{\overline{ℚ}_{ℓ}} ℂ_{ℓ} fixes a bilinear form on \overline{V}_{ℓ} coming from V_{ℓ} ⊗_{ℚ_{ℓ}} \overline{ℚ}_{ℓ}, then \overline{𝔥}_{ℓ}^{ss} fixes the form. So, we can extend scalars to ℂ_{ℓ} and will use the same notation \overline{𝔥}_{ℓ}, \overline{𝔤}_{ℓ} for the corresponding extentions of the Lie algebras.

(i). Let us consider first the symplectic case.

If \overline{𝔤}_{ℓ}^{ss} fixes 1-dimensional subspace χ ⊆ ⋀^{2}\overline{V}_{λ}, then χ is a 1-dimensional \overline{𝔤}_{ℓ}-sub-representation. The Hodge-Tate decomposition implies

⋀^{2}\overline{V}_{λ} = ⋀^{2}\overline{V}_{λ}(0) ⊕ (\overline{V}_{λ}(1) ⊗ \overline{V}_{λ}(0)) ⊕ ⋀^{2}\overline{V}_{λ}(1).

The Hodge-Tate weight of the terms on the right is 0, 1 and 2 respectively.

Since dim_{ℂ_{ℓ}}(χ) = 1, it is of Hodge-Tate weight 0, 1 or 2. χ^{g/2} = det(\overline{V}_{λ}) := ⋀^{g}\overline{V}_{λ} = ⋀^{m_{σ}} \overline{V}_{λ}(1) ⊗ ⋀^{m_{ρ}}\overline{V}_{λ}(0), the Hodge-Tate weight of the RHS is m_{σ}. Hence the Hodge- Tate weight of χ^{g/2} is m_{σ}. On the other hand, the weight of χ^{g/2} is g/2times the weight of χ, hence is equal to 0, g/2 or g. So m_{σ} = 0, g/2 or g. The cases m_{σ} = 0 or g correspond to the 𝕜-signature (0, g) or (g, 0). In either case the abelian variety is isogenous to a product of CM elliptic curves ([Sh 1, Proposition 14]). In the case m_{σ} = g/2 we have m_{σ} = m_{ρ} and the Lie algebra \overline{𝔤}_{ℓ} = \overline{𝔤}_{ℓ}^{ss} (as well as \overline{𝔥}_{ℓ} = \overline{𝔥}_{ℓ}^{ss}) is non-self-dual (cf. 0.7.1), hence this χ does not exist!

(ii). The orthogonal case is a direct consequence of the symplectic and 0.4.3

(take s = 2 : Sym^{2}(\overline{V}_{λ}) ↪ ⋀^{2}(2\overline{V}_{λ})). □

Remark. Another way to conclude that the Hodge-Tate weight of χ is 1 is to use a result of Raynaud that the Galois action on 1-dimensional subrepresentations of (co)homology is by (powers of) the cyclotomic character χ_{ℓ}. Either way, the result is a consequence of the existence of the Hodge-Tate decomposition.

1.2. We can apply this consideration of the Hodge-Tate decomposition to abelian varieties with D = 𝕜 and 𝕜-signature (m_{σ}, m_{ρ}) such that gcd(m_{σ}, m_{ρ}) = 1 (we call them Ribet-type abelian varieties, cf. [Ri]), then an argument of Serre ([S 1, §4]) implies the Tate conjecture in that case. Indeed, Ribet’s proof of loc. cit., Theorem 3, verbatim provides the Tate cycles are generated by divisors and hence the following theorem.

Theorem. If A is a Ribet-type abelian variety, then the Tate cycles (on the abelian variety, and all its self-products) are generated by divisors and hence the Tate, the Hodge and the Mumford-Tate conjectures hold. □

Remark. Applicability of this argument to the Tate cycles was undoubtly known to Ribet and was also noticed in [C 1] and [MZ].

2. Abelian 4-folds

2.0. The first non-trivial case of the Hodge, the Tate and the Mumford-Tate conjectures is that of abelian fourfolds. In that case 1 ≤ (D : ℚ) ≤ 8 and the dimensions of the irreducible subrepresentations of 𝔥, 𝔤_{ℓ} are ≤ 8.

The case of a simple 4-fold with (D : ℚ) ≥ 2 was studied in [MZ]. In this section we survey the 4-dimensional case just indicating the ideas involved. For details (in particular, proof of Theorem 2.4 below) see [MZ].

2.1. If (D : ℚ) ≥ 2, then the dimensions ≤ 4 and the restrictions imposed on 𝔤_{ℓ}, 𝔥 (cf. 0.1.2, 0.3.1, 0.5) force the representations to be unique, hence abelian varieties in these cases verify MT.

Moreover, in“most cases” these Lie algebras are the “largest possible,” viz., coincide with the Lie algebra of the Lefschetz group. Thus for any such abelian variety (excluding type III, cf. 0.1.5) the Hodge and the Tate conjectures hold.

2.2. The case D = ℚ is slightly more subtle: the choice for 𝔤_{ℓ}, 𝔥 is not unique anymore. In fact, both possibilities, 𝔰𝔩_{2} × 𝔰𝔩_{2} × 𝔰𝔩_{2} and 𝔰𝔭_{8}, do occur.

The first one, viz., \overline{𝔤}_{ℓ} = \overline{𝔥}_{ℓ} ≅ 𝔰𝔭_{8}, is the generic case (cf. [Ab 1], [Ma]). Abelian varieties with \overline{𝔥} ≅ 𝔰𝔩_{2} × 𝔰𝔩_{2} × 𝔰𝔩_{2}, were constructed by Mumford in [M 2].

However, imposing some extra conditions on a simple 4-fold with D = ℚ we still can conclude MT (cf. 5.2).

2.2.1. In the generic case \overline{𝔤}_{ℓ} = \overline{𝔥}_{ℓ} = \overline{𝔩}_{ℓ}, where \overline{𝔩}_{ℓ} := Lie(L(A)) ⊗_{ℚ} \overline{ℚ}_{ℓ}, the Lie algebra of the Lefschetz group, and all the conjectures follow from 0.1.5.

2.2.2. In the Mumford case, \overline{𝔥} ≅ 𝔰𝔩_{2} × 𝔰𝔩_{2} × 𝔰𝔩_{2}, again \overline{𝔤}_{ℓ} = \overline{𝔥}_{ℓ} and MT holds. However, \overline{𝔩} ≅ 𝔰𝔭_{8} and not all the Hodge/Tate cycles (on self-products!) are divisorial.

2.2.3. Meanwhile, whatever the case, 𝔰𝔩_{2}×𝔰𝔩_{2}×𝔰𝔩_{2} or 𝔰𝔭_{8}, [Ta, Lemma 4.10] implies that all the Hodge and the Tate cycles on the abelian variety itself are divisorial. However, calculations show (cf. [H 1, Lemma 5.2, (5.2.2)]) that on the “square” of the Mumford 4-fold not all the Hodge cycles are divisorial.

2.3.1. If D = ℚ, then either \overline{𝔤}_{ℓ} = \overline{𝔥}_{ℓ} and MT holds, or \overline{𝔤}_{ℓ} ≠ \overline{𝔥}_{ℓ} and then \overline{𝔤}_{ℓ} ≅ 𝔰𝔩_{2} × 𝔰𝔩_{2} × 𝔰𝔩_{2}, \overline{𝔥}_{ℓ} ≅ 𝔰𝔭_{8}. In the latter case the (Lie algebra of the) Hodge group is equal to the (Lie algebra of the) Lefschetz group, thus all the Hodge cycles on the self-products of the abelian variety are divisorial, see also §7.

2.3.2. For the Weil case (cf. 0.7.3) generically the ring of Hodge cycles is not generated by divisors ([W], also [MZ]). So, if there are any doubts about the Hodge conjecture (and hence the Tate conjecture), the Weil abelian varieties are the ones to look at. Recently, C. Schoen ([Sc], also [vG]) succeded in proving the Hodge conjecture for one family of Weil 4-folds admitting an action of ℚ(μ_{3}).

2.3.3. 2.1 answers a question of Tate (cf. [T 2], p. 82) on whether the Tate conjecture is true for the Schoen family.

2.4. We summarize the above discussion in the following theorem.

Theorem. 1. If A is any 4-dimensional abelian variety, then the rings of the Tate cycles and the Hodge cycles coincide (hence, the Hodge and the Tate conjectures for this variety are equivalent).

2. If, additionally, End^{∘}(A) ≠ ℚ, then MT holds.

Remarks. 1. Later (Theorem 5.2) we will see that even when End^{∘}(A) = ℚ, MT holds under some reduction conditions.

2. Recall (0.4) that MT implies that the Hodge and the Tate conjectures are equivalent for an abelian variety and all its self-products.

3. If A is non-simple, then the second hypothesis of the theorem is satisfied.

Proof. The only “real” case to consider is that of A simple. The proof can be found in [MZ] (in fact, Moonen-Zarhin considered a deeper problem of “when and why” a simple 4-fold has an exceptional Weil class).

If A is a non-simple abelian 4-fold, say A is isogenous to A_{1}×A_{2}, then dim(A_{i}) ≤ 3. Hence, by the (1,1)-theorems and duality, all the Hodge cycles on A_{i} are divisorial. The embeddings 𝔤_{ℓ} ⊂ 𝔥_{ℓ} ↪ 𝔰𝔭(V_{ℓ}) factor through the sub-representations corresponding to the simple components of A. The dimensions of the sub-represen-tations are ≤ 6, and there is not “enough room” for 𝔤_{ℓ} and 𝔥_{ℓ} to be different, i.e., 𝔤_{ℓ} = 𝔥_{ℓ}, hence MT holds. □

2.5. Let us indicate what is the situation regarding the Hodge and the Tate conjectures for non-simple abelian 4-folds. As above, let A be isogenous to A_{1} × A_{2}. Then all the Hodge and the Tate cycles on the A_{i}’s are divisorial.

2.5.1. We can also say that all the Hodge cycles (and hence the Tate cycles) on A and all its self-products are generated by divisors in the following cases:

1. Neither of the A_{i}’s is of type IV ([H 2, Theorem 0.1]).

2. A_{1} is not of type IV, A_{2} is of CM-type (loc. cit., Proposition 3.1).

3. If the A_{i}’s are non-CM, type IV abelian surfaces, then according to [Sh 1, Theorem 5, Propositions 17, 19], the A_{i}’s are products of CM elliptic curves. Hence so is A = A_{1} × A_{2} and for such abelian varieties the result stated above is known ([Im]; [H 1, Theorem 2.7]).

4. If the A_{i}’s are isogenous CM surfaces, then by remark 0.1.2(1) and 0.1.5, Hg(A) = Hg(A_{1}), L(A) = L(A_{1}). By 0.1.5(1) L(A_{1}) = Hg(A_{1}), hence L(A) = Hg(A), and applying 0.1.5(1) once again we conclude the result.

2.5.2. For the remaining case, viz., both the A_{i}’s are non-isogenous CM abelian varieties, let us just mention that Shioda constructed an example of a product of a simple CM 3-fold, A_{1}, with a CM elliptic curve, A_{2}, such that on A = A_{1} × A_{2} there exist exceptional, non-divisorial, Hodge cycles, [Shi, Example 6.1]. In this example, however, the Hodge (hence the Tate) conjecture holds.

II. Abelian varieties with reduction conditions

3. Bad reduction and monodromy action

3.0. Let A be an abelian variety defined over a number field K. Assume A has bad reduction at a prime ℘ of O_{K}. Let Ã be the identity component of the special fiber of the Néron model of A. Then Ã is semi-abelian:

0 → H → Ã → B → 0,

where H is the affine subgroup of Ã, B is the abelian quotient.

3.0.1. Since we are concerned with the Lie algebra of the (image of) Galois, we may pass to a finite extension of K (cf. Remark 1 in 0.3). So, according to the semi-stable reduction theorem ([G, Théorèm 3.6]), by extending the base field if necessary, we may assume that the reduction is semi-stable (i.e., H is a torus) and split (i.e., H is split: H ≅ 𝔾_{m}^{r}).

The dimension r of H we call the toric rank of (the reduction of) A.

3.0.2. D = End^{∘}(A) as before, there is a homomorphism D → End^{∘}(H), 1_{A} ↦ 1_{H}. But End^{∘}(𝔾_{m}^{r}) = M_{r}(ℤ) ⊗_{ℤ} ℚ, hence (D : ℚ)|r.

3.1. Consider the corresponding “specialization sequence”

0 → V_{ℓ}^{I} → V_{ℓ}(A) → U → 0,

where V_{ℓ}(A) is the Tate module of A, I := I(℘) is the inertia group at ℘, V_{ℓ}^{I} := V_{ℓ}(A)^{I} is the submodule of I-invariants and U is a trivial I-module (cf. [G, Proposition 3.5]). We have dim_{ℚ_{ℓ}}(V_{ℓ}(A)) = 2g, dim_{ℚ_{ℓ}}(V_{ℓ}^{I}) = 2g − r, dim_{ℚ_{ℓ}}(U) = r.

3.2. The above sequence is a sequence of of I-modules. The I-action is called the local monodromy action.

3.3. The reduction map at ℘ induces an isomorphism V_{ℓ}^{I} ≅ V_{ℓ}(Ã), the Tate module of Ã, and takes a submodule W ⊆ V_{ℓ}^{I} to the Tate module V_{ℓ}(H) ⊆ V_{ℓ}(Ã) of the toric part H of Ã (cf. [ST, Lemma 2], [G, 2.3], [I], [O]). In fact, according to the “Igusa-Grothendieck Orthogonality Theorem,” W = (V^{I})^{⊥} with respect to the Weil pairing on V (cf. [I, Theorem 1], [G, Théorème 2.4], also [O, Theorem (3.1)]).

3.4. The monodromy action on V_{ℓ}(A) is, in general, quasi-unipotent (e.g., [G], [ST], [O]). However, since (we assumed that) the reduction of Ais semi-stable and split, this action is, in fact, unipotent (cf. [G, Corollaire 3.8]).

3.4.0. Picking a vector subspace T of V_{ℓ}(A) specializing to U, we get the matrix form of the monodromy action:

\begin{matrix}V_{ℓ}^{I} & ^{\{ W \{} \\ T & \{ \end{matrix} ( \begin{matrix}1_{r} & 0 & ∗_{r} \\ 0 & 1_{2g−2r} & 0 \\ 0 & 0 & 1_{r} \end{matrix} ) .

3.4.1. Passing to the Lie algebra 𝔦 := Lie(I), we conclude the existence of nilpotents, τ ∈ 𝔦 ⊂ 𝔤_{ℓ}, of order 2, i.e., τ^{2} = 0, and rank (with respect to V_{ℓ}) rk_{V_{ℓ}}(τ) ≤ r, where rk_{V_{ℓ}}(τ) := dim_{ℚ_{ℓ}}(τV_{ℓ}) = rank of the matrix of τ ∈ 𝔤𝔩(V_{ℓ}).

3.4.2. The Neron-Ogg-Shafarevich criterion ensures that ∃ τ ≠ 0, since A has bad reduction.

3.4.3. Moreover, if N is given by the above matrix, then τ = N − 1_{2g} is the logarithm of the monodromy corresponding to the monodromy filtration (cf. [G, 4.1 and also Corollaire 4.4]; also [Il, 2.6])

(0) ⊂ W ⊂ V_{ℓ}^{I} ⊂ V_{ℓ},

and τ maps V_{ℓ} → W, V_{ℓ}^{I} → 0, inducing an isomorphism of the quotients V_{ℓ}/V_{ℓ}^{I} →^{∼} W, or, in our notation,

τ : T →^{∼} W

(cf. [G, 4.1.2], [Il, (2.6.3)]; see also our Remark in 0.3 for why we omit the Tate twist in this formula). In particular, rk_{V_{ℓ}}(τ) = r.

3.5. By extending scalars to \overline{ℚ}_{ℓ} we get the corresponding nilpotents (of the same order) in each irreducible component of V_{ℓ} ⊗ \overline{ℚ}_{ℓ} with the sum of the ranks with respect to each of the components being equal to the rank with respect to V_{ℓ}.

4. Minimal reduction

4.0. We say that an abelian variety A over a number field has minimal bad reduction at a prime ℘ of this field (or, just minimal reduction, for short) if the reduction is bad and the rank of the toric part H of Ã (cf. 3.0) is the minimal possible.

4.1. Let us go back to the case D = 𝕜, in which \overline{𝔤}_{ℓ} ⊆ \overline{𝔥}_{ℓ} ⊆ 𝔤𝔩(W_{λ}) →^{Δ} 𝔰𝔭(W_{λ} ⊕ W̌_{λ}) (cf. 0.6.2). The toric rank should be even (cf. 3.0.2), say, 2r. If τ′ = Δ(τ) ∈ Δ(\overline{𝔤}_{ℓ}) is a nilpotent of rank rk_{V_{ℓ}⊗\overline{ℚ}_{ℓ}}(τ′) = 2r (cf. 3.4.3), then τ^{2} = 0, rk_{W_{λ}}(τ) = r.

4.1.1. 3.0.2 and 3.4.3 imply that in the case D = 𝕜 the minimal toric rank is 2.

4.1.2. Hence in the minimal reduction case ∃ τ ∈ \overline{𝔤}_{ℓ} ⊆ \overline{𝔥}_{ℓ} ⊆ 𝔤𝔩(W_{λ}) such that τ^{2} = 0, rk_{W_{λ}}(τ) = 1. The same, clearly, holds if we replace the Lie algebras with their semi-simple components, since all nilpotents live in these components.

Let U := W_{λ}, and (as in 0.4.4) 𝔤̃ = \overline{𝔤}_{ℓ}^{ss}, 𝔥̃ = \overline{𝔥}_{ℓ}^{ss}, and rewrite the above as:

τ ∈ 𝔤̃ ⊆ 𝔥̃ ⊆ 𝔰𝔩(U), τ^{2} = 0, rk_{U}(τ) = 1,

𝔤̃, 𝔥̃ : semi-simple irreducible representations.

Such an element τ of rank 1 and order 2 is called a transvection.

4.2. It is a very restrictive condition for an irreducible representation of a semisimple Lie algebra to contain a transvection.

Lemma. If 𝔞 ↪ 𝔰𝔩(U) is a semi-simple faithful irreducible representation, τ ∈ 𝔞, τ^{2} = 0, rk_{U}(τ) = 1, then 𝔞 is simple and, moreover, it is either 𝔰𝔭(U) or 𝔰𝔩(U).

Proof. This is proved in [McL] (cf. also [PS]). □

4.3. We will also need the following simple fact.

Lemma. If rk_{U}(τ) is prime to dim(U), then 𝔞 is simple. □

5. Applications of minimal reduction

5.1. An immediate application of 4.1.2 (existence of rank 1 quadratic nilpotents in 𝔤) and Lemma 4.2 is the following theorem.

Theorem. If A is a simple abelian variety with D ⊆ 𝕜, having minimal reduction, then MT holds. Moreover, if D = ℚ, then all the Hodge and the Tate cycles are divisorial, hence the Hodge and the Tate conjectures hold.

Proof. 1. If D = ℚ, then the Tate module V_{ℓ}(A) is absolutely irreducible and symplectic. The minimality of reduction implies that the rank of a correspondent nilpotent is 1. The result now follows from 4.2.

2. If D = 𝕜, the result follows from 4.1.2, 4.2 and 1.1 (cf. 0.1.2). □

Remarks. 1. Such abelian varieties exist and, moreover, form a subset dense in the complex topology in the corresponding moduli space (cf. [L]).

2. The importance of the Weil type abelian varieties is not limited to the fact that they (may) have non-divisorial Weil cycles (cf. 4.4). Proving the algebraicity of the Weil cycles is a critical ingredient in proving the Tate conjecture (cf. [D 3, §§ 4∼6]; also [An], [Ab 2, § 6]).

5.2. Using the same method we can now extend Theorem 2.4 in the following way.

Theorem. If A is a simple abelian 4-fold with D = ℚ admitting bad but not purely multiplicative reduction, then all the Hodge and the Tate cycles are divisorial, hence the Hodge conjecture, the Tate conjecture and MT hold.

Proof. The possible values of the toric rank in this case are 1, 2, 3 (4 corresponds to the purely multiplicative reduction).

First recall (2.2) that the only choices for 𝔤̃ and 𝔥̃ are 𝔰𝔭_{8} or 𝔰𝔩_{2} × 𝔰𝔩_{2} × 𝔰𝔩_{2}. But 𝔰𝔩_{2}×𝔰𝔩_{2}×𝔰𝔩_{2} does not contain quadratic nilpotents of rank 1, 2 or 3. So, 𝔤̃ = 𝔥̃ ≅ 𝔰𝔭_{8} and 0.1.5 finishes the proof. □

Remarks. 1. Applying results/methods of [Z 5], [LZ] one can also verify all the conjectures for a simple abelian 4-fold admitting certain types of good reductions, further restricting the class of 4-folds for which the conjectures are not yet known.

2. G. Mustafin [Mus] has proved a “geometric analog” of MT for (families of) abelian varieties with purely multiplicative reduction (i.e., the algebraic envelope of the image of the monodromy coincides with the Hodge group for a “sufficiently general” abelian variety in the family), cf. also [H 3]. It appears, though, that his methods cannot be transplanted to the arithmetic situation.

6. Another type of bad reduction

6.0. Now we want to establish a result analogous to Theorem 5.1 for another type of bad reduction. Namely, consider an abelian variety A admitting bad (semi-stable, split) reduction of toric rank r such that r/(D : ℚ) is prime to 2 dim(A)/(D : ℚ), where as before D = End^{∘}(A). Assume also D is commutative. In this case 𝔤̃, 𝔥̃ (notation as in 4.1.2) are simple (cf. 4.3) and contain nilpotents of rank prime to the dimension of the representations (cf. 3.4.3). In place of Lemma 4.2 we use the results of Premet-Suprunenko [PS] on classification of quadratic elements (= nilpotents of order 2) and quadratic modules (= representations containing non-trivial quadratic elements) of simple Lie algebras.^{†}

We use the fact that the representations of 𝔤̃, 𝔥̃ are minuscule and 𝔤̃ is not exceptional (0.5.1).

We may assume that the dimensions of the representations is > 4.

6.1. One of our key tools replacing Lemma 4.2 is the following result.

Theorem. If 𝔞 ⊂ 𝔟 ↪ 𝔰𝔩(U), 𝔞 ≠ 𝔟, both Lie algebras are simple and the representations are (faithful) irreducible and minuscule, then 𝔟 is classical and (its highest weight is) ϖ_{1}.

Proof. Since any minuscule representation is quadratic (cf. [B, Ch VIII, §7.3]), we can apply [PS, Theorem 3], and exclude non-minuscule cases. □

6.2. We will be interested in 2 cases: D = 𝕜 and D = ℚ. In the former case we know that 𝔤̃ and 𝔥̃ are both non-self-dual (cf. Theorem 1.1) and if they satisfy the conditions of the theorem, then

– 𝔥̃ is classical, ϖ_{1} and non-self-dual, hence 𝔥̃ = (A_{n}, ϖ_{1})

– 𝔤̃ is classical, minuscule and non-self-dual, hence 𝔤̃ = (A_{m}, ϖ_{s}), m ≠ 2s. If D = ℚ, we know that 𝔤̃, 𝔥̃ are symplectic and we again have a unique possi-

bility for 𝔥̃, viz., 𝔥̃ = (C_{n}, ϖ_{1}). However, there are several a priori possible choices for 𝔤̃.

†This terminology is apparently standard in the finite groups theory, cf. [Th].

To eliminate (as many as we can) possibilities of 𝔤̃ ≠ 𝔥̃ we use the existence in the representations of a quadratic nilpotent of rank prime to the dimension of the representation.

So, we consider a slightly more general situation. As above • 𝔤̃ ⊊ 𝔥̃ ⊂ 𝔰𝔩(U), 𝔤̃, 𝔥̃ are classical simple Lie algebras,

• 𝔤̃ is minuscule, 𝔥̃ is ϖ_{1},

• there exists τ ∈ 𝔤̃ ⊂ 𝔥̃ ⊂ 𝔰𝔩(U) with τ^{2}(U) = 0, rk_{U}(τ) = r, dim(U) = n

and gcd(r, n) = 1.

We add the following condition wich is satisfied in both our cases:

• 𝔤̃, 𝔥̃ are non-self-dual, or orthogonal, or symplectic simultaneously.

6.3. First we exclude the cases 𝔤̃ = (D_{m}, ϖ_{m−1}), (D_{m}, ϖ_{m}) for m > 4.

Lemma. If τ ∈ 𝔤̃ ↪ 𝔰𝔩(U), 𝔤̃ = (D_{m}, ϖ_{m−1}) or (D_{m}, ϖ_{m}), τ is a quadratic element, then gcd(r, n) > 2.

Proof. [PS, Lemma 21, Note 2, Lemma 17] imply r = 2^{m−3} or 2^{m−2}, while n = 2^{m−1}. □

𝔤̃ ≠ (D_{4}, ϖ_{3}), (D_{4}, ϖ_{4}) either. It is enough to show this for ϖ_{4}, since they are (graph)isomorphic.

Proposition. (D_{4}, ϖ_{4}) ↛ (classical, ϖ_{1}).

Proof. Note that n = 8 in here. We do this case by case:

1. (D_{4}, ϖ_{4}) ↛ (B_{•}, ϖ_{1}), since the dimension of the RHS is odd.

2. (D_{4}, ϖ_{4}) ↛ (C_{•}, ϖ_{1}), since the LHS is orthogonal while the RHS is symplectic (cf. [B, table 1]).

3. (D_{4}, ϖ_{4}) ↛ (D_{4}, ϖ_{1}) (e.g., [Z 2, §5, Key lemma], although this is overkill).

4. (D_{4}, ϖ_{4}) ↛ (A_{•}, ϖ_{1}), since the LHS is orthogonal, the RHS is not. □

Proposition. If 𝔤̃ = (D_{m}, ϖ_{1}), then 𝔤̃ = 𝔥̃.

Proof. 𝔥̃ = A_{•}, C_{•} are excluded: 𝔤̃ is orthogonal, 𝔥̃ is not; 𝔥̃ = B_{•} is excluded by a dimensional reason. □

Proposition. If 𝔤̃ = (C_{m}, ϖ_{1}), then 𝔤̃ = 𝔥̃.

Proof. 𝔥̃ = A_{•}, B_{•}, D_{•} are not symplectic ... □

Proposition. If 𝔤̃ = (B_{m}, ϖ_{1}), then 𝔤̃ = 𝔥̃.

Proof. 𝔥̃ = A_{•}, C_{•} are not orthogonal... (D_{•}, ϖ_{1}) is even-dimensional... □

Proposition. If 𝔤̃ = (A_{m}, ϖ_{s}), then 𝔥̃ must be (A_{n−1}, ϖ_{1}).

Proof. First note that the only self-dual representation of A_{m} is ϖ_{s} with s = \frac{m+1}{2} (there is no such a representation if m is even). So, if 𝔥̃ is self-dual, then so is 𝔤̃ (cf. 6.2) and we may assume s = \frac{m+1}{2}, m : odd. But then dim(A_{m}, ϖ_{s}) = \binom{2s}{s} : even, thus 𝔥̃ ≠ B_{•}. To exclude the other cases (i.e., C_{•}, D_{•}) we use the fact that r = rk(τ) = \binom{m−1}{s−1} ([PS, §2 & Lemma 18]):

r = \binom{2(s − 1)}{s − 1}, n = \binom{2s}{s} = \binom{2(s − 1)}{s − 1}\frac{(2s − 1)2s}{s^{2}} = r \frac{2(2s − 1)}{s};

gcd(n, r) = 1 ⇒ r|s, which is not true: n > 4 ⇒ s > 3 ⇒ \binom{2(s−1)}{(s−1)} > s. □

Remark. If the 𝕜-signature of the abelian variety is (m_{σ}, m_{ρ}) with m_{σ} ≠ m_{ρ}, then, even if 𝔤̃ is not simple, simple components of 𝔤̃ are of type A (cf. [Y]).

So, the only possibility for 𝔤̃ ⫋ 𝔥̃ is 𝔤̃ = (A_{m}, ϖ_{s}) for some s, 𝔥̃ = (A_{n−1}, ϖ_{1}). In this case we can say the following.

Proposition. Let 𝔤̃ = (A_{m}, ϖ_{s}) ↪ 𝔥̃ = (A_{n−1}, ϖ_{1}) ≅ 𝔰𝔩(U) (fix the isomorphism), 2 ≤ s < \frac{m+1}{2}, τ ∈ 𝔤̃, r = rk_{U}(τ), gcd(n, r) = 1. Then either s = 3, m = 7, or s = 2.

Proof. r = \binom{m−1}{s−1} ([PS, §2 & Lemma 18]), n = \binom{m+1}{s} = \binom{m−1}{s−1} \frac{m(m+1)}{s(m+1−s)}, gcd(r, n) = 1 ⇒ r | s(m + 1 − s) and the result follows from the following simple observation:

\binom{m−1}{s−1}|s(m + 1 − s) if and only if (m, s) = (7, 3) or s = 2. □

Remark. If s = 2, then r = \binom{m−1}{s−1} = m−1, n = \binom{m+1}{s} = \frac{m(m+1)}{2}. Since gcd(m, m− 1) = 1 and gcd(m − 1, m + 1) = 1 or 2, gcd(r, n) = 1 if and only if m is even or m ≡ 1 (mod 4) (i.e., m ≢ 3 (mod 4)).

6.4. So, if D = 𝕜 we have the following result.

Theorem. If A is a simple abelian variety with D = 𝕜, g = dim(A) having bad reduction with the toric rank 2r and r is prime to g, then MT holds if (g, r) is neither (56, 15) nor of the form (\frac{m(m+1)}{2}, m − 1). □

6.5. If D = ℚ, then 𝔤̃, 𝔥̃ are symplectic, hence the theorem holds with no exceptions.

Theorem. If A is an abelian variety with D = ℚ, g = dim(A) having bad reduction with the toric rank r prime to 2g, then MT holds. □

6.6. Using the same methods one can handle the case of quadratic elements of rank 2. Namely, the following result holds.

Theorem. Let A be a simple abelian variety with D = ℚ. If A has bad reduction with toric rank 2, then MT holds.

Proof. (Sketch) Let dim(A) = g. We can assume g ≥ 4.

One can check (cf. [PS, §2, Lemma 18]) that if 𝔟 ↪ 𝔰𝔩(U) is a semi-simple irreducible classical Lie algebra, dim(U) ≥ 8 and ∃ τ ∈ 𝔟, τ^{2} = 0, rk_{U}(τ) = 2, then either 𝔟 is simple, and hence 𝔟 = 𝔰𝔩(U), 𝔰𝔭(U) or 𝔰𝔬(U) (since dim(U) ≥ 8, 𝔟 ≅ (A_{3}, ϖ_{2})), or 𝔟 = 𝔞 × 𝔰𝔩_{2}, where 𝔞 ≅ 𝔰𝔩_{g} or 𝔰𝔭_{g}. Since 𝔤̃, 𝔥̃ are symplectic, the only possibility is 𝔰𝔭(U) ! □

6.7. All the varieties considered in §6 exist and dense in the (complex topology in the) corresponding moduli spaces (cf. [L]).

6.8. 1. The idea of using special element(s) in the representation of the Hodge group has been used before. However, to our knowledge, in those earlier cases the element was semi-simple of low rank (e.g., [Z 1]) and the results then follow from a theorem of Kostant [Ko] (cf. also [Z 3]).

2. Katz used special unipotent elements to show that certain monodromy groups are large. However, the unipotents he considered were of the maximal possible rank, i.e., having only one Jordan block ([Ka 1, Ch. 7]).^{∗}

Katz was also using semi-simple elements for similar purposes, [Ka 2].

7. A curious result

7.1. Let us mention another application of Theorem 6.1.

Theorem. If A is a simple abelian variety with (semi-simple parts of) 𝔥, 𝔤_{ℓ} simple, satisfying one of the following conditions:

- the variety is of type I or II,

- D = 𝕜, the variety is of non-Weil type (i.e., the 𝕜-signature is (m_{σ}, m_{ρ})

with m_{σ} ≠ m_{ρ}), then one of the following must hold:

- MT holds for A,

- all the Hodge cycles (on A and all its self-products) are generated by divisor classes (hence the Hodge conjecture holds).

Proof. As we mentioned in 6.0, the representations of 𝔥_{ℓ}, 𝔤_{ℓ} are minuscule, hence quadratic (cf. [B, Ch VIII, §7.3, Proposition 7]), then so is 𝔩_{ℓ} := 𝔩 ⊗_{ℚ} ℚ_{ℓ}, where 𝔩 is the Lie algebra of the Lefschetz group (cf. 0.1.5), and 𝔤̃ ⊂ 𝔥̃ ⊂ 𝔩̃ ⊆ 𝔰𝔩(U) (cf. 0.2.3; here 𝔩̃:= \overline{𝔩}_{ℓ}^{ss}, \overline{𝔩}_{ℓ} = 𝔩_{ℓ} ⊗_{ℚ_{ℓ}} \overline{ℚ}_{ℓ}). If 𝔤̃ ⫋ 𝔥̃ (i.e., MT does not hold), then by Theorem 6.1 𝔥̃ ↪ 𝔰𝔩(U) is classical and ϖ_{1}. Thus 𝔩̃ is also simple, classical and ϖ_{1}. We want to show that in this case \overline{𝔥}_{ℓ} = \overline{𝔩}_{ℓ} and the theorem then follows from 0.1.5.

∗As an application of Katz’s classification of representations containing such unipotents (cf. loc.

cit., 11.5∼11.7) one can find modular curves for which the image of Galois in the corresponding ℓ-adic representation is large.

Consider first the case of an abelian variety of type I or II. The Lie algebras \overline{𝔥}_{ℓ} and \overline{𝔩}_{ℓ} are both symplectic, simple, classical and ϖ_{1}. Hence \overline{𝔥}_{ℓ} = 𝔰𝔭(U) (resp. 𝔰𝔬(U)) = \overline{𝔩}_{ℓ}.

If D = 𝕜, then C_{\overline{𝔥}_{ℓ}} is 1-dimensional (0.7.2), hence C_{\overline{𝔥}_{ℓ}} = C_{\overline{𝔩}_{ℓ}} and 𝔥̃ = 𝔩̃ = 𝔰𝔩(U) (cf. 1.1; also [Mu, Lemma 2.3]). The theorem follows. □

7.2. Remarks. 1. As one can see from the proof, if there is a way to assure that 𝔥 is simple and ϖ_{1}, then 𝔥 = 𝔩, hence (for types I or II) the Hodge conjecture holds (e.g., 2.3).

2. Abelian varieties with bad reduction as in 6.4∼6.5 have simple (semi-simple parts of) Hodge and Galois groups (cf. 4.3).

3. For the Weil type varieties, i.e., m_{σ} = m_{ρ}, C_{\overline{𝔥}_{ℓ}} = \{0\} ≠ C_{\overline{𝔩}_{ℓ}} generically. This is (a restatement of) the main result of [W].

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Alex Lesin, Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA 94720

E-mail address: shura@msri.org