The Mumford–Tate Conjecture for the Product

of an Abelian Surface and a K3 Surface

Johan Commelin

Received: February 9, 2016 Revised: October 25, 2016

Communicated by Takeshi Saito

Abstract. The Mumford–Tate conjecture is a precise way of saying that the Hodge structure on singular cohomology conveys the same information as the Galois representation on ℓ-adic étale cohomology, for an algebraic variety over a finitely generated field of characteristic 0. This paper presents a proof of the Mumford–Tate conjecture in degree 2 for the product of an abelian surface and a K3 surface.

2010 Mathematics Subject Classification: Primary 14C15; Secondary 14C30, 11G10, 14J28. Keywords and Phrases: Mumford–Tate conjecture, abelian surface, K3 surface.

1 Introduction

The main result of this paper is the following theorem. The next paragraph recalls the Mumford–Tate conjecture; and §1.5 gives an outline of the proof.

1.1 Theorem. — Let K be a finitely generated subfield of ℂ. If A is an abelian surface over K and X is a K3 surface over K, then the Mumford–Tate conjecture is true for H^{2}(A × X)(1).

1.2 The Mumford–Tate conjecture. — Let K be a finitely generated field of characteristic 0; and let K ↪ ℂ be an embedding of K into the complex numbers. Let K̄ be the algebraic closure of K in ℂ. Let X/K be a smooth projective variety. One may attach several cohomology groups to X. For the purpose of this article we are interested in two cohomology theories: Betti cohomology and ℓ-adic étale cohomology (for a prime number ℓ). We will write H_{B}^{w}(X) for the ℚ-Hodge structure H_{sing}^{w}(X(ℂ), ℚ) in weight w. Similarly, we write H_{ℓ}^{w}(X) for the Gal(K̄/K)-representation H_{ét}^{W} (XK̄, ℚ_{ℓ}).

The Mumford–Tate conjecture was first posed for abelian varieties, by Mumford in [22], and later extended to general algebraic varieties. The conjecture is a precise way of saying that the cohomology groups H_{B}^{w}(X) and H_{ℓ}^{w}(X) contain the same information about X. To make this precise, let G_{B}(H_{B}^{w}(X)) be the Mumford–Tate group of the Hodge structure H_{B}^{w}(X), and let G_{ℓ}^{∘}(H_{ℓ}^{w}(X)) be the connected component of the Zariski closure of Gal(K̄/K) in GL(H_{ℓ}^{w}(X)). The comparison theorem by Artin, comparing singular cohomology with étale cohomology, canonically identifies GL(H_{B}^{w}(X)) ⊗ ℚ_{ℓ} with GL(H_{ℓ}^{w}(X)). The Mumford–Tate conjecture (for the prime ℓ, and the embedding K ↪ ℂ) states that under this identification

G_{B}(H_{B}^{w}(X)) ⊗ ℚ_{ℓ} ≅ G_{ℓ}^{∘}(H_{ℓ}^{w}(X)).

1.3 Notation and terminology. — Like above, let K be a finitely generated field of characteristic 0; and fix an embedding K ↪ ℂ. In this article we use the language of motives in the sense of André, [2]. To be precise, our category of base pieces is the category of smooth projective varieties over K, and our reference cohomology is Betti cohomology, H_{B}(_). We stress that H_{B}(_) depends on the chosen embedding K ↪ ℂ, but the resulting category of motives does not depend on K ↪ ℂ, see proposition 2.3 of [2]. We write H^{w}(X) for the motive of weight w associated with a smooth projective variety X/K. The Mumford–Tate conjecture naturally generalises to motives, as follows: Let M be a motive. We will write H_{B}(M) for its Hodge realisation; H_{ℓ}(M) for its ℓ-adic realisation; G_{B}(M) for its Mumford–Tate group (i.e., the Mumford– Tate group of H_{B}(M)); and G_{ℓ}^{∘}(M) for G_{ℓ}^{∘}(H_{ℓ}(M)). We will use the notation MTC_{ℓ}(M) for the conjectural statement

G_{B}(M) ⊗ ℚ_{ℓ} ≅ G_{ℓ}^{∘}(M),

Mumford–Tate for Product of Abelian Surface and K3 1693

and MTC(M) for the assertion ∀ℓ : MTC_{ℓ}(M). The following remark allows us to take finitely generated extensions of the base field, whenever needed. If K ⊂ L is an extension of finitely generated subfields of ℂ, and if M is a motive over K, then MTC(M) ⟺ MTC(M_{L}); see proposition 1.3 of [19]. In this paper, we never use specific properties of the chosen embedding K ↪ ℂ, and all statements are valid for every such embedding. In particular, we will speak about subfields of ℂ, where the embedding is implicit. In this paper, we will use compatible systems of ℓ-adic representations. For more information we refer to the book [27] by Serre, the letters of Serre to Ribet (see [28]), or the work of Larsen and Pink [16, 17]. Throughout this paper, A is an abelian variety, over some base field. (Outside section 5, it is even an abelian surface.) Assume A is absolutely simple; and choose a polarisation on A. Let (D, †) denote its endomorphism ring End^{0}(A) together with the Rosati involution associated with the polarisation. The simple algebra D together with the positive involution † has a certain type in the Albert classification that does not depend on the chosen polarisation (see theorem 2 on page 201 of [23]). We say that A is of type x if (D, †) is of type x, where x runs through {i, …, iv}. If E denotes the center of D, with degree e = [E : ℚ], then we also say that A is of type x(e). Whenever we speak of simple groups or simple Lie algebras, we mean noncommutative simple groups, and non-abelian simple Lie-algebras. Let T be a type of Dynkin diagram (e.g., A_{n}, B_{n}, C_{n} or D_{n}). Let 𝔤 be a semisimple Lie algebra over K. We say that T does not occur in the Lie type of 𝔤, if the Dynkin diagram of 𝔤K̄ does not have a component of type T. For a semisimple group G over K, we say that T does not occur in the Lie type of G, if T does not occur in the Lie type of Lie(G).

1.4 — Let K be a finitely generated subfield of ℂ. Let A/K be an abelian surface, and let X/K be a K3 surface. Since H^{1}(X) = 0, Künneth’s theorem gives H^{2}(A × X) ≅ H^{2}(A) ⊕ H^{2}(X). Recall that the Mumford–Tate conjecture for A is known in degree 1, and hence in all degrees. (This is classical, but see corollary 4.4 of [18] for a reference.) The Mumford–Tate conjecture for X (in degree 2) is true as well, by [30, 31, 1]. Still, it is not a formal consequence that the Mumford–Tate conjecture for A × X is true in degree 2: for two motives M_{1} and M_{2}, it is a formal fact that G_{B}(M_{1} ⊕ M_{2}) ↪ G_{B}(M_{1}) × G_{B}(M_{2}), with surjective projections onto G_{B}(M_{1}) and G_{B}(M_{2}). However, this inclusion need not be surjective. For example, if M_{1} = M_{2}, then map in question is the diagonal inclusion. Similar remarks hold for G_{ℓ}^{∘}(_).

1.5 Outline of the proof. — Roughly speaking, the proof of theorem 1.1 as presented in this paper, is as follows:

1. We replace H^{2}(A)(1) and H^{2}(X)(1) by their transcendental parts M_{A} and M_{X}. It suffices to prove MTC(M_{A} ⊕ M_{X}), see lemma 7.2.

2. It suffices to show that G_{ℓ}^{∘}(M_{A} ⊕ M_{X})^{der} ≅ G_{ℓ}^{∘}(M_{A})^{der} × G_{ℓ}^{∘}(M_{X})^{der}, see

lemma 7.3. Write 𝔤_{A} for Lie(G_{B}(M_{A})^{der}). There exists a number field F_{A} acting on 𝔤_{A}, such that 𝔤_{A} viewed as F_{A}-Lie algebra is a direct sum of absolutely simple factors. Analogously we find 𝔤_{X} and F_{X}; see §6.4. We explicitly compute 𝔤_{A}, 𝔤_{X}, F_{A}, and F_{X} in remark 6.3 and remark 6.6.

3. Using Čebotarëv’s density theorem, we show that (2.) can be applied if F_{A} ≅ F̸_{X}; see lemma 7.4.

4. By Goursat’s lemma, (2.) can also be applied if 𝔤_{A} ⊗ ℂ and 𝔤_{X} ⊗ ℂ do not have a common simple factor; see lemma 7.6.

5. The previous two items cover almost all cases. The remaining cases (listed in §7.5) fall into two subcases.

(a) If dim(M_{X}) ≤ 5, we show that we may replace X by its Kuga–Satake abelian variety B. We then prove MTC(A × B) using techniques of Lombardo, developed in [18]; see lemma 7.7. Note: In this case the condition in (2.) might not hold (for example if X is the Kummer variety associated with A).

(b) In the final case F_{A} ≅ F_{X} is totally real quadratic. We show that if the condition in (2.) does not hold, then there is a prime number ℓ for which H_{ℓ}(M_{X}) is not an irreducible Galois representation. However, by assembling results from [4], [6], and [34], we see that there exist places for which the characteristic polynomial of Frobenius acting on H_{ℓ}(M_{X}) is an irreducible polynomial. This leads to a contradiction, and thus (2.) can be applied in this final case; see lemma 7.9.

1.6 Acknowledgements. — I first and foremost thank Ben Moonen, my supervisor, for his inspiration and help with critical parts of this paper. Part of this work was done while I was visiting Matteo Penegini at the University of Milano; and I thank him for the hospitality and the inspiring collaboration. I thank Bert van Geemen, Davide Lombardo, Milan Lopuhaä, and Lenny Taelman for useful discussions about parts of the proof. I thank Ronald van Luijk for telling me about [6]. All my colleagues in Nijmegen who provided encouraging or insightful remarks during the process of research and writing also deserve my thanks. Further thanks goes to grghxy, Guntram, and Mikhail Borovoi on mathoverflow.net^{1}. I sincerely thank the referee for all the valuable comments and structural remarks that helped improve this article. This research has been financially supported by the Netherlands Organisation for Scientific Research (NWO) under project no. 613.001.207 (Arithmetic and motivic aspects of the Kuga–Satake construction).

2 Galois theoretical preliminaries

Recall the following elementary result from finite group theory.

2.1 Lemma. — Let T be a set with a transitive action by a finite group G. Fix n ∈ ℤ_{≥0}, and let C ⊂ G be the set of elements g ∈ G that fix at least n points of T. If n · |C| ≥ |G|, then |T | = n. Proof. By computing the cardinality of {(g, t) ∈ G × T | gt = t} in two distinct ways, one gets the formula |G| · |G\T | = ∑_{g∈G}|T^{g}|. From this we derive,

1 = |G\T | = \frac{1}{|G|} ∑_{g∈G}|T^{g}| ≥ \frac{n · |C|}{|G|} ≥ 1.

Hence n · |C| = |G| and all elements in C have exactly n fixed points. In particular the identity element has n fixed points, which implies |T | = n. □ Note that, in the setting of the previous lemma, if the action of G on T is faithful, then |G| = n, and T is principal homogeneous under G.

2.2 Lemma. — Let F_{1} be a Galois extension of ℚ. Let F_{2} be a number field such that for all prime numbers ℓ, the product of local fields F_{1} ⊗ ℚ_{ℓ} is a factor of F_{2} ⊗ ℚ_{ℓ}. Then F_{1} ≅ F_{2}.

1A preliminary version of lemma 2.2 arose from a question on MathOverflow titled “How simple does a ℚ-simple group remain after base change to ℚ_{ℓ}?” (http://mathoverflow.net/ q/214603/78087). The answers also inspired lemma 2.1.

Documenta Mathematica 21 (2016) 1691–1713

Proof. Let L be a Galois closure of F_{2}, and let G be the Galois group Gal(L/ℚ), which acts naturally on the set of field embeddings Σ = Hom(F_{2}, L). Let n be the degree of F_{1}, and let C be the set {g ∈ G | |Σ^{g}| ≥ n} of elements in G that have at least n fixed points in Σ. By Čebotarëv’s density theorem (Satz VII.13.4 of [24]), the set of primes that split completely in F_{1}/ℚ has density 1/n. Therefore, the set of primes ℓ for which F_{2}⊗ℚ_{ℓ} has a semisimple factor isomorphic to (ℚ_{ℓ})^{n} must have density ≥ 1/n. In other words, n · |C| ≥ |G|. By lemma 2.1, this implies |Σ| = n, and since G acts faithfully on Σ, we find that F_{2}/ℚ is Galois of degree n. Because Galois extensions of number fields can be recovered from their set of splitting primes (Satz VII.13.9 of [24]), we conclude that F_{1} ≅ F_{2}. □

2.3 Lemma. — Let F_{1} be a quadratic extension of ℚ. Let F_{2} be a number field of degree ≤ 5 over ℚ. If for all prime numbers ℓ, the products of local fields F_{1} ⊗ ℚ_{ℓ} and F_{2} ⊗ ℚ_{ℓ} have an isomorphic factor, then F_{1} ≅ F_{2}. Proof. Let L be a Galois closure of F_{2}, and let G be the Galois group Gal(L/ℚ), which acts naturally on the set of field embeddings Σ = Hom(F_{2}, L). Observe that G acts transitively on Σ, and we identify G with its image in 𝔖(Σ). Write n for the degree of F_{2} over ℚ, which also equals |Σ|. The order of G is divisible by n. Hence, if n is prime, then G must contain an n-cycle. Suppose that G contains an n-cycle. By Čebotarëv’s density theorem there must be a prime number ℓ that is inert in F_{2}/ℚ. By our assumption F_{2} ⊗ ℚ_{ℓ} also contains a factor of at most degree 2 over ℚ_{ℓ}. This shows that n = 2. If n = 4, then G does not contain an n-cycle if and only if it is isomorphic to V_{4} or A_{4}. If G ≅ V_{4}, then only the identity element has fixed points, and by Čebotarëv’s density theorem this means that the set of primes ℓ for which F_{2} ⊗ ℚ_{ℓ} has a factor ℚ_{ℓ} has density 1/4, whereas the set of primes splitting in F_{1}/ℚ has density 1/2. On the other hand, if G ≅ A_{4}, then only 3 of the 12 elements have a 2-cycle in the cycle decomposition, and by Čebotarëv’s density theorem this means that the set of primes ℓ for which F_{2} ⊗ ℚ_{ℓ} has a factor isomorphic to a quadratic extension of ℚ_{ℓ} has density 1/4, whereas the set of primes inert in F_{1}/ℚ has density 1/2. This gives a contradiction. We conclude that n must be 2; and therefore F_{1} ≅ F_{2}, by lemma 2.2. □

3 A result on semisimple groups over number fields

Throughout this section K is a field of characteristic 0.

3.1 Remark. — Let 𝔥 ⊂ 𝔤_{1} ⊕ 𝔤_{2} be Lie algebras over K such that 𝔥 projects surjectively onto 𝔤_{1} and 𝔤_{2}. Assume that 𝔤_{1} and 𝔤_{2} are finite-dimensional and semisimple. Then there exist semisimple Lie algebras 𝔰_{1}, 𝔱, and 𝔰_{2} such that 𝔤_{1} ≅ 𝔰_{1} ⊕ 𝔱, 𝔤_{2} ≅ 𝔱 ⊕ 𝔰_{2}, and 𝔥 ≅ 𝔰_{1} ⊕ 𝔱 ⊕ 𝔰_{2}. (To see this, recall that a finite-dimensional semisimple Lie algebra is the sum of its simple ideals.)

3.2 Lemma. — Let K ⊂ F be a finite field extension, and let G/F be an algebraic group. If Lie(G)_{K} denotes the Lie algebra Lie(G) viewed as K-algebra, then Lie(Res_{F/K} G) is naturally isomorphic to Lie(G)_{K}.

Proof. The Lie algebra Lie(G) is the kernel of the natural map G(F [ε]) → G(F); where the Lie bracket may be described as follows: Consider the map

\begin{matrix}G(F [ε_{1}]) × G(F [ε_{2}]) & ⟶ & G(F [ε_{1}, ε_{2}]) \\ (X_{1}, X_{2}) & ⟼ & X_{1}X_{2}X_{1}^{−1}X_{2}^{−1} \end{matrix}

If (X_{1}, X_{2}) ∈ Lie(G) × Lie(G), then a calculation shows that its image lands in G(F [ε_{1} · ε_{2}]) ≅ G(F [ε]). What is more, it lands in Lie(G) ⊂ G(F [ε]). We denote this image with [X_{1}, X_{2}], the Lie bracket of X_{1} and X_{2}. Let G_{F/K} denote Res_{F/K} G. The following diagram shows that Lie(G_{F/K}) is canonically identified with Lie(G)_{K} as K-vector space, and that the Lie bracket is preserved.

\begin{matrix}0 & Lie(G_{F/K}) & G_{F/K}(K[ε]) & G_{F/K}(K) & 0 \\ 0 & Lie(G) & G(F [ε]) & G(F) & 0 \end{matrix} ; ≃ ; ≃ □

3.3 — Let F/K be a finite field extension. Let 𝔤 be a Lie algebra over F, and write (𝔤)_{K} for the Lie algebra 𝔤 viewed as K-algebra. If 𝔤 is an F -simple Lie algebra, then (𝔤)_{K} is K-simple: indeed, if (𝔤)_{K} ≅ 𝔥 ⊕ 𝔥′, and X ∈ 𝔥, then [fX, X′] = 0 for all f ∈ F and X′ ∈ 𝔥′. This implies fX ∈ 𝔥, hence 𝔥 = 𝔤.

3.4 Lemma. — For i = 1, 2, let K ⊂ F_{i} be a finite field extension, and let 𝔤_{i}/F_{i} be a product of absolutely simple Lie algebras (cf. our conventions in §1.3). If (𝔤_{1})_{K} and (𝔤_{2})_{K} have an isomorphic factor, then F_{1} ≅_{K} F_{2}. Proof. By the remark before this lemma, the K-simple factors of (𝔤_{i})_{K} are all of the form (𝔱_{i})_{K}, where 𝔱_{i} is an F_{i}-simple factor of 𝔤_{i}. So if (𝔤_{1})_{K} and (𝔤_{2})_{K} have an isomorphic factor, then there exist F_{i}-simple factors 𝔱_{i} of 𝔤_{i} for which there exists an isomorphism f: (𝔱_{1})_{K} → (𝔱_{2})_{K}. Let K̄ be an algebraic closure of K. Observe that

(𝔱_{i})_{K} ⊗_{K} K̄ ≅ ⨁_{σ∈Hom_{K}(F_{i},K̄)} 𝔱_{i} ⊗_{F_{i},σ} K̄,

and note that Gal(K̄/K) acts transitively on Hom_{K}(F_{i}, K̄). By assumption, the 𝔱_{i} are absolutely simple, hence the 𝔱_{i} ⊗_{F_{i},σ} K̄ are precisely the simple ideals of (𝔱_{i})_{K} ⊗_{K} K̄. Thus the isomorphism f gives a Gal(K̄/K)-equivariant bijection between the simple ideals of (𝔱_{1})_{K} ⊗_{K} K̄ and (𝔱_{2})_{K} ⊗_{K} K̄; and therefore between Hom_{K}(F_{1}, K̄) and Hom_{K}(F_{2}, K̄) as Gal(K̄/K)-sets. This proves the result. □

3.5 Lemma. — For i = 1, 2, let F_{i} be a number field, and let G_{i}/F_{i} be an almost direct product of connected absolutely simple F_{i}-groups. Let ℓ be a prime number, and let ι_{ℓ}: G ↪ (Res_{F_{1}/ℚ} G_{1})_{ℚ_{ℓ}} × (Res_{F_{2}/ℚ} G_{2})_{ℚ_{ℓ}} be a subgroup over ℚ_{ℓ}, with surjective projections onto both factors. If ι_{ℓ} is not an isomorphism, then F_{1} ⊗ ℚ_{ℓ} and F_{2} ⊗ ℚ_{ℓ} have an isomorphic simple factor.

Proof. Note that (Res_{F_{i}/ℚ} G_{i}) ⊗ ℚ_{ℓ} ≅ ∏_{λ|ℓ} Res_{F_{i,λ}/ℚ_{ℓ}}(G_{i} ⊗_{F_{i}} F_{λ}). By remark 3.1 there exist places λ_{i} of F_{i} over ℓ such that Lie(Res_{F_{1,λ_{1}} /ℚ_{ℓ}}(G_{1} ⊗_{F_{1}} F_{1,λ_{1}})) and Lie(Res_{F_{2,λ_{2}} /ℚ_{ℓ}}(G_{2} ⊗_{F_{2}} F_{2,λ_{2}})) have an isomorphic factor. By lemmas 3.2 and 3.4, this implies that F_{1,λ_{1}} ≅_{ℚ_{ℓ}} F_{2,λ_{2}}, which proves the lemma. □

4 Several results on abelian motives

4.1 Definition. — A motive M over a field K ⊂ ℂ is abelian (or an abelian motive) if it satisfies one of the following three equivalent conditions:

1. M is isomorphic to an object in the Tannakian subcategory generated by all abelian varieties over K.

2. There exists an abelian variety A over K such that M is isomorphic to an object in the Tannakian subcategory ⟨H(A)⟩^{⊗} generated by H(A). (Note that ⟨H(A)⟩^{⊗} = ⟨H^{1}(A)⟩^{⊗}.)

3. There is an isomorphism M ≅ ⨁_{i} M_{i}, and for each M_{i} there exists an abelian variety A_{i} such that M_{i} is a subobject of H(A_{i})(n_{i}).

4.2 Theorem. — The Hodge realisation functor H_{B}(_) is fully faithful on the subcategory of abelian motives over ℂ. As a consequence, if K is a finitely generated subfield of ℂ, and if M is an abelian motive over K, then the natural inclusion G_{B}(M) ⊂ G(M)^{∘} is an isomorphism, and G_{ℓ}^{∘}(M) ⊂ G_{B}(M) ⊗ ℚ_{ℓ}. Proof. This is an immediate consequence of théorème 0.6.2 of [2]. □

4.3 Remark. — All the motives in this article are abelian motives. For the motives coming from abelian varieties, this is obvious. The motive H^{2}(X), where X is a K3 surface, is also an abelian motive, by théorème 0.6.3 of [2].

4.4 Proposition. — The Mumford–Tate conjecture on centres is true for abelian motives. In other words, let M be an abelian motive. Let Z_{B}(M) be the centre of the Mumford–Tate group G_{B}(M), and let Z_{ℓ}(M) be the centre of G_{ℓ}^{∘}(M). Then Z_{ℓ}(M) ≅ Z_{B}(M) ⊗ ℚ_{ℓ}. Proof. The result is true for abelian varieties (see theorem 1.3.1 of [33] or corollary 2.11 of [32]). By definition of abelian motive, there is an abelian variety A such that M is contained in the Tannakian subcategory of motives generated by H(A). This yields a surjection G_{B}(A) ↠ G_{B}(M). Since G_{B}(A) is reductive, Z_{B}(M) is the image of Z_{B}(A) under this map. The same is true on the ℓ-adic side. (Note that G_{ℓ}^{∘}(A) is reductive, by Satz 3 in §5 of [10].) Thus we obtain a commutative diagram with solid arrows

\begin{matrix}Z_{ℓ}(A) & Z_{ℓ}(M) & G_{ℓ}^{∘}(M) \\ Z_{B}(A) ⊗ ℚ_{ℓ} & Z_{B}(M) ⊗ ℚ_{ℓ} & G_{B}(M) ⊗ ℚ_{ℓ} \end{matrix} ≃ ≃

which shows that the dotted arrow exists and is an isomorphism. (The vertical arrow on the right exists and is an inclusion, by theorem 4.2.) □

4.5 Lemma. — Let M be an abelian motive. Assume that the ℓ-adic realisations of M form a compatible system of ℓ-adic representations. If there is one prime ℓ for which the absolute rank of G_{ℓ}^{∘}(M) is equal to the absolute rank of G_{B}(M), then the Mumford–Tate conjecture is true for M. Proof. The absolute rank of G_{ℓ}^{∘}(M) does not depend on ℓ (see proposition 6.12 of [16] or Serre’s letter to Ribet). Since M is an abelian motive, we have G_{ℓ}^{∘}(M) ⊂ G_{B}(M) ⊗ ℚ_{ℓ} (see theorem 4.2). The Mumford–Tate conjecture follows from Borel–de Siebenthal theory^{2}: since G_{ℓ}^{∘}(M) ⊂ G_{B}(M) ⊗ ℚ_{ℓ} has maximal rank, it is equal to the connected component of the centraliser of its centre. By proposition 4.4, we know that the centre of G_{ℓ}^{∘}(M) equals the centre of G_{B}(M) ⊗ ℚ_{ℓ}. Hence G_{ℓ}^{∘}(M) ≅ G_{B}(M) ⊗ ℚ_{ℓ}. □

4.6 — Let K be a finitely generated subfield of ℂ. A pair (A, X), consisting of an abelian surface A and a K3 surface X over K, is said to satisfy condition 4.6 for ℓ if

G_{ℓ}^{∘}(H^{2}(A × X)(1))^{der} ⟶ G_{ℓ}^{∘}(H^{2}(A)(1))^{der} × G_{ℓ}^{∘}(H^{2}(X)(1))^{der}

is an isomorphism.

4.7 — Let ℓ be a prime number. Let G_{1} and G_{2} be connected reductive groups over ℚ_{ℓ}. By a (G_{1}, G_{2})-tuple over K we shall mean a pair (A, X), where A is an abelian surface over K, and X is a K3 surface over K such that G_{ℓ}^{∘}(H^{2}(A)(1)) ≅ G_{1} and G_{ℓ}^{∘}(H^{2}(X)(1)) ≅ G_{2}. We will show in section 7 that there exist groups G_{1} and G_{2} that satisfy the hypothesis of the following lemma, namely that condition 4.6 for ℓ is satisfied for all (G_{1}, G_{2})-tuples over number fields.

4.8 Lemma. — Let ℓ be a prime number. Let G_{1} and G_{2} be connected reductive groups over ℚ_{ℓ}. If for all number fields K, all (G_{1}, G_{2})-tuples (A, X) over K satisfy condition 4.6 for ℓ, then for all finitely generated subfields L of ℂ, all (G_{1}, G_{2})-tuples (A, X) over L satisfy condition 4.6 for ℓ. Proof. Let (A, X) be a (G_{1}, G_{2})-tuple over a finitely generated field L. Then there exists a (not necessarily proper) integral scheme S/ℚ, with generic point η, an abelian scheme A/S, and a K3 surface X /S, such that L is isomorphic to the function field of S, A_{η} ≅ A, and X_{η} ≅ X. By a result of Serre (using Hilbert’s irreducibility theorem, see Serre’s letters to Ribet [28], or section 10.6 of [26]), there exists a closed point c ∈ S such that G_{ℓ}^{∘}(H_{ℓ}^{2}(A_{c}×X_{c})(1)) ≅ G_{ℓ}^{∘}(H_{ℓ}^{2}(A×X)(1)). Since G_{ℓ}^{∘}(H_{ℓ}^{2}(A×X)(1)) surjects onto G_{ℓ}^{∘}(H_{ℓ}^{2}(A)(1)), we find that G_{ℓ}^{∘}(H_{ℓ}^{2}(A_{c})(1)) ≅ G_{ℓ}^{∘}(H_{ℓ}^{2}(A)(1)), and similar for X. The following diagram shows that shows that (A, X) satisfies condition 4.6 for ℓ if (A_{c}, X_{c}) satisfies it.

2The original article [5], by Borel and de Siebenthal deals with the case of compact Lie groups. See http://www.math.ens.fr/~gille/prenotes/bds.pdf for a proof in the case of reductive algebraic groups.

Documenta Mathematica 21 (2016) 1691–1713

G_{ℓ}^{∘}(H^{2}(A_{c} × X_{c})(1))^{der}  G_{ℓ}^{∘}(H^{2}(A_{c})(1))^{der} × G_{ℓ}^{∘}(H^{2}(X_{c})(1))^{der}

G_{ℓ}^{∘}(H^{2}(A × X)(1))^{der}  G_{ℓ}^{∘}(H^{2}(A)(1))^{der} × G_{ℓ}^{∘}(H^{2}(X)(1))^{der}

≃

≃
□

5 Some remarks on the Mumford–Tate conjecture for abelian varieties

5.1 — For the convenience of the reader, we copy some results from [18]. Before we do that, let us recall the notion of the Hodge group, Hdg_{B}(A), of an abelian variety. Let A be an abelian variety over a finitely generated field K ⊂ ℂ. By definition, the Mumford–Tate group of an abelian variety is G_{B}(A) = G_{B}(H_{B}^{1}(A)) ⊂ GL(H_{B}^{1}(A)), and we put

Hdg_{B}(A) = (G_{B}(A) ∩ SL(H_{B}^{1}(A)))^{∘} and Hdg_{ℓ}(A) = (G_{ℓ}(A) ∩ SL(H_{ℓ}^{1}(A)))^{∘}.

The equivalence

MTC_{ℓ}(A) ⟺ Hdg_{B}(A) ⊗ ℚ_{ℓ} ≅ Hdg_{ℓ}(A).

is a consequence of proposition 4.4.

5.2 Definition (1.1 in [18]). — Let A be an absolutely simple abelian variety of dimension g over K. The endomorphism ring D = End^{0}(A) is a division algebra. Write E for the centre of D. The ring E is a field, either tr (totally real) or cm. Write e for [E : ℚ]. The degree of D over E is a perfect square d^{2}. The relative dimension of A is

reldim(A) = \begin{cases}\frac{g}{de}, & if A is of type i, ii, or iii, \\ \frac{2g}{de}, & if A is of type iv. \end{cases}

Note that d = 1 if A is of type i, and d = 2 if A is of type ii or iii. In definition 2.22 of [18], Lombardo defines when an abelian variety is of general Lefschetz type. This definition is a bit unwieldy, and its details do not matter too much for our purposes. What matters are the following results, that prove that certain abelian varieties are of general Lefschetz type, and that show why this notion is relevant for us.

5.3 Lemma. — Let A be an absolutely simple abelian variety over a finitely generated subfield of ℂ. Assume that A is of type i or ii. If reldim(A) is odd, or equal to 2, then A is of general Lefschetz type. Proof. If reldim(A) is odd, then this follows from theorems 6.9 and 7.12 of [3]. Lombardo notes (remark 2.25 in [18]) that the proof of [3] also works if reldim(A) = 2, and also refers to theorem 8.5 of [7] for a proof of that fact. □

5.4 Lemma. — Let K be a finitely generated subfield of ℂ. Let A_{1} and A_{2} be two abelian varieties over K that are isogenous to products of abelian varieties of general Lefschetz type. If D_{4} does not occur in the Lie type of Hdg_{ℓ}(A_{1}) and Hdg_{ℓ}(A_{2}), then either

Hom_{K}(A_{1}, A_{2}) ≠ 0, or Hdg_{ℓ}(A_{1} × A_{2}) ≅ Hdg_{ℓ}(A_{1}) × Hdg_{ℓ}(A_{2}).

Proof. This is remark 4.3 of [18], where Lombardo observes that, under the assumption of the lemma, theorem 4.1 of [18] can be applied to products of abelian varieties of general Lefschetz type. □

5.5 Lemma. — Let A be an abelian variety over a finitely generated field K ⊂ ℂ. Let L ⊂ ℂ be a finite extension of K for which A_{L} is isogenous over L to a product of absolutely simple abelian varieties ∏ A_{i}^{k_{i}}. Assume that for all i the following conditions are satisfied:

(a) either A_{i} is of general Lefschetz type or A_{i} is of cm type;

(b) the Lie type of Hdg_{ℓ}(A_{i}) does not contain D_{4};

(c) the Mumford–Tate conjecture is true for A_{i}. Under these conditions the Mumford–Tate conjecture is true for A. Proof. Recall that MTC(A) ⟺ MTC(A_{L}). Note that MTC(A_{L}) is equivalent to MTC(∏ A_{i}), since H^{1}(A_{L}) and H^{1}(∏ A_{i}) = ⨁ H^{1}(A_{i}) generate the same Tannakian subcategory of motives. Observe that

Hdg_{ℓ}(∏ A_{i}) ⊂ Hdg_{B}(∏ A_{i}) ⊗ ℚ_{ℓ} ⊂ ∏ Hdg_{B}(A_{i}) ⊗ ℚ_{ℓ} = ∏ Hdg_{ℓ}(A_{i}),

where the first inclusion is Deligne’s “Hodge = absolute Hodge” theorem (see proposition 2.9 and theorem 2.11 of [9], or see theorem 4.2); the second inclusion follows from the fact that the Hodge group of a product is a subgroup of the product of the Hodge groups (with surjective projections); and the last equality is condition (c). If we ignore the factors that are cm, then an inductive application of the previous lemma yields Hdg_{ℓ}(A) = ∏ Hdg_{ℓ}(A_{i}). If we do not ignore the factors that are cm, then we actually get Hdg_{ℓ}(A)^{der} = ∏ Hdg_{ℓ}(A_{i})^{der}. Together with proposition 4.4, this proves Hdg_{ℓ}(A) = Hdg_{B}(A) ⊗ ℚ_{ℓ}. □ As an illustrative application of this result, Lombardo observes in corollary 4.5 of [18] that the Mumford–Tate conjecture is true for arbitrary products of elliptic curves and abelian surfaces.

6 Hodge theory of K3 surfaces and abelian surfaces

6.1 — In this section we recall some results of Zarhin that describe all possible Mumford–Tate groups of Hodge structures of K3 type, i.e., Hodge structures of weight 0 with Hodge numbers of the form (1, n, 1). The canonical example of a Hodge structure of K3 type is the Tate twist of the cohomology in degree 2 of a complex K3 surface X. Namely the Hodge structure H_{B}^{2}(X)(1) has Hodge numbers (1, 20, 1). Another example is provided by abelian surfaces, which is the content of remark 6.6 below.

6.2 Theorem. — Let (V, ψ) be a polarised irreducible Hodge structure of K3 type.

1. The endomorphism algebra E of V is a field.

2. The field E is tr (totally real) or cm.

3. If E is tr, then dim_{E}(V) ≥ 3.

4. Let E_{0} be the maximal totally real subfield of E. Let ψ̃ be the unique E-bilinear (resp. hermitian) form such that ψ = tr_{E/ℚ} ∘ψ̃ if E is tr (resp. cm). The Mumford–Tate group of V is

G_{B}(V) ≅ \begin{cases}Res_{E/ℚ} SO(ψ̃), & if E is tr; \\ Res_{E_{0}/ℚ} U(ψ̃), & if E is cm. \end{cases}

(Here U(ψ̃) is the unitary group over E_{0} associated with the hermitian form ψ̃.)

Proof. The first (resp. second) claim is theorem 1.6.a (resp. theorem 1.5) of [35]; the third claim is observed by Van Geemen, in lemma 3.2 of [13]; and the final claim is a combination of theorems 2.2 and 2.3 of [35]. (We note that [35] deals with Hodge groups, but because our Hodge structure has weight 0, the Mumford–Tate group and the Hodge group coincide.) □

6.3 Remark. — Let V, E and ψ̃ be as in theorem 6.2. If E is cm, and dim_{E}(V) = 1, then U(ψ̃)^{der} = SU(ψ̃) = 1; while if dim_{E}(V) > 1, then U(ψ̃)^{der} = SU(ψ̃) is absolutely simple over E_{0}. If E is tr and dim_{E}(V) ≠ 4, then SO(ψ̃) is absolutely simple over E. Assume E is tr and dim_{E}(V) = 4. In this case SO(ψ̃) is not absolutely simple over E: it has Lie type D_{2} = A_{1} ⊕ A_{1}. In this remark we will take a close look at this special case, because a good understanding of it will play a crucial rôle in the proof of lemma 7.9. Geometrically we find SO(ψ̃)Ē ≅ (SL_{2,Ē} × SL_{2,Ē})/⟨(−1, −1)⟩. We distinguish the following two cases:

1. SO(ψ̃) is not simple over E. The fact most relevant to us is the existence of a quaternion algebra D/E such that SO(ψ̃) ≅ (N × N^{op})/⟨(−1, −1)⟩ where N is the group over E of elements in D^{⋆} that have norm 1, and likewise for N^{op} ⊂ (D^{op})^{⋆}. One can read more about the details of this claim in section 8.1 of [19]. This situation is also described in section 26.B of [15], where the quaternion algebra is replaced by D × D viewed as quaternion algebra over E × E. This might be slightly more natural, but it requires bookkeeping of étale algebras which makes the proof in section 7 more difficult than necessary.

2. SO(ψ̃) is simple over E. This means that the action of Gal(Ē/E) on SO(ψ̃)Ē interchanges the two factors SL_{2,Ē}. The stabilisers of these factors are subgroups of index 2 that coincide. This subgroup fixes a quadratic extension F/E. From our description of the geometric situation, together with the description of the stabilisers, we see that Spin(ψ̃) = Res_{F/E} 𝒢 is a (2 : 1)-cover of SO(ψ̃), where 𝒢 is an absolutely simple, simply connected group of Lie type A_{1} over F.

What we have gained is that in all cases we have a description (up to isogeny) of G_{B}(V)^{der} as Weil restriction of a group that is an almost direct product of groups that are absolutely simple. This allows us to apply lemma 3.4, which will play an important rôle in section 7.

6.4 Notation and terminology. — Let V, E and ψ̃ be as in theorem 6.2. To harmonise the proof in section 7, we unify notation as follows:

F = \begin{cases}E_{0} & if E is cm, \\ E & if E is tr and dim_{E}(V) ≠ 4, \\ E & if E is tr, dim_{E}(V) = 4, and we are in case 6.3.1, \\ F & if E is tr, dim_{E}(V) = 4, and we are in case 6.3.2. \end{cases}

Similarly

𝒢 = \begin{cases}U(ψ̃) & if E is cm, \\ SO(ψ̃) & if E is tr and dim_{E}(V) ≠ 4, \\ SO(ψ̃) & if E is tr, dim_{E}(V) = 4, and we are in case 6.3.1, \\ 𝒢 & if E is tr, dim_{E}(V) = 4, and we are in case 6.3.2. \end{cases}

We stress that 𝒢^{der} is an almost direct product of absolutely simple groups over F. In section 7, most of the time it is enough to know that G_{B}(V) is isogenous to Res_{F/ℚ} 𝒢. When we need more detailed information, it is precisely the case that E is tr and dim_{E}(V) = 4. For this case we gave a description of 𝒢 in the previous remark.

6.5 — Let V, E and ψ̃ be as in theorem 6.2. Write n for dim_{E}(V). If E is tr, then we say that the group SO(ψ̃) over E is a group of type SO_{n,E}. We also say that G_{B}(V) is of type Res_{E/ℚ} SO_{n,E}. Similarly, if E is cm, with maximal totally real subfield E_{0}, then we say that the group U(ψ̃) over E_{0} is a group of type U_{n,E_{0}}, and that G_{B}(V) is of type Res_{E_{0}/ℚ} U_{n,E_{0}}.

6.6 Remark. — Let A be an abelian surface over ℂ. Recall that H_{B}^{2}(A)(1) has dimension 6. Let H be the transcendental part of H_{B}^{2}(A)(1) and let ρ denote the Picard number of A, so that dim_{ℚ}(H) + ρ = 6. Observe that H is an irreducible Hodge structure of K3 type. In this remark we explicitly calculate what Zarhin’s classification (theorem 6.2) means for H. If A is simple, then the Albert classification of endomorphism algebras of abelian varieties states that End(A) ⊗ ℚ can be one of the following:

1. The field of rational numbers, ℚ. In this case ρ = 1 and G_{B}(H) is of type SO_{5,ℚ}.

2. A real quadratic extension F/ℚ. In this case ρ = 2 and G_{B}(H) is of type SO_{4,ℚ}. By exemple 3.2.2(a) of [11], we see that Nm_{F/ℚ}(H^{1}(A)) ↪ ⋀^{2} H^{1}(A) ≅ H^{2}(A), where Nm(_) is the norm functor studied in [11].

This norm map identifies Nm_{F/ℚ}(H^{1}(A))(1) with the transcendental part H. Observe that consequently the Hodge group Hdg_{B}(H^{1}(A)) = Res_{F/ℚ} SL_{2,F} is a (2 : 1)-cover of G_{B}(H).

3. An indefinite quaternion algebra D/ℚ. (This means D ⊗_{ℚ} ℝ ≅ M_{2}(ℝ).) In this case ρ = 3 and G_{B}(H) is of type SO_{3,ℚ}.

4. A cm field E/ℚ of degree 4. In this case ρ = 2 and G_{B}(H) is of

type Res_{E_{0}/ℚ} U_{1,E_{0}}. (Note that the endomorphism algebra of A cannot be an imaginary quadratic field, by theorem 5 of [29].) If A is isogenous to the product of two elliptic curves Y_{1} × Y_{2}, then there are the following options:

5. The elliptic curves are not isogenous, and neither of them is of cm type, in which case ρ = 2 and G_{B}(H) is of type SO_{4,ℚ}. Indeed, Hdg_{B}(Y_{1}) and Hdg_{B}(Y_{2}) are isomorphic to SL_{2,ℚ}. Note that H = H_{B}^{2}(A)(1)^{tra} is isomorphic to the exterior tensor product (H_{B}^{1}(Y_{1})⊠H_{B}^{1}(Y_{2}))(1). We find that G_{B}(H) is the image of the canonical map SL_{2,ℚ} × SL_{2,ℚ} → GL(H). The kernel of this map is ⟨(−1, −1)⟩.

6. The elliptic curves are not isogenous, one has endomorphism algebra ℚ, and the other has cm by an imaginary quadratic extension E/ℚ. In this case ρ = 2 and G_{B}(H) is of type U_{2,ℚ}.

7. The elliptic curves are not isogenous, and Y_{i} (for i = 1, 2) has cm by an imaginary quadratic extension E_{i}/ℚ. Observe that E_{1} ≅ E̸_{2}, since Y_{1} and Y_{2} are not isogenous. Let E/ℚ be the compositum of E_{1} and E_{2}, which is a cm field of degree 4 over ℚ. In this case ρ = 2 and G_{B}(H) is of type Res_{E_{0}/ℚ} U_{1,E_{0}}.

8. The elliptic curves are isogenous and have trivial endomorphism algebra. In this case ρ = 3 and G_{B}(H) is of type SO_{3,ℚ}.

9. The elliptic curves are isogenous and have cm by an imaginary quadratic extension E/ℚ. In this case ρ = 4 and G_{B}(H) is of type U_{1,ℚ}.

7 Main theorem: the Mumford–Tate conjecture for the product of an abelian surface and a K3 surface

7.1 — Let K be a finitely generated subfield of ℂ. Let A be an abelian surface over K, and let M_{A} denote the transcendental part of the motive H^{2}(A)(1). (The Hodge structure H in remark 6.6 is the Betti realisation H_{B}(M_{A}) of M_{A}.) Let X be a K3 surface over K, and let M_{X} denote the transcendental part of the motive H^{2}(X)(1). Let E_{A} (resp. E_{X}) be the endomorphism algebra of M_{A} (resp. M_{X}). Recall from §6.4 that we associated a field F and a group 𝒢 with every Hodge structure V of K3 type. The important properties of F and G are that

» 𝒢^{der} is an almost direct product of absolutely simple groups over F; and » Res_{F/ℚ} 𝒢 is isogenous to G_{B}(V).

Let F_{A} and 𝒢_{A} be the field and group associated with H_{B}(M_{A}) as in §6.4. Similarly, let F_{X} and 𝒢_{X} be the field and group associated with H_{B}(M_{X}).

Concretely, for F_{A} this means that

F_{A} ≅ \begin{cases}End(A) ⊗ ℚ & in case 6.6.2 (so F_{A} is tr of degree 2) \\ E_{A,0} & in cases 6.6.4 and 6.6.7 (so F_{A} is tr of degree 2) \\ ℚ & otherwise. \end{cases}

We summarise the notation for easy review during later parts of this section: K finitely generated subfield of ℂ A abelian surface over K

M_{A} transcendental part of the motive H^{2}(A)(1) F_{A} field associated with the Hodge structure H_{B}(M_{A}), as in §6.4 𝒢_{A} group over F_{A} such that Res_{F_{A}/ℚ} 𝒢_{A} is isogenous to G_{B}(M_{A}), as in §6.4 X K3 surface over K

M_{X} transcendental part of the motive H^{2}(X)(1) F_{X} field associated with the Hodge structure H_{B}(M_{X}), as in §6.4 𝒢_{X} group over F_{X} such that Res_{F_{X}/ℚ} 𝒢_{X} is isogenous to G_{B}(M_{X}), as in §6.4 E_{X} the endomorphism algebra of M_{X} The proof of the main theorem (1.1) will take the remainder of this article. There are four main parts going into the proof, which are lemmas 7.4, 7.6, 7.7 and 7.9. The lemmas 7.2, 7.3 and 7.8 and corollary 7.10 are small reductions and intermediate results. Together lemmas 7.4, 7.6 and 7.7 deal with almost all combinations of abelian surfaces and K3 surfaces. Lemma 7.9 is rather technical, and is the only place in the proof where we use that M_{X} really is a motive coming from a K3 surface.

7.2 Lemma. — » The Mumford–Tate conjecture for H^{2}(A×X)(1) is equiv-

alent to MTC(M_{A} ⊕ M_{X}). » The ℓ-adic realisations of M_{A} ⊕ M_{X} form a compatible system of ℓ-adic

representations. Proof. The first claim is because H^{2}(A×X)(1) and M_{A}⊕M_{X} generate the same Tannakian subcategory of motives. By théorème 1.6 of [8], the H_{ℓ}^{2}(A × X)(1) form a compatible system of ℓ-adic representations and we only remove Tate classes to obtain H_{ℓ}(M_{A} ⊕ M_{X}); hence the ℓ-adic realisations of M_{A} ⊕ M_{X} also form a compatible system of ℓ-adic representations. □

7.3 Lemma. — If for some prime ℓ, the natural morphism

ι_{ℓ}: G_{ℓ}^{∘}(M_{A} ⊕ M_{X})^{der} ⟶ G_{ℓ}^{∘}(M_{A})^{der} × G_{ℓ}^{∘}(M_{X})^{der}

is an isomorphism (that is, if condition 4.6 holds), then the Mumford–Tate conjecture for M_{A} ⊕ M_{X} is true. Proof. The absolute rank of G_{B}(M_{A} ⊕M_{X}) is bounded from above by the sum of the absolute ranks of G_{B}(M_{A})^{der}, G_{B}(M_{X})^{der}, and the centre of G_{B}(M_{A} ⊕ M_{X}). By proposition 4.4, we know that the Mumford–Tate conjecture for M_{A} ⊕ M_{X} is true on the centres of G_{B}(M_{A} ⊕ M_{X}) ⊗ ℚ_{ℓ} and G_{ℓ}^{∘}(M_{A} ⊕ M_{X}). Hence if ι_{ℓ}: G_{ℓ}^{∘}(M_{A} ⊕ M_{X})^{der} ↪ G_{ℓ}^{∘}(M_{A})^{der} × G_{ℓ}^{∘}(M_{X})^{der} is an isomorphism, then the absolute rank of G_{ℓ}^{∘}(M_{A} ⊕ M_{X}) is bounded from below by the sum of the absolute ranks of G_{B}(M_{A})^{der}, G_{B}(M_{X})^{der}, and the centre of G_{B}(M_{A}⊕M_{X}). The result follows from lemma 4.5. □

7.4 Lemma. — If F_{A} ≅ F̸_{X}, then MTC(M_{A} ⊕ M_{X}) is true. Proof. By lemma 7.3 we are done if there is some prime number ℓ, for which ι_{ℓ}: G_{ℓ}^{∘}(M_{A} ⊕ M_{X})^{der} ↪ G_{ℓ}^{∘}(M_{A})^{der} × G_{ℓ}^{∘}(M_{X})^{der} is an isomorphism. Hence assume that for all ℓ, the morphism ι_{ℓ} is not an isomorphism. This will imply that F_{A} ≅ F_{X}. By lemma 3.5, we see that F_{A,ℓ} = F_{A} ⊗ ℚ_{ℓ} and F_{X,ℓ} = F_{X} ⊗ ℚ_{ℓ} have an isomorphic factor. If F_{A} is isomorphic to ℚ, then F_{X,ℓ} has a factor ℚ_{ℓ} for each ℓ, and we win by lemma 2.2. Next suppose that F_{A} ≅ ℚ̸, in which case F_{A} is a real quadratic extension of ℚ. If 𝒢_{X}^{der} is not an absolutely simple group, then it is of type SO_{4,E_{X}}. In particular dim_{E_{X}} (M_{X}) = 4 and F_{X} ≅ E_{X}. Since dim_{ℚ}(M_{X}) ≤ 22 we find [F_{X} : ℚ] ≤ 5, and we conclude by lemma 2.3. Finally, suppose that 𝒢_{X}^{der} is an absolutely simple group over F_{X}. We want to apply lemma 2.2, so we need to show that F_{A,ℓ} is a factor of F_{X,ℓ}, for all prime numbers ℓ. For the primes that are inert in F_{A,ℓ} this is obvious. We are thus left to show that F_{X,ℓ} has at least two factors ℚ_{ℓ} for every prime ℓ that splits in F_{A}. Note that 𝒢_{A}^{der} is an absolutely simple group over F_{A} of Lie type A_{1}. Using remark 3.1 we find, for each prime ℓ, semisimple Lie algebras 𝔰_{A,ℓ}, 𝔱_{ℓ} and 𝔰_{X,ℓ} such that

Lie(𝒢_{A}^{der}) ≅ Lie(G_{ℓ}^{∘}(M_{A})^{der}) ≅ 𝔰_{A,ℓ} ⊕ 𝔱_{ℓ}
Lie(𝒢_{X}^{der}) ≅ Lie(G_{ℓ}^{∘}(M_{X})^{der}) ≅  𝔱_{ℓ} ⊕ 𝔰_{X,ℓ}
Lie(G_{ℓ}^{∘}(M_{A} ⊕ M_{X})^{der}) ≅ 𝔰_{A,ℓ} ⊕ 𝔱_{ℓ} ⊕ 𝔰_{X,ℓ}.

The absolute ranks of G_{ℓ}^{∘}(M_{A})^{der}, G_{ℓ}^{∘}(M_{X})^{der}, and G_{ℓ}^{∘}(M_{A} ⊕ M_{X})^{der} do not depend on ℓ, by proposition 4.4 and remark 6.13 of [16] (or the letters of Serre to Ribet in [28]).^{3} Since the matrix

(\begin{matrix}1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 1 & 1 \end{matrix})

is invertible, we find that the absolute ranks of the Lie algebras 𝔰_{A,ℓ}, 𝔱_{ℓ}, and 𝔰_{X,ℓ} do not depend on ℓ, If ℓ is a prime that is inert in F_{A}, then 𝒢_{A}^{der} ⊗_{F_{A}} F_{A,ℓ} is an absolutely simple group. Since 𝔱_{ℓ} ≠ 0, we conclude that 𝔰_{A,ℓ} = 0. By the independence of the absolute ranks, 𝔰_{A,ℓ} = 0 for all primes ℓ. Consequently, if ℓ is a prime that splits in F_{A}, then 𝔱_{ℓ} has two simple factors that are absolutely simple Lie algebras over ℚ_{ℓ} of Lie type A_{1}. Since 𝒢_{X}^{der} is an absolutely simple group over F_{X}, we conclude that F_{X,ℓ} has at least two factors ℚ_{ℓ}, for every prime ℓ that splits in F_{A}. □

3More generally, Hui proved that for every semisimple system of compatible representations the semisimple rank does not depend on ℓ, see theorem 3.19 of [14].

Documenta Mathematica 21 (2016) 1691–1713

7.5 — From now on, we assume that F_{A} ≅ F_{X}, which we will simply denote with F. We single out the following cases, and prove the Mumford–Tate conjecture for M_{A} ⊕ M_{X} for all other cases in the next lemma.

1. G_{B}(M_{A}) and G_{B}(M_{X}) are both of type SO_{5,ℚ};

2. G_{B}(M_{A}) is of type SO_{3,ℚ}, or SO_{4,ℚ}, or U_{2,ℚ}, and the type of G_{B}(M_{X}) is also one of these types;

3. F is a real quadratic extension of ℚ, A is an absolutely simple abelian

surface with endomorphisms by F (so 𝒢_{A} ≅ SL_{2,F}), and 1. G_{B}(M_{X}) is of type SO_{3,F} or U_{2,F}; or

2. G_{B}(M_{X}) is non-simple of type SO_{4,F} as in case 6.3.1 of remark 6.3. Note that in the third case we did not forget case 6.3.2, since that is covered in case 2. We point out that in the first two cases dim(M_{X}) ≤ 5, which can be deduced from theorem 6.2.

7.6 Lemma. — If M_{A} and M_{X} do not fall into one of the cases listed in §7.5, then the Mumford–Tate conjecture for M_{A} ⊕ M_{X} is true. Proof. By lemma 7.3 we are done if there is some prime number ℓ, for which ι_{ℓ}: G_{ℓ}^{∘}(M_{A} ⊕ M_{X})^{der} ↪ G_{ℓ}^{∘}(M_{A})^{der} × G_{ℓ}^{∘}(M_{X})^{der} is an isomorphism. Recall that ℂ ≅ \overline{ℚ_{ℓ}}, as fields. If the Dynkin diagram of Lie(G_{ℓ}^{∘}(M_{A})^{der})_{ℂ} and the Dynkin diagram of Lie(G_{ℓ}^{∘}(M_{X})^{der})_{ℂ} have no common components, then ι_{ℓ} must be an isomorphism, and we win. Recall that MTC(M_{A}) and MTC(M_{X}) are known. Thus ι_{ℓ} is an isomorphism if the Dynkin diagram of Lie(G_{B}(M_{A})^{der})_{ℂ} and the Dynkin diagram of Lie(G_{B}(M_{X})^{der})_{ℂ} have no common components. By inspection of theorem 6.2 and remark 6.6, we see that this holds, except for the cases listed in §7.5. □

7.7 Lemma. — The Mumford–Tate conjecture for M_{A} ⊕ M_{X} is true if dim(M_{X}) ≤ 5. In particular, the Mumford–Tate conjecture is true for the first two cases listed in §7.5. Proof. If dim(M_{X}) = 2, then G_{B}(M_{X}) is commutative, and we are done by lemma 7.3. Let B be the Kuga–Satake variety associated with H_{B}(M_{X}). This is a complex abelian variety of dimension 2^{dim(M_{X})−2}. Up to a finitely generated extension of K, we may assume that B is defined over K. (In fact, B is defined over K, by work of Rizov, [25].) We may and do allow ourselves a finite extension of K, to assure that B is isogenous to a product of absolutely simple abelian varieties over K. By proposition 6.4.3 of [1] we deduce that M_{X} is a submotive of End(H^{1}(B)). (Alternatively, see proposition 6.3.3 of [12] for a direct argument that H_{B}(M_{X}) is a sub-ℚ-Hodge structure of End(H_{B}^{1}(B)); and use that M_{X} is an abelian motive together with theorem 4.2.) Consequently, MTC(A × B) implies MTC(M_{A} ⊕ M_{X}), for if the Mumford–Tate conjecture holds for a motive M, then it holds for all motives in the Tannakian subcategory ⟨M⟩^{⊗} generated by M.

Recall that the even Clifford algebra C^{+}(M_{X}) = C^{+}(H_{B}(M_{X})) acts faithfully on B. Theorem 7.7 of [12] gives a description of C^{+}(M_{X}); thus describing a subalgebra of End^{0}(B).

» If dim(M_{X}) = 3, then dim(B) = 2 and C^{+}(M_{X}) is a quaternion algebra

over ℚ. » If dim(M_{X}) = 4, then dim(B) = 4 and C^{+}(M_{X}) is either a product D×D,

where D is a quaternion algebra over ℚ; or C^{+}(M_{X}) is a quaternion algebra over a totally real quadratic extension of ℚ.

» If dim(M_{X}) = 5, then dim(B) = 8 and C^{+}(M_{X}) is a matrix alge-

bra M_{2}(D), where D is a quaternion algebra over ℚ. We claim that A × B satisfies the conditions of lemma 5.5. First of all, observe that A satisfies those conditions, which can easily be seen by reviewing remark 6.6. We are done if we check that B satisfies the conditions as well.

» If dim(M_{X}) = 3, then B is either a simple abelian surface, or isogenous to the square of an elliptic curve. In both cases, B satisfies the conditions of lemma 5.5.

» If dim(M_{X}) = 4, and C^{+}(M_{X}) is D × D for some quaternion algebra D over ℚ, then B splits (up to isogeny) as B_{1} ×B_{2}. In particular dim(B_{i}) = 2, since D cannot be the endomorphism algebra of an elliptic curve. Hence both B_{i} satisfy the conditions of lemma 5.5. On the other hand, if dim(M_{X}) = 4 and C^{+}(M_{X}) is a quaternion algebra over a totally real quadratic extension of ℚ, then there are two options.

» If B is not absolutely simple, then all simple factors have dimension ≤ 2; since End^{0}(B) is non-commutative. Indeed, the product of an elliptic curve and a simple abelian threefold has commutative endomorphism ring (see, e.g., section 2 of [21]).

» If B is absolutely simple, then it has relative dimension 1. This abelian fourfold must be of type ii(2), since type iii(2) does not occur (see proposition 15 of [29], or table 1 of [20] which also proves MTC(B)).

In both of these cases, B satisfies the conditions of lemma 5.5. » If dim(M_{X}) = 5, then B is the square of an abelian fourfold C whose

endomorphism algebra contains a quaternion algebra over ℚ.

» If C is not absolutely simple, then all simple factors must have di-

mension ≤ 2; since End^{0}(C) is non-commutative. » If C is simple, then we claim that C must be of type ii. Indeed, the

Mumford–Tate group of B surjects onto G_{B}(M_{X}), because H_{B}(M_{X}) is a sub-ℚ-Hodge structure of End(H_{B}^{1}(B)). Now dim(M_{X}) = 5, hence G_{B}(M_{X}) is of type SO_{5,ℚ}, with Lie type B_{2}. But §6.1 of [20] shows that if C is of type iii, then G_{B}(C) has Lie type D_{2} ≅ A_{1} ⊕A_{1}. This proves our claim. Since End^{0}(C) is a quaternion algebra and C is an abelian fourfold, table 1 of [20] shows that MTC(C) is true and D_{4} does not occur in the Lie type of G_{B}(C).

We conclude that MTC(A × B) is true, and therefore MTC(M_{A} ⊕ M_{X}) is true as well. □

The only cases left are those listed in case 7.5.3 of §7.5. Therefore, we may and do assume that F is a real quadratic field extension of ℚ; and that A is an absolutely simple abelian surface with endomorphisms by F (i.e., case 6.6.2). In particular 𝒢_{A} = SL_{2,F}.

7.8 Lemma. — If X falls in one of the subcases listed in case 7.5.3, then there exists a place λ of F such that 𝒢_{X}^{der} ⊗_{F} F_{λ} does not contain a split factor. Proof. In case 7.5.3.1, 𝒢_{X}^{der} is of Lie type A_{1}. In case 7.5.3.2, 𝒢_{X} ∼ N × N^{op}, where N is a form of SL_{2,F}, as explained in remark 6.3. By theorem 26.9 of [15], there is an equivalence between forms of SL_{2} over a field, and quaternion algebras over the same field. We find a quaternion algebra D over F corresponding to 𝒢_{X}^{der}, respectively N, in case 7.5.3.1, respectively case 7.5.3.2. In particular 𝒢_{X}^{der} contains a split factor if and only if the quaternion algebra is split. Let {σ, τ} be the set of embeddings Hom(F, ℝ). Since F acts on H_{B}(M_{X}), we see that F ⊗_{ℚ} ℝ ≅ ℝ^{(σ)} ⊕ ℝ^{(τ)} acts on

H_{B}(M_{X}) ⊗_{ℚ} ℝ ≅ W^{(σ)} ⊕ W^{(τ)}.

Here W^{(σ)} and W^{(τ)} are ℝ-Hodge structures of dimension dim_{F} (M_{X}). Observe that the polarisation form is definite on one of the terms, while it is non-definite on the other. Without loss of generality we may assume that the polarisation form is definite on W^{(σ)}, and non-definite on W^{(τ)}. Thus, the group G_{B}(M_{X}) ⊗_{ℚ} ℝ is the product of a compact group and a noncompact group; and therefore, Res_{F/ℚ} 𝒢_{X} ⊗_{ℚ} ℝ is the product of a compact group and a non-compact group. Indeed 𝒢_{X} ⊗_{F} ℝ^{(σ)} is compact, while 𝒢_{X} ⊗_{F} ℝ^{(τ)} is non-compact. By the first paragraph of the proof, this means that D ⊗_{F} ℝ^{(σ)} is non-split, while D ⊗_{F} ℝ^{(τ)} is split. Since the Brauer invariants of D at the infinite places do not add up to 0, there must be a finite place λ of F such that D_{λ} is non-split. At this place λ, the group 𝒢_{X}^{der} ⊗_{F} F_{λ} does not have a split factor. □

7.9 Lemma. — Assume that K is a number field. If X falls in one of the subcases listed in case 7.5.3, then there is a prime number ℓ for which the natural map

ι_{ℓ}: G_{ℓ}^{∘}(M_{A} ⊕ M_{X})^{der} ⟶ G_{ℓ}^{∘}(M_{A})^{der} × G_{ℓ}^{∘}(M_{X})^{der}

is an isomorphism. Proof. The absolute rank of G_{ℓ}^{∘}(M_{A} ⊕ M_{X})^{der} does not depend on ℓ, by proposition 4.4 and lemma 7.2 and remark 6.13 of [16] (or the letters of Serre to Ribet in [28], or theorem 3.19 of [14]). We now show that this absolute rank must be even, by looking at a prime ℓ that is inert in F. At such a prime ℓ all simple factors of Lie(G_{ℓ}^{∘}(M_{A})^{der} ×G_{ℓ}^{∘}(M_{X})^{der}) are ℚ_{ℓ}-Lie algebras with even absolute rank (since [F : ℚ] = 2). By remark 3.1, the Lie algebra of G_{ℓ}^{∘}(M_{A} ⊕ M_{X})^{der} is a summand of Lie(G_{ℓ}^{∘}(M_{A})^{der} × G_{ℓ}^{∘}(M_{X})^{der}), and therefore the absolute rank of G_{ℓ}^{∘}(M_{A} ⊕ M_{X})^{der} must be even.

Let λ be one of the places of F found in lemma 7.8, and let ℓ be the place of ℚ lying below λ. Since Lie(G_{ℓ}^{∘}(M_{A} ⊕ M_{X})^{der}) must surject to Lie(G_{ℓ}^{∘}(M_{A})^{der}) (which is split, and has absolute rank 2), and Lie(G_{ℓ}^{∘}(M_{A} ⊕ M_{X})^{der}) must also surject onto Lie(G_{ℓ}^{∘}(M_{X})^{der}), which has a non-split factor, by lemma 7.8, we conclude that the absolute rank of Lie(G_{ℓ}^{∘}(M_{A} ⊕ M_{X})^{der}) must be at least 3. By the previous paragraph, we find that the absolute rank must be at least 4. If dim_{E_{X}} (M_{X}) ≠ 4 (case 7.5.3.1) then 𝒢_{X}^{der} is a group of Lie type A_{1}, and therefore the product G_{ℓ}^{∘}(M_{A})^{der} × G_{ℓ}^{∘}(M_{X})^{der} has absolute rank 4. Hence G_{ℓ}^{∘}(M_{A} ⊕ M_{X})^{der} must have absolute rank 4, which means that ι_{ℓ} is an isomorphism, by remark 3.1. If dim_{E_{X}} (M_{X}) = 4 (case 7.5.3.2), then 𝒢_{X} is a group of Lie type D_{2} = A_{1} ⊕A_{1}. (Note that in this final case G_{B}(M_{A}) and G_{B}(M_{X}) are semisimple, and therefore we may drop all the superscripts (_)^{der} from the notation.) Since in this case G_{ℓ}^{∘}(M_{A})×G_{ℓ}^{∘}(M_{X}) has absolute rank 6, and the absolute rank of G_{ℓ}^{∘}(M_{A}⊕M_{X}) is ≥ 4, it must be 4 or 6 (since it is even). Suppose G_{ℓ}^{∘}(M_{A}⊕M_{X}) has absolute rank 4. We apply remark 3.1 to the current situation, and find Lie algebras 𝔱 and 𝔰_{2} over ℚ_{ℓ} such that Lie(G_{ℓ}^{∘}(M_{A})) ≅ 𝔱 and Lie(G_{ℓ}^{∘}(M_{A} ⊕ M_{X})) ≅ Lie(G_{ℓ}^{∘}(M_{X})) ≅ 𝔱 ⊕ 𝔰_{2}. In particular, Lie(G_{ℓ}^{∘}(M_{X})), which is isomorphic to Lie(𝒢_{X}) ⊗_{ℚ} ℚ_{ℓ}, has a split simple factor. By lemma 7.8 this means that ℓ splits in F as λ · λ′. Observe that F_{λ} ≅ ℚ_{ℓ} ≅ F_{λ′}. The fact that E_{X,ℓ} ≅ F_{ℓ} factors as F_{λ} × F_{λ′} has several implications. In the case under consideration we have G_{B}(M_{X}) ≅ Res_{F/ℚ} 𝒢_{X}, and since MTC(M_{X}) holds, this implies G_{ℓ}^{∘}(M_{X}) ≅ 𝒢_{X,λ} ×𝒢_{X,λ′}. Besides that, H_{ℓ}(M_{X}) decomposes as H_{λ}(M_{X}) ⊕ H_{λ′}(M_{X}). The group Gal(K̄/K) acts on H_{λ}(M_{X}) via 𝒢_{X,λ}, and on H_{λ′}(M_{X}) via 𝒢_{X,λ′}. To summarise, our situation is now as follows. The prime number ℓ splits in F as λ·λ′. The group 𝒢_{A} is isomorphic to SL_{2,F}, and is split and simply connected. The group 𝒢_{X,λ′} is split, of type SO_{4,ℚ_{ℓ}}, with Lie algebra 𝔱. The group 𝒢_{X,λ} is non-split, of type SO_{4,ℚ_{ℓ}}, with Lie algebra 𝔰_{2}. Recall the natural diagram:

G_{ℓ}^{∘}(M_{A} ⊕ M_{X}) G_{ℓ}^{∘}(M_{A}) × G_{ℓ}^{∘}(M_{X}) (SL_{2,ℚ_{ℓ}} × SL_{2,ℚ_{ℓ}})/⟨(−1, −1)⟩ ≅ G_{ℓ}^{∘}(M_{A}) G_{ℓ}^{∘}(M_{X}) ≅ 𝒢_{X,λ′} × 𝒢_{X,λ} ι_{ℓ}

We are now set for the attack. We claim that the Galois representations H_{ℓ}(M_{A}) and H_{λ′}(M_{X}) are isomorphic. Indeed, from the previous paragraph we conclude that G_{ℓ}^{∘}(M_{A} ⊕ M_{X}) ≅ Γ × 𝒢_{X,λ}, where Γ is a subgroup of G_{ℓ}^{∘}(M_{A}) × 𝒢_{X,λ′} with surjective projections. Thus H_{ℓ}(M_{A}) and H_{λ′}(M_{X}) are both orthogonal representations of Gal(K̄/K), and the action of Galois factors via Γ(ℚ_{ℓ}). We will now show that H_{ℓ}(M_{A}) and H_{ℓ}(M_{X}) are isomorphic as representations of Γ, which proves the claim. The Lie algebra of Γ is isomorphic to 𝔱, and Lie(Γ) is the graph of an isomorphism Lie(G_{ℓ}^{∘}(M_{A})) → Lie(𝒢_{X,λ′}). Since G_{ℓ}^{∘}(M_{A}) and 𝒢_{X,λ′} are both covered by Res_{F/ℚ} SL_{2,F} ≅ Hdg_{ℓ}(A) with kernels {±1}, and Γ is a subgroup of G_{ℓ}^{∘}(M_{A}) × 𝒢_{X,λ′}, we find that Γ also has a (2 : 1)-cover by Res_{F/ℚ} SL_{2,F}. Hence Γ is the graph of an isomorphism G_{ℓ}^{∘}(M_{A}) → 𝒢_{X,λ′}. Because H_{ℓ}(M_{A}) and H_{λ′}(M_{X}) are 4-dimensional faithful orthogonal representations of Γ, they must be isomorphic; for up to isomorphism, there is a unique such representation. As a consequence, for all places v of K, the characteristic polynomial of Frob_{v} acting on H_{ℓ}(M_{A}) coincides with its characteristic polynomial when acting on H_{λ′}(M_{X}). We conclude that charpol_{F_{λ′}} (Frob_{v}|H_{λ′}(M_{X})) has coefficients in ℚ. But then the same is true for charpol_{F_{λ}}(Frob_{v}|H_{λ}(M_{X})) since their product is charpol_{ℚ_{ℓ}}(Frob_{v}|H_{ℓ}(M_{X})), which has coefficients in ℚ. In conclusion, charpol_{ℚ_{ℓ}}(Frob_{v}|H_{ℓ}(M_{X})) factors over ℚ as

charpol_{F_{λ}}(Frob_{v}|H_{λ}(M_{X})) · charpol_{F_{λ′}} (Frob_{v}|H_{λ′}(M_{X})).

This leads to a contradiction with the following facts. Since we assumed that K is a number field, the following results hold.

» The main theorem of [4], which tells us that (up to a finite extension of K) there exists a set 𝒱 of places of K with density 1 such that X has good and ordinary reduction at places v ∈ 𝒱 .

» Theorem 1 (item 1) of [6], which tells us that (up to another finite extension of K) there exists a set 𝒱 of places of K with density 1 such that X has good reduction at places v ∈ 𝒱 , and the Picard number of the reduction X_{v} is the same as that of X (which, in our case is 22 − 8 = 14).

» Proposition 3.2 of [34], which says that if X has good and ordinary reduction at v, then the characteristic polynomial charpol_{ℚ_{ℓ}}(Frob_{v}|H_{ℓ}^{2}(X_{v})^{tra}) is an irreducible polynomial with coefficients in ℚ.

Thus charpol_{ℚ_{ℓ}}(Frob_{v}|H_{ℓ}(M_{X})) is irreducible for a density 1 subset of places v; which contradicts the factorisation found above. We conclude that the absolute rank of G_{ℓ}^{∘}(M_{A} ⊕ M_{X}) cannot be 4 and therefore it must be 6, which implies that ι_{ℓ} is an isomorphism. □

7.10 Corollary. — If X falls in one of the subcases listed in case 7.5.3, then the Mumford–Tate conjecture is true for M_{A} ⊕ M_{X}. Proof. This result follows from lemmas 4.8 and 7.9. □

7.11 Proof of theorem 1.1. — By lemma 7.2 the main theorem reduces to the Mumford–Tate conjecture for M_{A} ⊕ M_{X}. The theorem follows from lemmas 7.4, 7.6 and 7.7 and corollary 7.10. □

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Johan Commelin Institute for Mathematics, Astrophysics and Particle Physics Radboud Universiteit Heyendaalseweg 135 6525 AJ Nijmegen, the Netherlands

       http://www.math.ru.nl/~jcommelin
       j.commelin@math.ru.nl

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