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Let A be an abelian variety defined over a number field K, the number of torsion points rational over a finite extension L is bounded polynomially in terms of the degree [L : K]. When A is isogenous to a product of simple abelian varieties of type I or II in Albert classification and is "fully of Lefschetz type", i.e. whose Mumford-Tate group is the group of symplectic similitudes commuting with endomorphisms and which satisfy the Mumford-Tate conjecture, we compute the optimal exponent for this bound in terms of the dimensions of the abelian subvarieties of A and their rings of endomorphisms. The result is unconditional for a product of simple abelian varieties of type I or II with odd relative dimension. Extending work of Serre, Pink and Hall, we also prove that the Mumford-Tate conjecture is true for a few new cases for such abelian varieties., in French, Accepted for publication in ANT

Let $X$ be an smooth projective algebraic variety over a number field $F \subset \mathbb{C}$. Suppose that the Absolute Hodge(AH) motive $M:= h^i(X)$ is contained in the Tannakian category generated by the AH motives of abelian varieties. For every prime number $\ell$ the Galois group $\Gamma_F:= Gal(\bar{F}/F)$ acts on $H_\ell(M)$, the $\ell $-adic realization of $M$. Over a finite extension of $F$ this action factorizes as $\rho_{M,\ell}:\Gamma_F\rightarrow G_M(\ql)$, where $G_M$ is the Mumford-Tate group of $M$. Fix a valuation $v$ of $F$ and suppose $v(\ell)=0 $. The restriction $ \rho_{M,\ell} \vert _{\Gamma_{F_v}}$ defines a representation ${}'W_v \rightarrow G_{M/\ql}$ of the Weil-Deligne group of $F_v$. J-P Serre and J-M Fontaine (independently) have made conjectures that indicates that ${}'W_v \rightarrow G_{M/\ql}$ should be defined over $\mathbb{Q}$ for $\ell$ fixed and that these representations form a compatible system for variable $\ell $. Under certain additional hypothesis , we answer these questions in affirmative, when $X$ has good reduction or Semi-Stable reduction at $v$.

Let $A$ be an abelian variety defined over a number field $K$, the number of torsion points rational over a finite extension $L$ is bounded polynomially in terms of the degree $[L:K]$. When $A$ is isogenous to a product of simple abelian varieties of $\GSp$ type, i.e. whose Mumford-Tate group is "generic" (isomorphic to the group of symplectic similitudes) and which satisfy the Mumford-Tate conjecture, we compute the optimal exponent for this bound in terms of the dimensions of the abelian subvarieties of $A$. The result is unconditional for a product of simple abelian varieties with endomorphism ring $\Z$ and dimension outside an explicit exceptional set $\mathcal{S}=\{4,10,16,32,...\}$. Furthermore, following a strategy of Serre, we also prove that if the Mumford-Tate conjecture is true for some abelian varieties of $\GSp$ type, it is then true for a product of such abelian varieties., 31 pages, new section 5, accepted for publication in Journal de l'Institut de Mathematiques de Jussieu

We study certain spaces of nilpotent orbits in Hodge domains, and treat a number of examples. More precisely, we compute the Mumford-Tate group of the limit mixed Hodge structure of a generic such orbit. The result is used to present these spaces as iteratively fibered algebraic-group orbits in a minimal way. We conclude with two applications to variations of Hodge structure., 58 pages, 22 figures; final version, to appear in Duke Math. J.; section 6 substantially expanded

Let $K$ be a number field, $A/K$ be an absolutely simple abelian variety of CM type, and $\ell$ be a prime number. We give explicit bounds on the degree over $K$ of the division fields $K(A[\ell^n])$, and when $A$ is an elliptic curve we also describe the full Galois group of $K(A_{\text{tors}})/K$. This makes explicit previous results of Serre and Ribet, and strengthens a theorem of Banaszak, Gajda and Kraso\'n. Our bounds are especially sharp in case the CM type of $A$ is nondegenerate., Comment: v2: theorem 1.2 and exposition improved, fixed some minor mistakes

We classify the irreducible Hermitian real variations of Hodge structure admitting an infinitesimal normal function, and draw conclusions for cycle-class maps on families of abelian varieties with a given Mumford-Tate group.

In this paper, we investigate the action of the ℚ-cohomology of the compact dual \(\widehat X\) of a compact Shimura Variety S(Γ) on the ℚ-cohomology of S(Γ)> under a cup product. We use this to split the cohomology of S(Γ) into a direct sum of (not necessarily irreducible) ℚ-Hodge structures. As an application, we prove that for the class of arithmetic subgroups of the unitary groups U(p,q) arising from Hermitian forms over CM fields, the Mumford–Tate groups associated to certain holomorphic cohomology classes on S(Γ) are Abelian. As another application, we show that all classes of Hodge type (1,1) in H2 of unitary four-folds associated to the group U(2,2) are algebraic.

We collect evidence in support of a conjecture of Griffiths, Green and Kerr on the arithmetic of extension classes of limiting mixed Hodge structures arising from semistable degenerations over a number field. After briefly summarizing how a result of Iritani implies this conjecture for a collection of hypergeometric Calabi-Yau threefold examples studied by Doran and Morgan, the authors investigate a sequence of (non-hypergeometric) examples in dimensions 1 through 6 arising from Katz's theory of the middle convolution. A crucial role is played by the Mumford-Tate group (of type G2) of the family of 6-folds, and the theory of boundary components of Mumford-Tate domains., Comment: 31 pages, 4 figures

We (1) characterize the Schubert varieties that arise as variations of Hodge structure (VHS); (2) show that the isotropy orbits of the infinitesimal Schubert VHS `span' the space of all infinitesimal VHS; and (3) show that the cohomology classes dual the Schubert VHS form a basis of the invariant characteristic cohomology associated to the infinitesimal period relation (a.k.a. Griffiths transversality)., v3: typos corrected

Let $X$ be an smooth projective algebraic variety over a number field $F \subset \mathbb{C}$. Suppose that the Absolute Hodge(AH) motive $M:= h^i(X)$ is contained in the Tannakian category generated by the AH motives of abelian varieties. For every prime number $\ell$ the Galois group $\Gamma_F:= Gal(\bar{F}/F)$ acts on $H_\ell(M)$, the $\ell $-adic realization of $M$. Over a finite extension of $F$ this action factorizes as $\rho_{M,\ell}:\Gamma_F\rightarrow G_M(\ql)$, where $G_M$ is the Mumford-Tate group of $M$. Fix a valuation $v$ of $F$ and suppose $v(\ell)=0 $. The restriction $ \rho_{M,\ell} \vert _{\Gamma_{F_v}}$ defines a representation ${}'W_v \rightarrow G_{M/\ql}$ of the Weil-Deligne group of $F_v$. J-P Serre and J-M Fontaine (independently) have made conjectures that indicates that ${}'W_v \rightarrow G_{M/\ql}$ should be defined over $\mathbb{Q}$ for $\ell$ fixed and that these representations form a compatible system for variable $\ell $. Under certain additional hypothesis , we answer these questions in affirmative, when $X$ has good reduction or Semi-Stable reduction at $v$., Soit $X$ une variété algébrique lisse et projectif sur un corps de nombres $F \subset \mathbb{C}$. On suppose que le motif de Hodge absolu $h^i(X)$ appartient à la catégorie Tannakienne engendrée par les motifs des variétés abélienne sur $F$. Pour tout nombre premier $\ell$, le groupe de Galois $\Gamma_F:= Gal(\bar{F}/F)$ opère sur $H_{\ell}(M)$, la réalisation $\ell$-adique de $M$. Quitte à remplacer $F$ par une extension finie, on peut supposer que cette action se factorise par un morphisme $\rho_{M,\ell}: \Gamma_F\rightarrow G_M(\ql)$, où $G_M$ est le groupe de Mumford-Tate de $M$. Fixons une valuation $v$ de $F$ et supposons $v(\ell)=0 $. La restriction $\rho_{M,\ell} \vert_{ \Gamma_{F_v}}$ définit une représentation ${}'W_v \rightarrow G_{M/\ql}$ du groupe de Weil-Deligne de $F_v$. Des conjectures de J-P Serre et J-M Fontaine indiquent que pour tout $\ell $, la représentation ${}'W_v \rightarrow G_{M/\ql}$ est définie sur $\mathbb{Q}$ et pour $\ell$ variable elles forment un système compatible de représentations. Sous certaines hypothèses supplémentaires, nous montrons que ceci est vrai si $X$ a bonne réduction en $v$ où réduction semi-stable en $v$.

Dans cet article on etudie la structure et les degenerescences du groupe de Murnford-Tate MT(M) d'un 1-motif M defini sur C. Ce groupe est un Q-groupe algebrique qui agit sur la realisation de Hodge de M et qui est muni d'une filtration croissante W○. On prouve que le radical unipotent de MT(M), qui est W -1 (MT(M)), s'injecte dans un groupe de Heisenberg generalise. Ensuite on explique comment se reduire a l'etude du groupe de Mumford-Tate d'une somme directe de 1-motifs dont le groupe des caracteres du tore et dont le reseau sont de rang 1. Puis on classifie et on etudie les degenerescences de MT(M), i.e. les phenomenes qui causent la chute de la dimension de MT(M).

A conjecture by Yves Andre and Frans Oort says that closed subvarieties of Shimura varieties that contain a Zariski dense subset of special points are subvarieties of Hodge type. We prove this in the case where the subvariety is a curve that contains infinitely many special points that lie in one Hecke orbit. Via work of Juergen Wolfart, Paula Cohen and Gisbert Wuestholz, this result has an application to algebraicity of values of hypergeometric functions., 25 pages, published version

Let (K, @) be a primitive CM-type with [K: O] = 2n (for definitions and previous results see Section 1.1). Fix n, and consider the collection s(n) = {Rank(@)}, where Rank(@) counts the number of independent translates of Qi under the Galois action and (K, @) ranges over all primitive types. The smallest element of S(n), denoted by B(n), is referred to as the sharp lower bound for the rank in dimension n. As was brought to the author’s attention by Ribet, bounds on B(n) of the form p + 1, for p a prime dividing n, follow directly from the proof of Ribet’s Nondegeneracy Theorem [21]. This is recorded as Theorem 1.4. Ribet’s method also gives bounds of the form 2q for q a prime with q2 dividing n, as is observed in Theorem 1.12. When combined with the author’s constructions of Abelian varieties in [S, 91, we obtain the precise value of B(n) for many values of n (Corollaries 1.5 to 1.8 and 1.13). We recall that these constructions use the analytic method of Weil and Shimura, together with new results on the reflex field from an investigation suggested to the author by Shim’ura. The interest of the rank comes from the theory of complex multiplication. If A is an Abelian variety of CM-type (K, @), then the Kubota Rank of A is Rank (@), and controls properties of the classfields constructed from A as in Kubota [ 161 and Ribet [20]. This is connected with the fact that the rank of @ is also the dimension of the Mumford-Tate group of A, and with the relation of this group to the I-adic representations of A, as in Serre [24, 251. The main body of the paper contains results that provide information on S(n). The main result, Theorem 2.5, asserts that when n is odd there is a computable subset S,‘,,,(n) of S(n) that accounts for the ranks of many CM-types on most CM-fields. More precisely, let K, be the maximal totally real subfield of K, and Kg be the Galois closure of K,. We consider the per

We prove: Let A be an abelian variety over a number field K. Then K has a finite Galois extension L such that for almost all @[email protected]?Gal(L) there are infinitely many prime numbers l with A"l([email protected]?(@s)) 0. Here [email protected]? denotes the algebraic closure of K and [email protected]?(@s) the fixed field in [email protected]? of @s. The expression ''almost all @s'' means ''all but a set of @s of Haar measure 0''.