Hodge structures on abelian varieties of type IV

Let A be a general member of a PEL-family of abelian varieties with endomorphisms by an imaginary quadratic number field k, and let E be an elliptic curve with complex multiplications by k. We show that the usual Hodge conjecture for products of A with powers of E implies the general Hodge conjecture for all powers of A. We deduce the general Hodge conjecture for all powers of certain 5-dimensional abelian varieties. Mathematics Subject Classification (2000): Primary 14C30, 14K20.Research supported in part by a Research and Creative Activity Award for Summer 2001 from East Carolina University.     

Some remarks on the Hodge conjecture for abelian varieties

We show how the classical Hodge conjecture for the middle cohomology of an abelian variety is equivalent to the general Hodge conjecture for the middle cohomology of a smooth ample divisor in the abelian variety. This is best suited to abelian varieties with actions of imaginary quadratic fields.     

Lefschetz Motives and the Tate Conjecture

A Lefschetz class on a smooth projective variety is an element of the Q-algebra generated by divisor classes. We show that it is possible to define Q-linear Tannakian categories of abelian motives using the Lefschetz classes as correspondences, and we compute the fundamental groups of the categories. As an application, we prove that the Hodge conjecture for complex abelian varieties of CM-type implies the Tate conjecture for all Abelian varieties over finite fields, thereby reducing the latter to a problem in complex analysis.     

Quaternionic pryms and Hodge classes

Abelian varieties of dimension 2n on which a definite quaternion algebra acts are parametrized by symmetrical domains of dimension n(n−1)/2. Such abelian varieties have primitive Hodge classes in the middle dimensional cohomology group. In general, it is not clear that these are cycle classes. In this paper we show that a particular 6-dimensional family of such 8-folds are Prym varieties and we use the method of Schoen to show that all Hodge classes on the general abelian variety in this family are algebraic. We also consider Hodge classes on certain 5-dimensional subfamilies and relate these to the Hodge conjecture for abelian 4-folds.     

Tate twists of Hodge structures arising from abelian varieties of type IV

We show that certain abelian varieties $A$ have the property that for every Hodge structure $V$ in the cohomology of $A$, every effective Tate twist of $V$ occurs in the cohomology of some abelian variety. We deduce the general Hodge conjecture for certain non-simple abelian varieties of type IV.     

Visibility of Shafarevich-Tate Groups of Abelian Varieties

We investigate Mazur's notion of visibility of elements of Shafarevich-Tate groups of abelian varieties. We give a proof that every cohomology class is visible in a suitable abelian variety, discuss the visibility dimension, and describe a construction of visible elements of certain Shafarevich-Tate groups. This construction can be used to give some of the first evidence for the Birch and Swinnerton-Dyer conjecture for abelian varieties of large dimension. We then give examples of visible and invisible Shafarevich-Tate groups.     

Hodge Structures Associated to SU(p, 1)

Let A be an abelian variety over ? such that the semisimple part of the Hodge group of A is a product of copies of SU(p, 1) for some p > 1. We show that any effective Tate twist of a Hodge structure occurring in the cohomology of A is isomorphic to a Hodge structure in the cohomology of some abelian variety.     

Good reductions of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic. Part I

We prove the existence of good smooth integral models of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic (0, p). As a first application we provide a smooth solution (answer) to a conjecture (question) of Langlands for Shimura varieties of Hodge type. As a second application we prove the existence in arbitrary unramified mixed characteristic (0, p) of integral canonical models of projective Shimura varieties of Hodge type with respect to h-hyperspecial subgroups as pro-étale covers of Néron models; this forms progress towards the proof of conjectures of Milne and Reimann. Though the second application was known before in some cases, its proof is new and more of a principle.     

Réalisation de Hodge du polylogarithme d'un schéma abélien

The Hodge realization of the polylogarithm of an Abelian scheme is an extension of Hodge modules. In [A. Levin, Polylogarithmic currents on abelian varieties, in: A. Reznikov, N. Schappacher (Eds.), Regulators in Analysis, Geometry and Number Theory, in: Progr. Math., vol. 171, Birkhäuser, 2000, pp. 207–229], Levin constructs some currents (generalized Eisenstein series) and conjectures that they describe the polylogarithmic extension. Our main result (Thm 3.1 and Cor 3.2) is a proof of this conjecture. This provides a tool to study the Eisenstein classes (see Section 4), which have a motivic origin (see [G. Kings, K-theory elements for the polylogarithm of abelian schemes, J. Reine Angew. Math. 517 (1999) 103–116]); an example of its application will be given in a second Note. To cite this article: D. Blottière, C. R. Acad. Sci. Paris, Ser. I 344 (2007).     

Abelian varieties over finitely generated fields and the conjecture of Geyer and Jarden on torsion

In this paper we prove the Geyer‐Jarden conjecture on the torsion part of the Mordell‐Weil group for a large class of abelian varieties defined over finitely generated fields of arbitrary characteristic. The class consists of all abelian varieties with big monodromy, i.e., such that the image of Galois representation on ?‐torsion points, for almost all primes ?, contains the full symplectic group.     

Derived categories and Kummer varieties

We prove that if two abelian varieties have equivalent derived categories then the derived categories of the smooth stacks associated to the corresponding Kummer varieties are equivalent as well. The second main result establishes necessary and sufficient conditions for the existence of equivalences between the twisted derived categories of two Kummer surfaces in terms of Hodge isometries between the generalized transcendental lattices of the corresponding abelian surfaces.      

On the Birch and Swinnerton-Dyer conjecture for abelian varieties attached to Hilbert modular forms

In this paper we prove that if the Birch and Swinnerton-Dyer conjecture holds for abelian varieties attached to Hilbert newforms of parallel weight 2 with trivial central character, then the Birch and Swinnerton-Dyer conjecture holds for abelian varieties attached to Hilbert newforms of parallel weight 2 with trivial central character regarded over arbitrary totally real number fields.     

Motivic integral of K3 surfaces over a non-archimedean field

We prove a formula expressing the motivic integral (Loeser and Sebag, 2003) [34] of a K3 surface over C((t)) with semi-stable reduction in terms of the associated limit mixed Hodge structure. Secondly, for every smooth variety over a complete discrete valuation field we define an analogue of the monodromy pairing, constructed by Grothendieck in the case of abelian varieties, and prove that our monodromy pairing is a birational invariant of the variety. Finally, we propose a conjectural formula for the motivic integral of maximally degenerate K3 surfaces over an arbitrary complete discrete valuation field and prove this conjecture for Kummer K3 surfaces.     

On Galois representations for abelian varieties with complex and real multiplications

In this paper, we study the image of l-adic representations coming from Tate module of an abelian variety defined over a number field. We treat abelian varieties with complex and real multiplications. We verify the Mumford-Tate conjecture for a new class of abelian varieties with real multiplication.     

Generalized Greatest Common Divisors,Divisibility Sequences,and Vojta’s Conjecture for Blowups

We apply Vojta’s conjecture to blowups and deduce a number of deep statements regarding (generalized) greatest common divisors on varieties, in particular on projective space and on abelian varieties. Special cases of these statements generalize earlier results and conjectures. We also discuss the relationship between generalized greatest common divisors and the divisibility sequences attached to algebraic groups, and we apply Vojta’s conjecture to obtain a strong bound on the divisibility sequences attached to abelian varieties of dimension at least two.     

Generalized Greatest Common Divisors, Divisibility Sequences, and Vojta’s Conjecture for Blowups

We apply Vojta’s conjecture to blowups and deduce a number of deep statements regarding (generalized) greatest common divisors on varieties, in particular on projective space and on abelian varieties. Special cases of these statements generalize earlier results and conjectures. We also discuss the relationship between generalized greatest common divisors and the divisibility sequences attached to algebraic groups, and we apply Vojta’s conjecture to obtain a strong bound on the divisibility sequences attached to abelian varieties of dimension at least two.     

Modular Shimura varieties and forgetful maps

In this note we consider several maps that occur naturally between modular Shimura varieties, Hilbert-Blumenthal varieties and the moduli spaces of polarized abelian varieties when forgetting certain endomorphism structures. We prove that, up to birational equivalences, these forgetful maps coincide with the natural projection by suitable abelian groups of Atkin-Lehner involutions.

     

Heights and Degrees: A Note on a Paper of C. Liebendörfer and G. Rémond

We verify in the rational quaternionic case a conjecture of C. Liebend?rfer relating the degrees of abelian subvarieties of certain polarized abelian varieties to the (quaternionic) heights of their defining equations.     

Schubert cycles, differential forms and D-modules on varieties of unseparated flags

Proper homogeneous G-spaces (where G is semisimple algebraic group) over positive characteristic fields k can be divided into two classes, the first one being the flag varieties G/P and the second one consisting of varieties of unseparated flags (proper homogeneous spaces not isomorphic to flag varieties as algebraic varieties). In this paper we compute the Chow ring of varieites of unseparated flags, show that the Hodge cohomology of usual flag varieties extends to the general setting of proper homogeneous spaces and give an example showing (by geometric means) that the D -affinity of Beilinson and Bernstein fails for varieties of unseparated flags.