Shioda-Tate formula for an abelian fibered variety and applications

On the Neron-Severi groups of fibered varieties

We apply Tate's conjecture on algebraic cycles to study the Néron-Severi groups of varieties fibered over a curve. This is inspired by the work of Rosen and Silverman, who carry out such an analysis to derive a formula for the rank of the group of sections of an elliptic surface. For a semistable fibered surface, under Tate's conjecture we derive a formula for the rank of the group of sections of the associated Jacobian fibration. For fiber powers of a semistable elliptic fibration $E --> C$, under Tate's conjecture we give a recursive formula for the rank of the Néron-Severi groups of these f…

Arithmetic on Elliptic Threefolds

In a recent paper, Rosen and Silverman showed that Tate's conjecture on the order of vanishing of L(E,s) implies Nagao's formula, which gives the rank of an elliptic surface in terms of a weighted average of fibral Frobenius trace values. The aim of this article is to extend their result to the case of elliptic threefolds, and deduce, from Tate's conjecture, a Nagao-type formula for the rank of an elliptic threefold E. This will require a two-pronged approach: on the one hand, we need some cohomological results in order to derive a Shioda-Tate-like formula for elliptic threefolds; on the other…

Comparing the Brauer group and the Tate Shafarevich group

We give a formula relating the order of the Brauer group of a surface fibered over a curve over a finite field to the order of the Tate-Shafarevich group of the Jacobian of the generic fiber. The formula implies that the Brauer group of a smooth and proper surface over a finite field is a square if it is finite.

Supersingular O'Grady varieties of dimension six

O'Grady constructed a 6-dimensional irreducible holomorphic symplectic variety by taking a crepant resolution of some moduli space of stable sheaves on an abelian surface. In this paper, we naturally extend O'Grady's construction to fields of positive characteristic p greater than 2, called OG6 varieties. We show that a supersingular OG6 variety is unirational, its rational cohomology group is generated by algebraic classes, and its rational Chow motive is of Tate type. These results confirm in this case the generalized Artin--Shioda conjecture, the supersingular Tate conjecture and the supers…

Artin-Tate 猜想与 Birch-Swinnerton-Dyer 猜想

The conjectures of Artin-Tate and Birch-Swinnerton-Dyer

本文通过两种证明建立纤维化曲面 Artin-Tate 猜想与一般纤维 Jacobian 的 BSD 猜想等价，并顺带给出 Geisser 关于 Brauer 群和 Tate-Shafarevich 群阶关系定理的新证明。

Torsion points on isogenous abelian varieties

Investigating a conjecture of Zannier, we study irreducible subvarieties of abelian schemes that dominate the base and contain a Zariski dense set of torsion points that lie on pairwise isogenous fibers. If everything is defined over the algebraic numbers and the abelian scheme has maximal variation, we prove that the geometric generic fiber of such a subvariety is a union of torsion cosets. We go on to prove fully or partially explicit versions of this result in fibered powers of the Legendre family of elliptic curves. Finally, we apply a recent result of Galateau-Martínez to obtain uniform b…