On the arithmetic of abelian varieties

TL;DR

This paper introduces new algebraic objects related to abelian varieties over function fields, proves their finiteness properties, and establishes a specialization theorem that generalizes classical results.

Contribution

It defines discrete Selmer and Shafarevich-Tate groups, proves their finite generation, and extends Néron's specialization theorem to Galois cohomology of abelian varieties.

Findings

Discrete Selmer groups are finitely generated Z-modules.

In the isotrivial case, the discrete Shafarevich-Tate group vanishes.

The new specialization theorem generalizes Néron's classical result.

Abstract

We prove some new results on the arithmetic of abelian varieties over function fields of one variable over finitely generated (infinite) fields. Among other things, we introduce certain new natural objects `discrete Selmer groups' and `discrete Shafarevich-Tate groups', and prove that they are finitely generated $\Bbb Z$-modules. Further, we prove that in the isotrivial case, the discrete Shafarevich-Tate group vanishes and the discrete Selmer group coincides with the Mordell-Weil group. One of the key ingredients to prove these results is a new specialisation theorem \`a la N\'eron for first Galois cohomology groups, of the ($l$-adic) Tate module of abelian varieties which generalises N\'eron's specialisation theorem for rational points of abelian varieties.