On Euler characteristics of Selmer groups for abelian varieties over global function fields

TL;DR

This paper derives Euler characteristic formulas for Selmer groups of abelian varieties over global function fields in the context of $ ext{Gal}(K/F)$-modules, advancing understanding of their algebraic structure.

Contribution

It introduces explicit Euler characteristic formulas for Selmer groups in $ ext{Gal}(K/F)$-extensions over global function fields, a novel extension of existing theories.

Findings

Euler characteristic formulas for Selmer groups derived

Results applicable to $ ext{Gal}(K/F)$-modules over global function fields

Enhances understanding of Selmer group structure in positive characteristic

Abstract

Let $F$ be a global function field of characteristic $p>0$, $K/F$ an $\ell$-adic Lie extension ($\ell\neq p$) and $A/F$ an abelian variety. We provide Euler characteristic formulas for the $Gal(K/F)$-module $Sel_A(K)_\ell$.