On Selmer groups of abelian varieties over $\ell$-adic Lie extensions of global function fields

TL;DR

This paper investigates the structure of Selmer groups of abelian varieties over certain $ ext{l}$-adic Lie extensions of global function fields, showing under specific conditions that these groups lack nontrivial pseudo-null submodules.

Contribution

It proves that, under certain conditions, the dual Selmer group over $ ext{l}$-adic Lie extensions has no nontrivial pseudo-null submodules, advancing understanding of Selmer group structure.

Findings

Selmer groups have no nontrivial pseudo-null submodules under given conditions.

Results apply to $ ext{l}$-adic Lie extensions unramified outside finite primes.

Provides new insights into the structure of Selmer groups over function fields.

Abstract

Let $F$ be a global function field of characteristic $p>0$ and $A/F$ an abelian variety. Let $K/F$ be an $\l$-adic Lie extension ($\l\neq p$) unramified outside a finite set of primes $S$ and such that $\Gal(K/F)$ has no elements of order $\l$. We shall prove that, under certain conditions, $Sel_A(K)_\l^\vee$ has no nontrivial pseudo-null submodule.