Selmer groups for elliptic curves in Z_l^d-extensions of function fields of characteristic p

TL;DR

This paper investigates the structure of Selmer groups of elliptic curves over certain infinite extensions of function fields in characteristic p, extending control theorems and analyzing module properties.

Contribution

It extends Mazur's Control Theorem to Z_l^d-extensions of function fields and characterizes Selmer groups as cofinitely generated modules.

Findings

Selmer groups are cofinitely generated modules over the Iwasawa algebra.

The paper establishes conditions under which these modules are cotorsion.

It provides a framework for understanding Selmer groups in positive characteristic settings.

Abstract

Let $F$ be a function field of characteristic $p>0$, $\F/F$ a Galois extension with $Gal(\F/F)\simeq \Z_l^d$ (for some prime $l\neq p$) and $E/F$ a non-isotrivial elliptic curve. We study the behaviour of Selmer groups $Sel_E(L)_r$ ($r$ any prime) as $L$ varies through the subextensions of $\F$ via appropriate versions of Mazur's Control Theorem. As a consequence we prove that $Sel_E(\F)_r$ is a cofinitely generated (in some cases cotorsion) $\Z_r[[Gal(\F/F)]]$-module.