Control Theorems for l-adic Lie extensions of global function fields

TL;DR

This paper extends control theorems to l-adic Lie extensions of global function fields, analyzing the structure of Selmer groups of abelian varieties and establishing conditions for their finite generation and torsion properties.

Contribution

It generalizes Mazur's Control Theorem to a broader class of l-adic Lie extensions over global function fields, including cases where l=p.

Findings

Selmer groups are finitely generated modules over Iwasawa algebras under mild hypotheses.

Conditions are provided for Selmer groups to be torsion modules when Galois groups have specific subgroup structures.

The results apply to both cases l ≠ p and l = p, broadening the scope of Iwasawa theory in function fields.

Abstract

Let F be a global function field of characteristic p>0, K/F an l-adic Lie extension unramified outside a finite set of places S and A/F an abelian variety without complex multiplication. We study Sel_A(K)_l^\vee (the Pontrjagin dual of the Selmer group) and (under some mild hypotheses) prove that it is a finitely generated Z_l[[\Gal(K/F)]]-module via generalizations of Mazur's Control Theorem. If Gal(K/F) has no elements of order l and contains a closed normal subgroup H such that Gal(K/F)/H\simeq Z_l, we are able to give sufficient conditions for Sel_A(K)_l^\vee to be finitely generated as Z_l[[H]]-module and, consequently, a torsion Z_l[[\Gal(K/F)]]-module. We deal with both cases l\neq p and l=p.