Pontryagin duality for Iwasawa modules and abelian varieties

TL;DR

This paper establishes a functional equation relating dual systems of finite abelian p-groups and applies it to prove an algebraic functional equation for the Pontryagin dual of the Selmer group of an abelian variety over a global field in a Z_p^d-extension.

Contribution

It introduces a new functional equation framework for projective systems of finite abelian p-groups and applies it to Selmer groups of abelian varieties in Iwasawa theory.

Findings

Proves a functional equation for dual projective systems of finite abelian p-groups.

Establishes an algebraic functional equation for the Pontryagin dual of Selmer groups.

Extends Iwasawa theory results to higher-dimensional p-adic Lie extensions.

Abstract

We prove a functional equation for two projective systems of finite abelian $p$-groups, $\{\fa_n\}$ and $\{\fb_n\}$, endowed with an action of $\ZZ_p^d$ such that $\fa_n$ can be identified with the Pontryagin dual of $\fb_n$ for all $n$. Let $K$ be a global field. Let $L$ be a $\ZZ_p^d$-extension of $K$ ($d\geq 1$), unramified outside a finite set of places. Let $A$ be an abelian variety over $K$. We prove an algebraic functional equation for the Pontryagin dual of the Selmer group of $A$.