On the Iwasawa Main conjecture of abelian varieties over function fields

TL;DR

This paper investigates a geometric analogue of the Iwasawa Main Conjecture for abelian varieties over function fields, focusing on constant and semistable cases, and establishes a key pseudo-isomorphism related to Selmer groups.

Contribution

It introduces a new algebraic functional equation linking Selmer groups of abelian varieties and their duals over function fields, extending known results to geometric settings.

Findings

Established a pseudo-isomorphism between dual Selmer groups of abelian varieties and their duals.

Extended algebraic functional equations to the setting of function fields.

Provided new insights into the Iwasawa Main Conjecture in a geometric context.

Abstract

We study a geometric analogue of the Iwasawa Main Conjecture for abelian varieties in the two following cases: constant ordinary abelian varieties over $Z_p^d$-extensions of function fields ($d\geq 1$) ramified at a finite set of places, and semistable abelian varieties over the arithmetic $Z_p$-extension of a function field. One of the tools we use in our proof is a pseudo-isomorphism relating the duals of the Selmer groups of $A$ and its dual abelian variety $A^t$. This holds as well over number fields and is a consequence of a quite general algebraic functional equation.