On the Selmer groups of abelian varieties over function fields of characteristic p>0

TL;DR

This paper explores a geometric analogue of Iwasawa theory for abelian varieties over function fields of characteristic p, extending classical number field results and proposing new conjectures.

Contribution

It develops a p-adic geometric analogue of Iwasawa theory for abelian varieties over function fields, including new phenomena and conjectures.

Findings

Proved analogues of key Iwasawa theory results in the function field setting

Identified new phenomena not present in number field cases

Proposed a conjecture analogous to the main conjecture in number field Iwasawa theory

Abstract

In this paper, we study a (p-adic) geometric analogue for abelian varieties over a function field of characteristic p of the cyclotomic Iwasawa theory and the non-commutative Iwasawa theory for abelian varieties over a number field initiated by Mazur and Coates respectively. We will prove some analogue of the principal results obtained in the case over a number field and we study new phenomena which did not happen in the case of number field case. We propose also a conjecture which might be considered as a counterpart of the principal conjecture in the case over a number field. \par This is a preprint which is distributed since 2005 which is still in the process of submision. Following a recent modification of some technical mistakes in the previous version of the paper as well as an amelioration of the presentation of the paper, we decide wider distribution via the archive.