Selmer groups over $\Z_p^d$-extensions

TL;DR

This paper investigates how the algebraic structure of Selmer groups associated with abelian varieties over global fields changes when considering different intermediate $ ext{Z}_p^e$-extensions within a fixed $ ext{Z}_p^d$-extension, focusing on the variation of their characteristic ideals.

Contribution

It provides new insights into the behavior of the characteristic ideal of the dual Selmer group as the extension varies within a $ ext{Z}_p^d$-extension, extending previous Iwasawa theory results.

Findings

Characterizes the variation of the characteristic ideal across intermediate extensions.

Establishes relations between Selmer groups over different $ ext{Z}_p^e$-extensions.

Provides formulas connecting the Iwasawa modules in the tower.

Abstract

Consider an abelian variety $A$ defined over a global field $K$ and let $L/K$ be a $\Z_p^d$-extension, unramified outside a finite set of places of $K$, with $\Gal(L/K)=\Gamma$. Let $\Lambda(\Gamma):=\Z_p[[\Gamma]]$ denote the Iwasawa algebra. In this paper, we study how the characteristic ideal of the $\Lambda(\Gamma)$-module $X_L$, the dual $p$-primary Selmer group, varies when $L/K$ is replaced by a intermediate $\Z_p^e$-extension.