Selmer groups over ${\mathbb{Z}}_p^d$-extensions

Abstract

Consider an abelian variety $A$ defined over a global field $K$ and let $L/K$ be a ${\mathbb{Z}}_p^d$-extension, unramified outside a finite set of places of $K$, with $\Gal(L/K)=Γ$. Let $\Lambda (\Gamma ):={\mathbb{Z}}_p[[\Gamma ]]$ 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 ${\mathbb{Z}}_p^e$-extension.

33 pages