Growth of Sha in towers for isogenous curves

TL;DR

This paper investigates how the Tate-Shafarevich and Selmer groups of isogenous abelian varieties, especially elliptic curves, grow in towers of number fields, focusing on p-adic and l-adic Lie extensions.

Contribution

It provides new insights into the exponential growth patterns of these groups in specific p-adic and l-adic towers, extending Iwasawa theory results.

Findings

Growth is typically exponential in the studied towers.

Growth patterns depend on the mu-invariant in Iwasawa theory.

Results apply to cyclotomic and other Z_l-extensions.

Abstract

We study the growth of the Tate-Shafarevich and p-infinity Selmer groups for isogenous abelian varieties in towers of number fields, with an emphasis on elliptic curves. The growth types are usually exponential, as in the setting of `positive mu-invariant' in Iwasawa theory of elliptic curves. The towers we consider are p-adic and l-adic Lie extensions for l<>p, in particular cyclotomic and other Z_l-extensions.