Growth of in towers for isogenous curves

Abstract

We study the growth of$\unicode[STIX]{x0428}$and$p^{\infty }$-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 ‘positive${\it\mu}$-invariant’ setting in the Iwasawa theory of elliptic curves. The towers we consider are$p$-adic and$l$-adic Lie extensions for$l\neq p$, in particular cyclotomic and other$\mathbb{Z}_{l}$-extensions.