Growth of Sha in towers for isogenous curves

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.

28 pages

References in corpus (1)

• A remark on Tate's algorithm and Kodaira types

Cited by in corpus (7)

• A remark on Tate's algorithm and Kodaira types
• Some remarks on Kida's formula when $\mu \ne 0$
• Conductors of l-adic representations
• Duality invariance of Faltings heights, Hodge line bundles and global periods
• Local invariants of isogenous elliptic curves
• The algebra ${\mathbb{Z}}_{\ell }[[{\mathbb{Z}}_p^d]]$ and applications to Iwasawa theory
• Arithmetic of abelian varieties in Artin-Schreier extensions