Mordell-Weil ranks and Tate-Shafarevich groups of elliptic curves with mixed-reduction type over cyclotomic extensions

TL;DR

This paper studies the growth of Mordell-Weil ranks and Tate-Shafarevich groups of elliptic curves with mixed reduction types over cyclotomic extensions, extending Iwasawa theory techniques.

Contribution

It generalizes previous Iwasawa-theoretic results to elliptic curves with mixed reduction over cyclotomic extensions, defining multiply signed Selmer groups and analyzing their properties.

Findings

Mordell-Weil ranks are bounded over certain subextensions.

Asymptotic growth formulas for Tate-Shafarevich groups are derived.

Selmer groups are shown to be torsion under specific conditions.

Abstract

Let $E$ be an elliptic curve defined over a number field $K$ where $p$ splits completely. Suppose that $E$ has good reduction at all primes above $p$. Generalizing previous works of Kobayashi and Sprung, we define multiply signed Selmer groups over the cyclotomic $\mathbb{Z}_p$-extension of a finite extension $F$ of $K$ where $p$ is unramified. Under the hypothesis that the Pontryagin duals of these Selmer groups are torsion over the corresponding Iwasawa algebra, we show that the Mordell-Weil ranks of $E$ over a subextension of the cyclotomic $\mathbb{Z}_p$-extension are bounded. Furthermore, we derive an aysmptotic formula of the growth of the $p$-parts of the Tate-Shafarevich groups of $E$ over these extensions.