Rank Jumps and Growth of Shafarevich--Tate Groups for Elliptic Curves in $\mathbb{Z}/p\mathbb{Z}$-Extensions

TL;DR

This paper investigates how the rank of elliptic curves and the sizes of their associated groups grow in cyclic prime degree extensions, using Iwasawa theory to improve existing results.

Contribution

It introduces new techniques from Iwasawa theory to analyze rank jumps and growth of Shafarevich--Tate groups in cyclic degree-$p$ extensions, advancing prior understanding.

Findings

Demonstrates rank jumps in certain cyclic extensions

Provides bounds on growth of the $p$-primary Selmer group

Improves previous results on Shafarevich--Tate group growth

Abstract

In this paper, we use techniques from Iwasawa theory to study questions about rank jump of elliptic curves in cyclic extensions of prime degree. We also study growth of the $p$-primary Selmer group and the Shafarevich--Tate group in cyclic degree-$p$ extensions and improve upon previously known results in this direction.