Increasing the p-Selmer rank by twisting

Abstract

In this paper, we study the $p$-Selmer groups in the family of $p$-twists of an elliptic curve $E$ over a number field $K$. We prove that if $E/K$ is an elliptic curve over a number field $K$, and if $d$ is congruent to the dimension of the Selmer group of $E/K$ modulo $2$ and is greater than that dimension, then there exist infinitely many characters $\ddot{I}‡\in \text{Hom}(G_K,\hat{I}{¼}_p)$ such that ${\text{dim}}_{{\mathbb{F}}_p}({\text{Sel}}_p(E/K,\ddot{I}‡))=d$ under certain conditions.

To appear in International Journal of Number Theory