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 $\chi \in \text{Hom}(G_K, \mu_p)$ such that $\text{dim}_{\mathbb{F}_p}(\text{Sel}_p(E/K, \chi)) = d$ under certain conditions.