$p$-twisted Selmer near-companion curves

Abstract

Let $E_1$ and $E_2$ be elliptic curves over a number field $K$. In \cite{scc}, Mazur and Rubin define the concept of $n$-Selmer near-companions and conjecture that if $E_1$ and $E_2$ are $n$-Selmer near-companions over $K$, then $E_1[n]$ is $G_K$-isomorphic to $E_2[n]$. Yu proves the conjecture on $n$-Selmer near-companion curves in the case $n=2$. We we introduce the notion of $p$-twisted Selmer near-companions ($p$-TSNC) over $K$ and prove that if $E_1$ and $E_2$ are $p$-TSNC over $K$, then $K(E_1[p])=K(E_2[p])$ under certain conditions.