2-Selmer near-companion curves

TL;DR

This paper proves a conjecture by Mazur and Rubin regarding 2-Selmer near-companion curves, establishing a link between bounded Selmer rank differences and Galois module isomorphisms of 2-torsion points.

Contribution

It confirms the conjecture for the case n=2, showing that bounded 2-Selmer rank differences imply isomorphic Galois modules for elliptic curves.

Findings

Bounded 2-Selmer rank differences imply Galois isomorphism of 2-torsion points.

Proves Mazur-Rubin conjecture for n=2.

Establishes a criterion connecting Selmer ranks and Galois representations.

Abstract

Let $E$ and $A$ be elliptic curves over a number field $K$. Let $\chi$ be a quadratic character of $K$. We prove the conjecture posed by Mazur and Rubin on $n$-Selmer near-companion curves in the case $n=2$. Namely, we show if the difference of the $2$-Selmer ranks of $E^\chi$ and $A^\chi$ is bounded independent of $\chi$, there is a $G_K$-isomorphism $E[2] \cong A[2]$.