2-Selmer near-companion curves

Let $E$ and $A$ be elliptic curves over a number field $K$. Let $χ$ 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 $E^χ$ and $A^χ$ is bounded independent of $χ$, there is a $G_K$-isomorphism $E[2] \cong A[2]$.

No reviews yet.

No comments yet.