2-Selmer companion modular forms

Abstract

Let $N$ be a positive integer and $K$ be a number field. Suppose that $f_1,f_2\in S_k(\hat{I}{“}_0(N))$ are two newforms such that their residual Galois representations at $2$ are isomorphic. Let $\ddot{I}{‰}_2:G_{\mathbb{Q}}\to {\mathbb{Z}}_2^{\ast }$ be the $2$-adic cyclotomic character. Then, under suitable hypotheses, we have shown that for every quadratic character $\ddot{I}‡$ of $K$ and each critical twist $j$, the residual Greenberg $2$-Selmer groups of $f_1\ddot{I}‡\ddot{I}{‰}_2^{-j}$ and $f_2\ddot{I}‡\ddot{I}{‰}_2^{-j}$ over $K$ are isomorphic. This generalizes the corresponding result of Mazur-Rubin on $2$-Selmer companion elliptic curves. Conversely, if the difference of the residual Greenberg (respectively Bloch-Kato) $2$-Selmer ranks of $f_1\ddot{I}‡$ and $f_2\ddot{I}‡$ is bounded independent of every quadratic character $\ddot{I}‡$ of $K$, then under suitable hypotheses we have shown that the residual Galois representations at $2$ of $f_1$ and $f_2$ are isomorphic as $G_K$-modules. The corresponding result for elliptic curves was a conjecture of Mazur-Rubin, which was proved by M. Yu.