2-Selmer companion modular forms

TL;DR

This paper investigates the relationships between residual Galois representations and Selmer groups of modular forms, generalizing known results for elliptic curves to modular forms over number fields.

Contribution

It extends the concept of Selmer companions from elliptic curves to modular forms, establishing isomorphisms of residual Greenberg Selmer groups under certain conditions.

Findings

Residual Greenberg 2-Selmer groups are isomorphic for modular forms with isomorphic residual Galois representations.

Bounded differences in Selmer ranks imply isomorphism of residual Galois representations.

Generalizes Mazur-Rubin's results from elliptic curves to modular forms.

Abstract

Let $N$ be a positive integer and $K$ be a number field. Suppose that $f_1,f_2 \in S_k(\Gamma_0(N))$ are two newforms such that their residual Galois representations at $2$ are isomorphic. Let $\omega_2: G_{\mathbb Q} \rightarrow {\mathbb Z}^*_2$ be the $2$-adic cyclotomic character. Then, under suitable hypotheses, we have shown that for every quadratic character $\chi$ of $K$ and each critical twist $j$, the residual Greenberg $2$-Selmer groups of $f_1\chi\omega_2^{-j}$ and $f_2\chi\omega_2^{-j}$ 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 $f_1\chi$ and $f_2\chi$ is bounded independent of every quadratic character $\chi$ 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.