2-Selmer companion modular forms
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913993042427904 |
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| author | Abhishek Jha, Somnath Shekhar, Sudhanshu |
| author_facet | Abhishek Jha, Somnath Shekhar, Sudhanshu |
| contents | Let $N$ be a positive integer and $K$ be a number field. Suppose that $f_1,f_2 \in S_k(Γ_0(N))$ are two newforms such that their residual Galois representations at $2$ are isomorphic. Let $ω_2: G_{\mathbb Q} \rightarrow {\mathbb Z}^*_2$ be the $2$-adic cyclotomic character. Then, under suitable hypotheses, we have shown that for every quadratic character $χ$ of $K$ and each critical twist $j$, the residual Greenberg $2$-Selmer groups of $f_1χω_2^{-j}$ and $f_2χω_2^{-j}$ 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 $f_1χ$ and $f_2χ$ is bounded independent of every quadratic character $χ$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_23805 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | 2-Selmer companion modular forms Abhishek Jha, Somnath Shekhar, Sudhanshu Number Theory 11F33, 11R34, 11S25 Let $N$ be a positive integer and $K$ be a number field. Suppose that $f_1,f_2 \in S_k(Γ_0(N))$ are two newforms such that their residual Galois representations at $2$ are isomorphic. Let $ω_2: G_{\mathbb Q} \rightarrow {\mathbb Z}^*_2$ be the $2$-adic cyclotomic character. Then, under suitable hypotheses, we have shown that for every quadratic character $χ$ of $K$ and each critical twist $j$, the residual Greenberg $2$-Selmer groups of $f_1χω_2^{-j}$ and $f_2χω_2^{-j}$ 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 $f_1χ$ and $f_2χ$ is bounded independent of every quadratic character $χ$ 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. |
| title | 2-Selmer companion modular forms |
| topic | Number Theory 11F33, 11R34, 11S25 |
| url | https://arxiv.org/abs/2506.23805 |