On a parity result for the symmetric square of modular forms with congruent residual representations
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912208038920192 |
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| author | Ray, Jishnu |
| author_facet | Ray, Jishnu |
| contents | The parity of Selmer ranks for elliptic curves defined over the rational numbers $\mathbb{Q}$ with good ordinary reduction at an odd prime $p$ has been studied by Shekhar. The proof of Shekhar relies on proving a parity result for the $λ$-invariants of Selmer groups over the cyclotomic $\mathbb{Z}_p$-extension $\mathbb{Q}_\infty$ of $\mathbb{Q}$.
This has been further generalized for elliptic curves with supersingular reduction at $p$ by Hatley and for modular forms by Hatley--Lei. In this paper, we prove a parity result for the $λ$-invariants of Selmer groups over $\mathbb{Q}_\infty$ for the symmetric square representations associated to two modular forms with congruent residual Galois representations. We treat both the ordinary and the non-ordinary cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_03050 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On a parity result for the symmetric square of modular forms with congruent residual representations Ray, Jishnu Number Theory Primary: 11R23, 11F11, 11R18 Secondary: 11F85 The parity of Selmer ranks for elliptic curves defined over the rational numbers $\mathbb{Q}$ with good ordinary reduction at an odd prime $p$ has been studied by Shekhar. The proof of Shekhar relies on proving a parity result for the $λ$-invariants of Selmer groups over the cyclotomic $\mathbb{Z}_p$-extension $\mathbb{Q}_\infty$ of $\mathbb{Q}$. This has been further generalized for elliptic curves with supersingular reduction at $p$ by Hatley and for modular forms by Hatley--Lei. In this paper, we prove a parity result for the $λ$-invariants of Selmer groups over $\mathbb{Q}_\infty$ for the symmetric square representations associated to two modular forms with congruent residual Galois representations. We treat both the ordinary and the non-ordinary cases. |
| title | On a parity result for the symmetric square of modular forms with congruent residual representations |
| topic | Number Theory Primary: 11R23, 11F11, 11R18 Secondary: 11F85 |
| url | https://arxiv.org/abs/2406.03050 |