On a parity result for the symmetric square of modular forms with congruent residual representations

Jishnu Ray

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arXiv · 2406.03050

On a parity result for the symmetric square of modular forms with congruent residual representations

Jishnu Ray

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Abstract

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 $\lambda$-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 $\lambda$-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.

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Jishnu Ray. 2024-06-05. On a parity result for the symmetric square of modular forms with congruent residual representations. https://arxiv.org/abs/2406.03050

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