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

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.