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

The parity of Selmer ranks for elliptic curves defined over the rational numbers 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 Zₚ-extension Q_∞ of 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 Q_∞ 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.