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

Jishnu Ray

TL;DR

The paper establishes parity relations for the Iwasawa $\lambda$-invariants of Selmer groups attached to symmetric square representations of two modular forms with congruent mod-$p$ Galois representations, extending prior parity results from elliptic curves to a modular-form context. The ordinary case leverages Greenberg–Vatsal type parity via the local polynomials to define explicit finite correction sets $\mathcal{S}_{f,\psi}$, yielding a mod-$2$ congruence between the $\lambda$-invariants of the two symmetric-square Selmer groups. In the non-ordinary setting, the authors deploy signed Selmer groups and Coleman maps to obtain analogous parity statements, including a control-theorem-based argument showing that, under $\mu=0$, imprimitive $\lambda$-invariants coincide and a similar mod-$2$ parity holds. The results provide computable parity information for congruent modular forms, underpining parity phenomena in the symmetric-square context and contributing to a broader understanding of $p$-parity in families of Galois 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 $λ$-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.

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

TL;DR

The paper establishes parity relations for the Iwasawa -invariants of Selmer groups attached to symmetric square representations of two modular forms with congruent mod- Galois representations, extending prior parity results from elliptic curves to a modular-form context. The ordinary case leverages Greenberg–Vatsal type parity via the local polynomials to define explicit finite correction sets , yielding a mod- congruence between the -invariants of the two symmetric-square Selmer groups. In the non-ordinary setting, the authors deploy signed Selmer groups and Coleman maps to obtain analogous parity statements, including a control-theorem-based argument showing that, under , imprimitive -invariants coincide and a similar mod- parity holds. The results provide computable parity information for congruent modular forms, underpining parity phenomena in the symmetric-square context and contributing to a broader understanding of -parity in families of Galois representations.

Abstract

The parity of Selmer ranks for elliptic curves defined over the rational numbers with good ordinary reduction at an odd prime has been studied by Shekhar. The proof of Shekhar relies on proving a parity result for the -invariants of Selmer groups over the cyclotomic -extension of . This has been further generalized for elliptic curves with supersingular reduction at by Hatley and for modular forms by Hatley--Lei. In this paper, we prove a parity result for the -invariants of Selmer groups over 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.
Paper Structure (8 sections, 18 theorems, 75 equations)

This paper contains 8 sections, 18 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. The symmetric square representation
  3. The Greenberg Selmer group
  4. Computing the parity of $\delta_\ell(\mathbf{A}_{f,\psi}/\mathbb{Q}_\infty)$
  5. Congruent modular forms
  6. The non-ordinary case
  7. The cyclotomic and finite level.
  8. Data availability

Key Result

Theorem 2.1

There exists a Galois representation $\rho_f:G_\mathbb{Q} \longrightarrow GL_2(L_{\mathfrak{P}})$ such that Let $V_f$ denote the representation space of $\rho_f$. Since $G_\mathbb{Q}$ is compact, choose an $\mathcal{O}_{L_\mathfrak{P}}$- lattice $T_f$ which is invariant under $\rho_f$. Let be the residual representation attached to $\rho_f$.

Theorems & Definitions (37)

  • Theorem 2.1: Eichler, Shimura, Deligne, Mazur-Wiles, Wiles, etc.
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Corollary 2.6
  • proof
  • Lemma 2.7
  • proof
  • Lemma 2.8
  • ...and 27 more