On Signs of eigenvalues of Modular forms satisfying Ramanujan Conjecture

TL;DR

This paper investigates the sign distribution of eigenvalues of Siegel cusp forms satisfying the Ramanujan conjecture, providing a lower bound on the density of primes where their eigenvalues have opposite signs.

Contribution

It offers a new lower bound on the density of primes with eigenvalues of two Siegel cusp forms having opposite signs, under the assumption of the Ramanujan conjecture.

Findings

Established a lower bound for the density of primes with eigenvalue signs opposite for two forms.

Demonstrated the application of the Ramanujan conjecture to eigenvalue sign analysis.

Extended understanding of eigenvalue sign patterns in Siegel cusp forms.

Abstract

Let $F \in S_{k_1}(\Gamma^{(2)}(N_1))$ and $G \in S_{k_2}(\Gamma^{(2)}(N_2))$ be two Siegel cusp forms over the congruence subgroups $\Gamma^{(2)}(N_1)$ and $\Gamma^{(2)}(N_2)$ respectively. Assume that they are Hecke eigenforms in different eigenspaces and satisfy the Generalized Ramanujan Conjecture. Let $\lambda_F(p)$ denote the eigenvalue of $F$ with respect to the Hecke operator $T(p)$. In this article, we compute a lower bound for the density of the set of primes, $\{ p : \lambda_F(p) \lambda_G(p) < 0 \}.$