On signs of Hecke eigenvalues of Siegel eigenforms

TL;DR

This paper investigates the signs of Hecke eigenvalues of Siegel eigenforms of degree two, establishing results on their sign patterns and the distinctness of Satake parameters for almost all primes, using Galois representations.

Contribution

It provides new insights into the sign behavior of Hecke eigenvalues and the uniqueness of Satake parameters for Siegel eigenforms, advancing understanding of their arithmetic properties.

Findings

Sign patterns of Hecke eigenvalues can distinguish Siegel eigenforms.

Almost all primes have distinct Satake p-parameters for different eigenforms.

Results are achieved using Galois representations attached to Siegel eigenforms.

Abstract

In this article, we distinguish Siegel cuspidal eigenforms of degree two on the full symplectic group from the signs of their Hecke eigenvalues. To establish our theorem, we obtain a result towards simultaneous sign changes of eigenvalues of two Siegel eigenforms. In the course of the proof, we also prove that the Satake $p$-parameters of two different Siegel eigenforms are distinct for a set of primes $p$ of density 1. The main ingredient to prove the latter result is the theory of Galois representations attached to Siegel eigenforms.