On signs of Hecke eigenvalues of 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. © 2022 The Authors. The publishing rights in this article are licensed to University College London under an exclusive licence. Mathematika is published by the London Mathematical Society on behalf of University College London.