On Signs of Hecke eigenvalues of Siegel modular forms

Abstract

Let F ∈ Sk+n(Γ2n) be an Ikeda lift and λF(m) be the eigenvalue corresponding to Hecke operator T(m). For a fixed n and k, we show that λF(p) is positive for all large enough primes p. For n=2, we show that, given r there exists Cr such that λF(pr) > 0 for all p > Cr. Let F ∈ Sk1(Γ(2)(N1)) and G ∈ Sk2 (Γ(2)(N2)) be two eigenforms that satisfy the Generalized Ramanujan Conjecture. We compute a lower bound for the density of the set of primes, {p : λF(p)λG(p) < 0}.