Sign changes of Hecke eigenvalues of Siegel cusp forms of degree 2

Abstract

Abstract. Let µ(n), n > 0, be the sequence of Hecke eigenvalues of a cuspidal Siegel eigenform F of degree 2. It is proved that if F is not in the Maaß space, then there exist infinitely many primes p for which the sequence µ(p r ), r > 0, has infinitely many sign changes.

The First Negative Hecke Eigenvalue of Genus 2 Siegel Cuspforms With Level N ≥ 1

“…3). We show (essentially a result of [10]) that Breulmann's result generalizes to the level Γ 2 0 (N ) situation as well. See Theorem 3.1 for the precise result.…”

“…This result has been generalized in [10] for the case N > 1. Their result shows that if F is a Hecke eigenform for all T (n) with gcd(n, N ) = 1 such that π F is not CAP, then there exists an infinite set S F of prime numbers p N such that if p ∈ S F , then there are infinitely many r such that λ F (p r ) > 0 and infinitely many r such that λ F (p r ) < 0.…”

“…For a more detailed exposition of the material in this section the reader is urged to consult [10] or [11].…”

“…One should observe that since we are assuming F is without character, we can say that π F is CAP if and only if it is either a theta lift or a twist of a theta lift by a quadratic character. In such a case, we have the following characterization of the local representations π F,p for p N (see [10]). …”

On the congruences primes of Saito-Kurokawa lifts of odd square-free level

“…This is discussed in Corollary 5.10. There is also still the possibility that G arises as a CAP form that is a theta lift twisted by a quadratic character, see [14] for example. We do not address this possibility here.…”