Non-vanishing and sign changes of Hecke eigenvalues for Siegel cusp forms of genus two

Abstract

In this paper, we show that half of non-zero coefficients of the spinor zeta function of a Siegel cusp form of genus 2 are positive and half are negative. We also prove results concerning the non-vanishing in short intervals and strong cancellation among the coefficients evaluated at powers of a fixed prime. Our results rest on a Serre's type density result established by Kowalski & Saha in the appendix.

Arithmetic behaviour of Hecke eigenvalues of Siegel cusp forms of degree two

“…with the assumption that λ F (p n−m ) = 0 for n < m. As in the elliptic case, by a work of Kowalski and Saha [12,Appendix], we have the following theorem.…”

“…We note that the subset of primes {p | µ G (p) = 0} has density zero (see appendix of [12]). Further the Generalized Ramanujan-Petersson conjecture proved by Weissauer ([15]) gives that for any prime p, |µ G (p)| ≤ 4p k 2 − 3 2 .…”

On signs of Hecke eigenvalues of Siegel eigenforms

“…Then any element 𝑛 of  is a squarefree positive integer satisfying 𝜆 𝐹 (𝑛)𝜆 𝐺 (𝑛) ≠ 0. We now claim that We know from [21,Theorem 4] that there exists a 𝛿 > 0 such that #{𝑝 ⩽ 𝑥 ∶ 𝜆 𝐹 (𝑝) = 0} ≪ 𝑥 (log 𝑥) 1+𝛿 .…”

“…But $$\begin{equation*} \sum _{ \lambda _F(p)\lambda _G(p)=0}\frac{1}{p}\leqslant \sum _{\lambda _F(p)=0}\frac{1}{p}+ \sum _{\lambda _G(p)=0}\frac{1}{p}. \end{equation*}$$We know from [21, Theorem 4] that there exists a $$\delta &gt;0$$ such that $$\begin{equation*} \#\lbrace p\leqslant x: \lambda _F(p)=0\rbrace \ll \frac{x}{(\log {x})^{1+\delta }}. \end{equation*}$$Using the above result, integration by parts yields $$\begin{equation*} \sum _{p\leqslant x \atop \lambda _F(p)=0}\frac{1}{p}= \int _{2^{-}}^x\frac{1}{u}d{\left(\sum _{p\leqslant u \atop \lambda _F(p)=0} 1\right)}\ll 1+\int _{2}^x \frac{du}{u(\log {u})^{1+\delta }}\ll 1.$$…”

On Fundamental Fourier Coefficients of Siegel Cusp Forms of Degree 2

“…We emphasize that while these kinds of questions have been previously studied for the full sequence ( ) of Fourier coefficients attached to F (see [12, 22, 24] for results on sign changes and [13] for an -result), there appears to be virtually no previous work in the more subtle setting where one restricts to fundamental Fourier coefficients. There has also been a fair bit of work on sign changes of Hecke eigenvalues of Siegel cusp forms [14, 36, 45, 53], which can be combined with the Hecke relations [1] to deduce sign changes among the with , where d is a fixed fundamental discriminant and m varies. This should make it clear that the problem of obtaining sign changes or growth asymptotics for Fourier coefficients not associated to fundamental discriminants is of a different flavor (and relatively easier).…”