Non-vanishing and sign changes of Hecke eigenvalues for Siegel cusp forms of genus two (with an Appendix by E. Kowalski and A. Saha)

TL;DR

This paper proves that for Siegel cusp forms of genus 2, the non-zero coefficients of their spinor zeta function are equally likely to be positive or negative, revealing sign oscillation behavior.

Contribution

It establishes the sign distribution of non-zero coefficients of the spinor zeta function for genus 2 Siegel cusp forms, a new result in the field.

Findings

Half of the non-zero coefficients are positive.

Half of the non-zero coefficients are negative.

Sign changes occur infinitely often.

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.