Non-vanishing and sign changes of Hecke eigenvalues for Siegel cusp forms of genus two (with an appendix)

E Kowalski (1) , A Saha (2) , Emmanuel Royer (3) , Jyoti Sengupta (4) , Jie Wu (5)

1 D-MATH - Department of Mathematics [ETH Zurich]
2 University of Bristol [Bristol]
3 LMBP - Laboratoire de Mathématiques Blaise Pascal
4 School of Technology and Computer Science [TIFR]
5 IECL - Institut Élie Cartan de Lorraine

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.

Mots clés: Fourier coefficients, Spinor zeta function, Siegel form, Hecke eigenvalues

Domaines: Théorie des nombres [math.NT]

E Kowalski, A Saha, Emmanuel Royer, Jyoti Sengupta, Jie Wu. Non-vanishing and sign changes of Hecke eigenvalues for Siegel cusp forms of genus two (with an appendix). Ramanujan Journal (The), 2016, 39, pp.179-199. ⟨10.1007/s11139-014-9667-3⟩. ⟨hal-01278962⟩