Hecke eigenvalues and Fourier–Jacobi coefficients of Siegel cusp forms of degree $2$

Murugesan Manickam, Karam Deo Shankhadhar and Vasudevan Srivatsa

Vol. 332 (2024), No. 2, 243–259

DOI: 10.2140/pjm.2024.332.243

Abstract

The nonvanishing of the first Fourier–Jacobi coefficient of a Siegel eigenform  $F$ gives us that the vanishing of its $m$-th Fourier–Jacobi coefficient  $F|{\rho }_m$ implies the vanishing of its $m$-th eigenvalue ${\lambda }_F(m)$. Conversely, we prove that for any odd, squarefree $m$ if ${\lambda }_F(m)$ is zero then $F|{\rho }_m$ vanishes. While investigating this converse question and its important consequences, we generalize certain existing results of Kohnen and Skoruppa (1989) for index  $1$ Jacobi cusp forms to any arbitrary index, which are also of independent interest.

Keywords

Siegel modular forms, Hecke operators, Hecke eigenvalues, Fourier–Jacobi coefficients, Jacobi forms, modular forms of integral and half-integral weight

Mathematical Subject Classification

Primary: 11F11, 11F37, 11F46, 11F50

Milestones

Received: 13 June 2024

Revised: 3 October 2024

Accepted: 2 November 2024

Published: 6 December 2024