Nonvanishing modulo $\ell$ of Fourier coefficients of Jacobi forms

Abstract

Let $\varphi ={\sum }_{r^2\le 4mn}c(n,r)q^n{\zeta }^r$ be a Jacobi form of weight $k$ (with $k>2$ if $\varphi$ is not a cusp form) and index $m$ with integral algebraic coefficients which is an eigenfunction of all Hecke operators $T_p,(p,m)=1,$ and which has at least one nonvanishing coefficient $c(n,r)$ with $r$ prime to $m$. We prove that for almost all primes $\ell$ there are infinitely many fundamental discriminants $D=r^2-4mn<0$ prime to $m$ with ${\nu }_{\ell }(c(n,r))=0$, where ${\nu }_{\ell }$ denotes a continuation of the $\ell$-adic valuation on $\mathbb{Q}$ to an algebraic closure. As applications we show indivisibility results for special values of Dirichlet $L$-series and for the central critical values of twisted $L$-functions of even weight newforms.

arXiv comment18 pages