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

TL;DR

This paper proves that for Jacobi forms with certain properties, infinitely many Fourier coefficients are not divisible by almost all primes, leading to applications in number theory such as indivisibility of special L-values.

Contribution

It establishes the nonvanishing modulo  of Fourier coefficients of Jacobi forms for almost all primes, a result with implications for L-series and automorphic forms.

Findings

Infinitely many Fourier coefficients are -adically nonzero for almost all primes .

Applications include indivisibility results for special values of Dirichlet L-series.

Results extend to twisted L-functions of even weight newforms.

Abstract

Let $\phi = \sum_{r^{2} \leq 4mn}c(n,r)q^{n}\zeta^{r}$ be a Jacobi form of weight $k$ (with $k > 2$ if $\phi$ 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.