  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.
