Nonvanishing modulo ℓof Fourier coefficients of Jacobi forms

Markus Schwagenscheidt

doi:10.1007/s40993-015-0035-1

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.

Highlights

Introduction and main resultsLet k and m be positive integers
Ono and Skinner [11] and Bruinier [1] have shown, using different methods, that for a non-trivial Hecke eigenform f = n≥0 a(n)qn of weight k + 1/2 for 0(4m) with integral algebraic coefficients, which is not a linear combination of Shimura theta functions, almost all primes have the property that there are infinitely many square free n with ν (a(n)) = 0, where ν denotes a continuation of the -adic valuation on Q to a fixed algebraic closure
We adapt the methods from [1] to prove an analogous result for the Fourier coefficients of Jacobi forms: Theorem 1.1

Summary

We first decompose

Since φ is invariant under the slash operation of the first element on the right-hand side, the αb-part of φ|Tp is given by pk−4 φ b(p2) λ,μ(p) k,m. = pk−4 p−kφ((τ + b)/p2, (z + μ − Nλb)/p). If we replace μ − Nλb by μ and plug in the Fourier expansion of φ, the last line becomes p−3 c(n, r)e(nb/p2)e(rμ/p)qn/p2 ζ r/p. C(p2n , pr )qn ζ r n,r∈Z r2≤4mn p2|n,p|r n ,r ∈Z r 2≤4mn of the αb-part, where we replaced n = p2n and r = pr. Using the invariance of φ under the slash operation of the first element on the right-hand side we see that the βh-part of φ|Tp equals χ (p) times pk−4 φ h(p)∗ λ,μ(p) k,m.

Plugging in the Fourier expansion of φ we get

Dj p