Nonvanishing modulo of Fourier coefficients of Jacobi forms

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 ϕ 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 there are infinitely many fundamental discriminants D=r 2−4m n<0 prime to m with ν (c(n,r))=0, where ν denotes a continuation of the -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.

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Acknowledgements

I would like to thank Jan H. Bruinier for suggesting the topic of this work to me and for many helpful discussions.

This work was partially supported by DFG grant BR-2163/4-1.

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Correspondence to Markus Schwagenscheidt.

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Schwagenscheidt, M. Nonvanishing modulo of Fourier coefficients of Jacobi forms. Res. number theory 2, 4 (2016). https://doi.org/10.1007/s40993-015-0035-1

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Keywords

  • Nonvanishing
  • Indivisibility
  • Fourier coefficients
  • Jacobi forms
  • Special values of L-functions