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Nonvanishing modulo of Fourier coefficients of Jacobi forms

  • Markus Schwagenscheidt
Open Access
Research
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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.

Keywords

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

Notes

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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© The Author(s) 2016

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Fachbereich Mathematik, Technische Universität DarmstadtDarmstadtGermany

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