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On the non-vanishing of L-functions associated to cusp forms of half-integral weight

The Ramanujan Journal volume 51, pages 455–477 (2020)Cite this article

Abstract

We prove a strengthening of Muić’s integral non-vanishing criterion for Poincaré series on unimodular locally compact Hausdorff groups and use it to prove a result on non-vanishing of L-functions associated to cusp forms of half-integral weight.

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References

  1. Chen, J., Rubin, H.: Bounds for the difference between median and mean of gamma and Poisson distributions. Stat. Probab. Lett. 4(6), 281–283 (1986)

    MathSciNet  Article  Google Scholar 

  2. Knopp, M., Robins, S.: Easy proofs of Riemann’s functional equation for \(\zeta (s)\) and of Lipschitz summation. Proc. Am. Math. Soc. 129(7), 1915–1922 (2001)

    MathSciNet  Article  Google Scholar 

  3. Kohnen, W.: Nonvanishing of Hecke L-functions associated to cusp forms inside the critical strip. J. Number Theory 67(2), 182–189 (1997)

    MathSciNet  Article  Google Scholar 

  4. Kohnen, W., Raji, W.: Non-vanishing of L-functions associated to cusp forms of half-integral weight in the plus space. Res. Number Theory (2017). https://doi.org/10.1007/s40993-017-0072-z

  5. Miyake, T.: Modular Forms. Springer Monographs in Mathematics. Springer, Berlin (2006)

    MATH  Google Scholar 

  6. Muić, G.: On a construction of certain classes of cuspidal automorphic forms via Poincaré series. Math. Ann. 343(1), 207–227 (2009)

    MathSciNet  Article  Google Scholar 

  7. Muić, G.: On the cuspidal modular forms for the Fuchsian groups of the first kind. J. Number Theory 130(7), 1488–1511 (2010)

    MathSciNet  Article  Google Scholar 

  8. Muić, G.: On the non-vanishing of certain modular forms. Int. J. Number Theory 7(2), 351–370 (2011)

    MathSciNet  Article  Google Scholar 

  9. Muić, G.: On the analytic continuation and non-vanishing of L-functions. Int. J. Number Theory 8(8), 1831–1854 (2012)

    MathSciNet  Article  Google Scholar 

  10. Niculescu, C., Persson, L.E.: Convex Functions and Their Applications: A Contemporary Approach. CMS Books in Mathematics. Springer, New York (2006)

    Book  Google Scholar 

  11. Raghuram, A.: Non-vanishing of L-functions of cusp forms inside the critical strip. In: Number theory, Ramanujan Math. Soc. Lect. Notes Ser. No. 1, Ramanujan Math. Soc., Mysore, pp. 97–105 (2005)

  12. Rainville, E.D.: Special Functions. Macmillan, New York (1960)

    MATH  Google Scholar 

  13. Ramakrishnan, B., Shankhadhar, K.D.: Nonvanishing of L-functions associated to cusp forms of half-integral weight. In: Automorphic Forms. Springer Proceedings in Mathematics & Statistics, vol. 115, pp. 223–231 (2014)

  14. Shimura, G.: On modular forms of half integral weight. Ann. Math. (2) 97(3), 440–481 (1973)

    MathSciNet  Article  Google Scholar 

  15. Zhi, Y.: Strict monotonicity of nonnegative strictly concave function vanishing at the origin. Teach. Math. 19(2), 68–75 (2016)

    Google Scholar 

  16. Žunar, S.: On the non-vanishing of Poincaré series on the metaplectic group. Manuscripta Math. 158(1–2), 1–19 (2019)

    MathSciNet  Article  Google Scholar 

  17. Žunar, S.: On Poincaré series of half-integral weight. Glas. Mat. 53(2), 239–264 (2018)

    MathSciNet  Article  Google Scholar 

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Acknowledgements

This paper grew out of my PhD thesis. I would like to thank my advisor, Goran Muić, for his encouragement, support, and many discussions. I would also like to thank Petar Bakić for some useful comments.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička 30, 10000, Zagreb, Croatia

    Sonja Žunar

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  1. Sonja Žunar
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Correspondence to Sonja Žunar.

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The author acknowledges support from the Croatian Science Foundation Grant No. 9364.

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Žunar, S. On the non-vanishing of L-functions associated to cusp forms of half-integral weight. Ramanujan J 51, 455–477 (2020). https://doi.org/10.1007/s11139-019-00135-2

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  • DOI: https://doi.org/10.1007/s11139-019-00135-2

Keywords

  • L-functions
  • Cusp forms of half-integral weight
  • Non-vanishing of Poincaré series
  • Metaplectic cover of \(\mathrm {SL}_2({\mathbb {R}})\)

Mathematics Subject Classification

  • 11F37