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Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna


IMPACT FACTOR 2018: 0.867

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Volume 30, Issue 3

Issues

On Hecke eigenvalues of Siegel modular forms in the Maass space

Sanoli Gun
  • Corresponding author
  • Institute of Mathematical Sciences, Homi Bhabha National Institute, C.I.T. Campus, Taramani, Chennai 600 113, India
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/ Biplab Paul
  • Institute of Mathematical Sciences, Homi Bhabha National Institute, C.I.T. Campus, Taramani, Chennai 600 113, India
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/ Jyoti Sengupta
Published Online: 2017-10-07 | DOI: https://doi.org/10.1515/forum-2017-0092

Abstract

In this article, we prove an Omega result for the Hecke eigenvalues λF(n) of Maass forms F which are Hecke eigenforms in the space of Siegel modular forms of weight k, genus two for the Siegel modular group Sp2(). In particular, we prove

λF(n)=Ω(nk-1exp(clognloglogn)),

when c>0 is an absolute constant. This improves the earlier result

λF(n)=Ω(nk-1(lognloglogn))

of Das and the third author. We also show that for any n3, one has

λF(n)nk-1exp(c1lognloglogn),

where c1>0 is an absolute constant. This improves an earlier result of Pitale and Schmidt. Further, we investigate the limit points of the sequence {λF(n)/nk-1}n and show that it has infinitely many limit points. Finally, we show that λF(n)>0 for all n, a result proved earlier by Breulmann by a different technique.

Keywords: Omega result; Hecke eigenvalues; Maass forms

MSC 2010: 11F46; 11F30

References

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    S. Breulmann, On Hecke eigenforms in the Maass space, Math. Z. 232 (1999), no. 3, 527–530. CrossrefGoogle Scholar

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    S. Das and J. Sengupta, An Omega-result for Saito–Kurokawa lifts, Proc. Amer. Math. Soc. 142 (2014), no. 3, 761–764. Google Scholar

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    A. Pitale and R. Schmidt, Ramanujan-type results for Siegel cusp forms of degree 2, J. Ramanujan Math. Soc. 24 (2009), no. 1, 87–111. Google Scholar

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About the article


Received: 2017-04-24

Revised: 2017-09-13

Published Online: 2017-10-07

Published in Print: 2018-05-01


Citation Information: Forum Mathematicum, Volume 30, Issue 3, Pages 775–783, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2017-0092.

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