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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

# 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
• Email
• Other articles by this author:
/ Biplab Paul
• Institute of Mathematical Sciences, Homi Bhabha National Institute, C.I.T. Campus, Taramani, Chennai 600 113, India
• Email
• Other articles by this author:
/ 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 ${\lambda }_{F}\left(n\right)$ 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 $S{p}_{2}\left(ℤ\right)$. In particular, we prove

${\lambda }_{F}\left(n\right)=\mathrm{\Omega }\left({n}^{k-1}\mathrm{exp}\left(c\frac{\sqrt{\mathrm{log}n}}{\mathrm{log}\mathrm{log}n}\right)\right),$

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

${\lambda }_{F}\left(n\right)=\mathrm{\Omega }\left({n}^{k-1}\left(\frac{\sqrt{\mathrm{log}n}}{\mathrm{log}\mathrm{log}n}\right)\right)$

of Das and the third author. We also show that for any $n\ge 3$, one has

${\lambda }_{F}\left(n\right)\le {n}^{k-1}\mathrm{exp}\left({c}_{1}\sqrt{\frac{\mathrm{log}n}{\mathrm{log}\mathrm{log}n}}\right),$

where ${c}_{1}>0$ is an absolute constant. This improves an earlier result of Pitale and Schmidt. Further, we investigate the limit points of the sequence ${\left\{{\lambda }_{F}\left(n\right)/{n}^{k-1}\right\}}_{n\in ℕ}$ and show that it has infinitely many limit points. Finally, we show that ${\lambda }_{F}\left(n\right)>0$ 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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A. N. Andrianov, Euler products that correspond to Siegel’s modular forms of genus 2, Russian Math. Surveys 29 (1974), no. 3, 45–116.

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T. Barnet-Lamb, D. Geraghty, M. Harris and R. Taylor, A family of Calabi–Yau varieties and potential automorphy II, Publ. Res. Inst. Math. Sci. 47 (2011), no. 1, 29–98.

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

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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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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,

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