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On Hecke eigenvalues of Siegel modular forms in the Maass space

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Published/Copyright: October 7, 2017
Forum Mathematicum
From the journal Forum Mathematicum Volume 30 Issue 3

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 S⁢p2⁢(ℤ). In particular, we prove

λF⁢(n)=Ω⁢(nk-1⁢exp⁡(c⁢log⁡nlog⁡log⁡n)),

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

λF⁢(n)=Ω⁢(nk-1⁢(log⁡nlog⁡log⁡n))

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

λF⁢(n)≤nk-1⁢exp⁡(c1⁢log⁡nlog⁡log⁡n),

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∈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

Communicated by Jan Bruinier

Acknowledgements

The third author would like to thank the Institute of Mathematical Sciences for providing an excellent working atmosphere where the work was done. The authors would like to thank Purusottam Rath for asking for the possibility of infinitude of limit points of the sequence considered in Theorem 1.7 and going through an earlier version of the paper.

References

[1] A. N. Andrianov, Euler products that correspond to Siegel’s modular forms of genus 2, Russian Math. Surveys 29 (1974), no. 3, 45–116. 10.1070/RM1974v029n03ABEH001285Search in Google Scholar

[2] 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. 10.2977/PRIMS/31Search in Google Scholar

[3] S. Breulmann, On Hecke eigenforms in the Maass space, Math. Z. 232 (1999), no. 3, 527–530. 10.1007/PL00004769Search in Google Scholar

[4] S. Das and J. Sengupta, An Omega-result for Saito–Kurokawa lifts, Proc. Amer. Math. Soc. 142 (2014), no. 3, 761–764. 10.1090/S0002-9939-2013-11797-1Search in Google Scholar

[5] H. Iwaniec and E. Kowalski, Analytic Number Theory, American Mathematical Society, Providence, 2004. 10.1090/coll/053Search in Google Scholar

[6] T. Oda, On the poles of Andrianov L-functions, Math. Ann. 256 (1981), no. 3, 323–340. 10.1007/BF01679701Search in Google Scholar

[7] A. Pitale, Classical interpretation of the Ramanujan conjecture for Siegel cusp forms of genus n, Manuscripta Math. 130 (2009), no. 2, 225–231. 10.1007/s00229-009-0287-ySearch in Google Scholar

[8] 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. Search in Google Scholar

[9] M. Ram Murty, Oscillations of Fourier coefficients of modular forms, Math. Ann. 262 (1983), no. 4, 431–446. 10.1007/BF01456059Search in Google Scholar

[10] R. A. Rankin, An Ω-result for the coefficients of cusp forms, Math. Ann. 203 (1973), 239–250. 10.1007/BF01629259Search in Google Scholar

[11] G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Cambridge Stud. Adv. Math. 46, Cambridge University Press, Cambridge, 1995. Search in Google Scholar

[12] R. Weissauer, Endoscopy for GSp⁢(4) and the cohomology of Siegel modular threefolds, Lecture Notes in Math. 1968, Springer, Berlin, 2009. 10.1007/978-3-540-89306-6Search in Google Scholar

Received: 2017-4-24
Revised: 2017-9-13
Published Online: 2017-10-7
Published in Print: 2018-5-1

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. A generalized uniqueness theorem and the graded ideal structure of Steinberg algebras
  3. Positive solutions for nonlinear nonhomogeneous parametric Robin problems
  4. Period relations for cusp forms of GSp4
  5. A generalization of a graph theory Mertens’ theorem: Galois covering case
  6. Commutators of the fractional integrals for second-order elliptic operators on Morrey spaces
  7. Vojta’s conjecture on rational surfaces and the abc conjecture
  8. The least unramified prime which does not split completely
  9. The ω-inequality problem for concatenation hierarchies of star-free languages
  10. Higher weight on GL(3). I: The Eisenstein series
  11. Fourier transforms of powers of well-behaved 2D real analytic functions
  12. Parabolic conformally symplectic structures I; definition and distinguished connections
  13. The quasi-arithmetic means and Cartan barycenters of compactly supported measures
  14. Estimates of lattice points in the discriminant aspect over abelian extension fields
  15. On Hecke eigenvalues of Siegel modular forms in the Maass space
  16. Harmonicity and minimality of complex and quaternionic radial foliations
https://doi.org/10.1515/forum-2017-0092
Keywords for this article
Omega result; Hecke eigenvalues; Maass forms

Articles in the same Issue

  1. Frontmatter
  2. A generalized uniqueness theorem and the graded ideal structure of Steinberg algebras
  3. Positive solutions for nonlinear nonhomogeneous parametric Robin problems
  4. Period relations for cusp forms of GSp4
  5. A generalization of a graph theory Mertens’ theorem: Galois covering case
  6. Commutators of the fractional integrals for second-order elliptic operators on Morrey spaces
  7. Vojta’s conjecture on rational surfaces and the abc conjecture
  8. The least unramified prime which does not split completely
  9. The ω-inequality problem for concatenation hierarchies of star-free languages
  10. Higher weight on GL(3). I: The Eisenstein series
  11. Fourier transforms of powers of well-behaved 2D real analytic functions
  12. Parabolic conformally symplectic structures I; definition and distinguished connections
  13. The quasi-arithmetic means and Cartan barycenters of compactly supported measures
  14. Estimates of lattice points in the discriminant aspect over abelian extension fields
  15. On Hecke eigenvalues of Siegel modular forms in the Maass space
  16. Harmonicity and minimality of complex and quaternionic radial foliations
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