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Fourth power moment of coefficients of automorphic L-functions for GL(m)

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Published/Copyright: September 22, 2016
Forum Mathematicum
From the journal Forum Mathematicum Volume 29 Issue 5

Abstract

Let π be a unitary cuspidal automorphic representation for GLm⁢(𝔸ℚ), and let L⁢(s,π) be the automorphic L-function attached to π, which has a Dirichlet series expression in the half-plane ℜ⁡s>1, i.e.

L⁢(s,π)=∑n=1∞λπ⁢(n)ns.

In this paper we are interested in the upper bound of the fourth power moment of λπ⁢(n), i.e. ∑n≤xλπ⁢(n)4. If m=2, we are able to consider the sixteenth power moment of λπ⁢(n). As an application, we consider the lower bound of ∑n≤x|λπ⁢(n)|, which improves previous results.

Keywords: Maass cusp forms; automorphic representation.
MSC 2010: 11F30; 11F66; 11M41

Communicated by Freydoon Shahidi

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11531008

Funding source: Ministry of Education of the People’s Republic of China

Award Identifier / Grant number: IRT 1264

Funding statement: This work is supported in part by the key project of the National Natural Science Foundation of China (11531008) and IRT 1264 from the Ministry of Education of the People’s Republic of China.

Acknowledgements

The authors would like to thank the reviewer for his/her detailed comments and invaluable suggestions.

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Received: 2016-4-16
Revised: 2016-8-26
Published Online: 2016-9-22
Published in Print: 2017-9-1

© 2017 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Weyl asymptotics for Hanoi attractors
  3. Reconstruction of graded groupoids from graded Steinberg algebras
  4. Regularity of weak solutions to linear and quasilinear parabolic systems of non-divergence type with non-smooth in time principal matrix: A(t)-caloric method
  5. Mutations of simple-minded systems in Calabi–Yau categories generated by a spherical object
  6. Some homological properties of category 𝒪. IV
  7. Subgroups of direct products closely approximated by direct sums
  8. Takahasi semigroups
  9. Integration of controlled rough paths via fractional calculus
  10. Qualitative properties of positive solutions of quasilinear equations with Hardy terms
  11. Fourth power moment of coefficients of automorphic L-functions for GL(m)
  12. Reversible homogeneous Finsler metrics with positive flag curvature
https://doi.org/10.1515/forum-2016-0097
Keywords for this article
Maass cusp forms; automorphic representation.

Articles in the same Issue

  1. Frontmatter
  2. Weyl asymptotics for Hanoi attractors
  3. Reconstruction of graded groupoids from graded Steinberg algebras
  4. Regularity of weak solutions to linear and quasilinear parabolic systems of non-divergence type with non-smooth in time principal matrix: A(t)-caloric method
  5. Mutations of simple-minded systems in Calabi–Yau categories generated by a spherical object
  6. Some homological properties of category 𝒪. IV
  7. Subgroups of direct products closely approximated by direct sums
  8. Takahasi semigroups
  9. Integration of controlled rough paths via fractional calculus
  10. Qualitative properties of positive solutions of quasilinear equations with Hardy terms
  11. Fourth power moment of coefficients of automorphic L-functions for GL(m)
  12. Reversible homogeneous Finsler metrics with positive flag curvature
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