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On the geometric trace of a generalized Selberg trace formula

  • András Biró and Dávid Tóth EMAIL logo
Published/Copyright: April 24, 2024
Forum Mathematicum
From the journal Forum Mathematicum Volume 37 Issue 2

Abstract

A certain generalization of the Selberg trace formula was proved by the first named author in 1999. In this generalization instead of considering the integral of K ⁢ ( z , z ) (where K ⁢ ( z , w ) is an automorphic kernel function) over the fundamental domain, one considers the integral of K ⁢ ( z , z ) ⁢ u ⁢ ( z ) , where u ⁢ ( z ) is a fixed automorphic eigenfunction of the Laplace operator. This formula was proved for discrete subgroups of PSL ⁢ ( 2 , ℝ ) , and just as in the case of the classical Selberg trace formula it was obtained by evaluating in two different ways (“geometrically” and “spectrally”) the integral of K ⁢ ( z , z ) ⁢ u ⁢ ( z ) . In the present paper we work out the geometric side of a further generalization of this generalized trace formula: we consider the case of discrete subgroups of PSL ⁢ ( 2 , ℝ ) n where n > 1 . Many new difficulties arise in the case of these groups due to the fact that the classification of conjugacy classes is much more complicated for n > 1 than in the case n = 1 .

Keywords: Selberg trace formula
MSC 2020: 11F72

Communicated by Jan Frahm

Funding statement: The research reported in this paper is supported by the “TKP2020, National Challenges Program” of the National Research, Development and Innovation Office (BME NC TKP2020) and by the Ministry of Innovation and Technology and the National Research, Development and Innovation Office within the Artificial Intelligence National Laboratory of Hungary. It is also supported by the MTA–RI Lendület “Momentum” Analytic Number Theory and Representation Theory Research Group, by the NKFIH (National Research, Development and Innovation Office) grants FK 135218 (Dávid Tóth), K135885 and K 143876 (András Biró) and by the Rényi Intézet Lendület Automorphic Research Group.

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Received: 2023-09-25
Revised: 2024-03-12
Published Online: 2024-04-24
Published in Print: 2025-02-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Triangles with one fixed side–length, a Furstenberg-type problem, and incidences in finite vector spaces
  3. Existence and multiplicity of solutions for fractional Schrödinger-p-Kirchhoff equations in ℝ N
  4. Estimates of Picard modular cusp forms
  5. Some q-supercongruences from a q-analogue of Watson's 3 F 2 summation
  6. Existence of strong solutions for one-dimensional reflected mixed stochastic delay differential equations
  7. Free groups generated by two unipotent maps
  8. Degenerate Schrödinger--Kirchhoff {(p,N)}-Laplacian problem with singular Trudinger--Moser nonlinearity in ℝ N
  9. Colored multizeta values in positive characteristic
  10. Simultaneous nonvanishing of central L-values with large level
  11. Laplace convolutions of weighted averages of arithmetical functions
  12. Cohomological properties of maximal pro-p Galois groups that are preserved under profinite completion
  13. On the geometric trace of a generalized Selberg trace formula
  14. Elementary properties of free lattices
  15. Weighted estimates for product singular integral operators in Journé’s class on RD-spaces
  16. Small generators of abelian number fields
  17. Pointwise convergence and nonlinear smoothing of the generalized Zakharov–Kuznetsov equation
  18. Weighted bilinear multiplier theorems in Dunkl setting via singular integrals
https://doi.org/10.1515/forum-2023-0344
Keywords for this article
Selberg trace formula

Articles in the same Issue

  1. Frontmatter
  2. Triangles with one fixed side–length, a Furstenberg-type problem, and incidences in finite vector spaces
  3. Existence and multiplicity of solutions for fractional Schrödinger-p-Kirchhoff equations in ℝ N
  4. Estimates of Picard modular cusp forms
  5. Some q-supercongruences from a q-analogue of Watson's 3 F 2 summation
  6. Existence of strong solutions for one-dimensional reflected mixed stochastic delay differential equations
  7. Free groups generated by two unipotent maps
  8. Degenerate Schrödinger--Kirchhoff {(p,N)}-Laplacian problem with singular Trudinger--Moser nonlinearity in ℝ N
  9. Colored multizeta values in positive characteristic
  10. Simultaneous nonvanishing of central L-values with large level
  11. Laplace convolutions of weighted averages of arithmetical functions
  12. Cohomological properties of maximal pro-p Galois groups that are preserved under profinite completion
  13. On the geometric trace of a generalized Selberg trace formula
  14. Elementary properties of free lattices
  15. Weighted estimates for product singular integral operators in Journé’s class on RD-spaces
  16. Small generators of abelian number fields
  17. Pointwise convergence and nonlinear smoothing of the generalized Zakharov–Kuznetsov equation
  18. Weighted bilinear multiplier theorems in Dunkl setting via singular integrals
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