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Upper bounds for geodesic periods over rank one locally symmetric spaces

  • Jan Frahm ORCID logo EMAIL logo and Feng Su ORCID logo
Published/Copyright: January 13, 2018
Forum Mathematicum
From the journal Forum Mathematicum Volume 30 Issue 5

Abstract

We prove upper bounds for geodesic periods of automorphic forms over general rank one locally symmetric spaces. Such periods are integrals of automorphic forms restricted to special totally geodesic cycles of the ambient manifold and twisted with automorphic forms on the cycles. The upper bounds are in terms of the Laplace eigenvalues of the two automorphic forms, and they generalize previous results for real hyperbolic manifolds to the context of all rank one locally symmetric spaces.

Keywords: Locally symmetric spaces; automorphic forms; geodesic periods; totally geodesic cycles; unitary representations; reductive Lie groups
MSC 2010: 11F70; 22E46; 53C35

Communicated by Jan Bruinier

A Integral formulas

For α,β>0 and 0<Re⁡λ<2⁢Re⁡(μ+ν), we have (see [11, equation 3.259 (3)])

(A.1)∫0∞xλ-1⁢(1+α⁢x2)-μ⁢(1+β⁢x2)-ν⁢𝑑x=12⁢α-λ2⁢B⁢(λ2,μ+ν-λ2)⁢F12⁢(ν,λ2;μ+ν;1-βα).

The Euler transformation formula holds (see [1, (2.2.7)]):

(A.2)F12⁢(α,β;γ;x)=(1-x)γ-α-β⁢F12⁢(γ-α,γ-β;γ;x).

The Euler integral representation holds for Re⁡(γ-β),Re⁡β>0 (see [1, (2.3.17)]):

(A.3)F12⁢(α,β;γ;1-x)=Γ⁢(γ)Γ⁢(γ-β)⁢Γ⁢(β)⁢∫0∞tβ-1⁢(1+t)α-γ⁢(1+x⁢t)-α⁢𝑑t.

The following integral formula holds for 0≤β<1 and 0<Re⁡μ<Re⁡(λ-2⁢ν) (see [11, equation 3.254 (2)] for λ=0 and u=1):

(A.4)∫1∞(x-1)μ-1⁢(x2+β)ν⁢𝑑x=Γ⁢(μ)⁢Γ⁢(-μ-2⁢ν)Γ⁢(-2⁢ν)⁢F12⁢(-μ2-ν,1-μ2-ν;12-ν;-β).

By using Pfaff’s transformation formula (see [1, (2.2.6)])

F12⁢(α,β;γ;x)=(1-x)-α⁢F12⁢(α,γ-β;γ;xx-1)

and the integral formula

∫01xρ-1⁢(1-x)σ-1⁢F12⁢(α,β;γ;x)⁢𝑑x=Γ⁢(ρ)⁢Γ⁢(σ)Γ⁢(ρ+σ)⁢F23⁢(α,β,ρ;γ,ρ+σ;1)

which holds for Re⁡ρ,Re⁡σ,Re⁡(γ+σ-α-β)>0 (see [11, equation 7.512 (5)]), it is easy to see by a simple substitution that for Re⁡ρ,Re⁡(α-σ-ρ+1),Re⁡(β-σ-ρ+1)>0 we have

(A.5)∫0∞xρ-1⁢(1+x)σ-1⁢F12⁢(α,β;γ;-x)⁢𝑑x=Γ⁢(ρ)⁢Γ⁢(α-σ-ρ+1)Γ⁢(α-σ+1)⁢F23⁢(α,γ-β,ρ;γ,α-σ+1;1).

For Re⁡(γ-α-β)>0, we have the Gauß special value (see [1, Theorem 2.2.2])

(A.6)F12⁢(α,β;γ;1)=Γ⁢(γ)⁢Γ⁢(γ-α-β)Γ⁢(γ-α)⁢Γ⁢(γ-β).

For Re⁡μ,Re⁡ν>0, we have (see [11, equation 3.621 (5)])

(A.7)∫0π2sinμ-1⁡θ⁢cosν-1⁡θ⁢d⁢θ=12⁢B⁢(μ2,ν2).

Acknowledgements

The authors thank the referee for many helpful comments and remarks that helped to improve the exposition.

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Received: 2017-09-04
Revised: 2017-11-29
Published Online: 2018-01-13
Published in Print: 2018-09-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Upper bounds for geodesic periods over rank one locally symmetric spaces
  3. Additive bases with coefficients of newforms
  4. A note on split extensions of bialgebras
  5. Petersson norm of cusp forms associated to real quadratic fields
  6. A realization theorem for sets of lengths in numerical monoids
  7. On the non-existence of the Mackey topology for locally quasi-convex groups
  8. The Davies method revisited for heat kernel upper bounds of regular Dirichlet forms on metric measure spaces
  9. The cup product of Brooks quasimorphisms
  10. Heat kernel estimates for time fractional equations
  11. Extreme non-Arens regularity of the group algebra
  12. Rational homology and homotopy of high-dimensional string links
  13. Global integrability for solutions to some anisotropic problem with nonstandard growth
  14. Hardy operators on Musielak–Orlicz spaces
  15. Contractibility of the stability manifold for silting-discrete algebras
  16. Order of the canonical vector bundle over configuration spaces of spheres
  17. An endpoint version of uniform Sobolev inequalities
  18. Space-time L2 estimates, regularity and almost global existence for elastic waves
  19. k-spaces and duals of non-archimedean metrizable locally convex spaces
  20. Path homology theory of multigraphs and quivers
https://doi.org/10.1515/forum-2017-0185
Keywords for this article
Locally symmetric spaces; automorphic forms; geodesic periods; totally geodesic cycles; unitary representations; reductive Lie groups

Articles in the same Issue

  1. Frontmatter
  2. Upper bounds for geodesic periods over rank one locally symmetric spaces
  3. Additive bases with coefficients of newforms
  4. A note on split extensions of bialgebras
  5. Petersson norm of cusp forms associated to real quadratic fields
  6. A realization theorem for sets of lengths in numerical monoids
  7. On the non-existence of the Mackey topology for locally quasi-convex groups
  8. The Davies method revisited for heat kernel upper bounds of regular Dirichlet forms on metric measure spaces
  9. The cup product of Brooks quasimorphisms
  10. Heat kernel estimates for time fractional equations
  11. Extreme non-Arens regularity of the group algebra
  12. Rational homology and homotopy of high-dimensional string links
  13. Global integrability for solutions to some anisotropic problem with nonstandard growth
  14. Hardy operators on Musielak–Orlicz spaces
  15. Contractibility of the stability manifold for silting-discrete algebras
  16. Order of the canonical vector bundle over configuration spaces of spheres
  17. An endpoint version of uniform Sobolev inequalities
  18. Space-time L2 estimates, regularity and almost global existence for elastic waves
  19. k-spaces and duals of non-archimedean metrizable locally convex spaces
  20. Path homology theory of multigraphs and quivers
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