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Forum Mathematicum

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Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna


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Volume 28, Issue 5

Issues

The hyperbolic lattice point problem in conjugacy classes

Dimitrios Chatzakos
  • Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom of Great Britain and Northern Ireland
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/ Yiannis N. Petridis
  • Corresponding author
  • Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom of Great Britain and Northern Ireland
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Published Online: 2016-01-10 | DOI: https://doi.org/10.1515/forum-2015-0102

Abstract

For Γ a cocompact or cofinite Fuchsian group, we study the hyperbolic lattice point problem in conjugacy classes, which is a modification of the classical hyperbolic lattice point problem. We use large sieve inequalities for the Riemann surfaces Γ\ to obtain average results for the error term, which are conjecturally optimal. We give a new proof of the error bound O(X2/3), due to Good. For SL2() we interpret our results in terms of indefinite quadratic forms.

Keywords: Lattice points; hyperbolic space; geodesic arcs

MSC 2010: 11F72; 37C35; 37D40

References

  • [1]

    Bruin P., Explicit bounds on automorphic and canonical Green functions of Fuchsian groups, Mathematika 60 (2014), no. 2, 257–306. Google Scholar

  • [2]

    Chamizo F., Topics in analytic number theory, Ph.D. thesis, Universdad Autónoma de Madrid, 1994. Google Scholar

  • [3]

    Chamizo F., Some applications of the large sieve in Riemann surfaces, Acta Arith. 77 (1996), no. 4, 315–337. Google Scholar

  • [4]

    Chamizo F., The large sieve in Riemann surfaces, Acta Arith. 77 (1996), no. 4, 303–313. Google Scholar

  • [5]

    Duke W., Rudnick Z. and Sarnak P., Density of integer points on affine homogeneous varieties, Duke Math. J. 71 (1993), no. 1, 143–179. Google Scholar

  • [6]

    Erdélyi A., Magnus W., Oberhettinger F. and Tricomi F. G., Higher Transcendental Functions, Volume I, McGraw–Hill, New York, 1953. Google Scholar

  • [7]

    Eskin A. and McMullen C., Mixing, counting, and equidistribution in Lie groups, Duke Math. J. 71 (1993), no. 1, 181–209. Google Scholar

  • [8]

    Fay J. D., Fourier coefficients of the resolvent for a Fuchsian group, J. Reine Angew. Math. 293/294 (1997), 143–203. Google Scholar

  • [9]

    Garbin D., Jorgenson J. and Munn M., On the appearance of Eisenstein series through degeneration, Comment. Math. Helv. 83 (2008), no. 4, 701–721. Google Scholar

  • [10]

    Good A., Local Analysis of Selberg’s Trace Formula, Lecture Notes in Math. 1040, Springer, Berlin, 1983. Google Scholar

  • [11]

    Gradshteyn I. S. and Ryzhik I. M., Table of Integrals, Series, and Products, 7th ed., Academic Press, Amsterdam, 2007. Google Scholar

  • [12]

    Günther P., Gitterpunktprobleme in symmetrischen Riemannschen Räumen vom Rang 1, Math. Nachr. 94 (1980), 5–27. Google Scholar

  • [13]

    Hill R. and Parnovski L., The variance of the hyperbolic lattice point counting function, Russ. J. Math. Phys. 12 (2005), no. 4, 472–482. Google Scholar

  • [14]

    Huber H., Über eine neue Klasse automorpher Funktionen und ein Gitterpunktproblem in der hyperbolischen Ebene. I, Comment. Math. Helv. 30 (1956), 20–62. Google Scholar

  • [15]

    Huber H., Ein Gitterpunktproblem in der hyperbolischen Ebene, J. Reine Angew. Math. 496 (1998), 15–53. Google Scholar

  • [16]

    Iwaniec H., Spectral Methods of Automorphic Forms, 2nd ed., Grad. Stud. Math. 53, American Mathematical Society, Providence, 2002. Google Scholar

  • [17]

    Martin K., McKee M. and Wambach E., A relative trace formula for a compact Riemann surface, Int. J. Number Theory 7 (2011), no. 2, 389–429. Google Scholar

  • [18]

    Parkkonen J. and Paulin F., On the hyperbolic orbital counting problem in conjugacy classes, Math. Z. 279 (2015), no. 3–4, 1175–1196. Google Scholar

  • [19]

    Patterson S. J., A lattice-point problem in hyperbolic space, Mathematika 22 (1975), no. 1, 81–88. Google Scholar

  • [20]

    Phillips R. and Rudnick Z., The circle problem in the hyperbolic plane, J. Funct. Anal. 121 (1994), no. 1, 78–116. Google Scholar

  • [21]

    Sarnak P., Class numbers of indefinite binary quadratic forms, J. Number Theory 15 (1982), no. 2, 229–247. Google Scholar

  • [22]

    Selberg A., Equidistribution in discrete groups and the spectral theory of automorphic forms, http://publications.ias.edu/selberg/section/2491.

  • [23]

    Tsuzuki M., Spectral square means for period integrals of wave functions on real hyperbolic spaces, J. Number Theory 129 (2009), no. 10, 2387–2438. Google Scholar

About the article


Received: 2015-05-28

Revised: 2015-11-04

Published Online: 2016-01-10

Published in Print: 2016-09-01


The first author was supported by a DTA from EPSRC during his PhD studies at UCL.


Citation Information: Forum Mathematicum, Volume 28, Issue 5, Pages 981–1003, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2015-0102.

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