Hyperbolic lattice-point counting and modular symbols

Yiannis N. Petridis[1]; Morten S. Risager[2]

  • [1] Department of Mathematics University College London Gower Street London WC1E 6BT The Graduate Center Mathematics Ph.D. Program 365 Fifth Avenue Room 4208 New York, NY 10016-4309
  • [2] Department of Mathematical Sciences University of Aarhus Ny Munkegade Building 530 8000 Aarhus, Denmark Department of Mathematical Sciences University of Copenhagen Universitetsparken 5 2100 Copenhagen Ø, Denmark

Journal de Théorie des Nombres de Bordeaux (2009)

  • Volume: 21, Issue: 3, page 721-734
  • ISSN: 1246-7405

Abstract

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For a cocompact group Γ of SL 2 ( ℝ ) we fix a real non-zero harmonic 1 -form α . We study the asymptotics of the hyperbolic lattice-counting problem for Γ under restrictions imposed by the modular symbols γ , α . We prove that the normalized values of the modular symbols, when ordered according to this counting, have a Gaussian distribution.

How to cite

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Petridis, Yiannis N., and Risager, Morten S.. "Hyperbolic lattice-point counting and modular symbols." Journal de Théorie des Nombres de Bordeaux 21.3 (2009): 721-734. <http://eudml.org/doc/10908>.

@article{Petridis2009,
abstract = {For a cocompact group $\{\Gamma \}$ of $\{\hbox\{SL\}_2( \{\mathbb\{R\}\})\} $ we fix a real non-zero harmonic $1$-form $\alpha $. We study the asymptotics of the hyperbolic lattice-counting problem for $\{\Gamma \}$ under restrictions imposed by the modular symbols $\left\langle \gamma ,\{\alpha \} \right\rangle $. We prove that the normalized values of the modular symbols, when ordered according to this counting, have a Gaussian distribution.},
affiliation = {Department of Mathematics University College London Gower Street London WC1E 6BT The Graduate Center Mathematics Ph.D. Program 365 Fifth Avenue Room 4208 New York, NY 10016-4309; Department of Mathematical Sciences University of Aarhus Ny Munkegade Building 530 8000 Aarhus, Denmark Department of Mathematical Sciences University of Copenhagen Universitetsparken 5 2100 Copenhagen Ø, Denmark},
author = {Petridis, Yiannis N., Risager, Morten S.},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {lattice points; modular symbols; trace formula; hyperbolic surface; Gaussian distribution},
language = {eng},
number = {3},
pages = {721-734},
publisher = {Université Bordeaux 1},
title = {Hyperbolic lattice-point counting and modular symbols},
url = {http://eudml.org/doc/10908},
volume = {21},
year = {2009},
}

TY - JOUR
AU - Petridis, Yiannis N.
AU - Risager, Morten S.
TI - Hyperbolic lattice-point counting and modular symbols
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2009
PB - Université Bordeaux 1
VL - 21
IS - 3
SP - 721
EP - 734
AB - For a cocompact group ${\Gamma }$ of ${\hbox{SL}_2( {\mathbb{R}})} $ we fix a real non-zero harmonic $1$-form $\alpha $. We study the asymptotics of the hyperbolic lattice-counting problem for ${\Gamma }$ under restrictions imposed by the modular symbols $\left\langle \gamma ,{\alpha } \right\rangle $. We prove that the normalized values of the modular symbols, when ordered according to this counting, have a Gaussian distribution.
LA - eng
KW - lattice points; modular symbols; trace formula; hyperbolic surface; Gaussian distribution
UR - http://eudml.org/doc/10908
ER -

References

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