For a cocompact group of we fix a real non-zero harmonic -form . We study the asymptotics of the hyperbolic lattice-counting problem for under restrictions imposed by the modular symbols . We prove that the normalized values of the modular symbols, when ordered according to this counting, have a Gaussian distribution.
Soit un sous-groupe de cocompact et soit une forme harmonique réelle (non nulle). Nous étudions le comportement asymptotique de la fonction comptant des points du réseau hyperbolique sous hypothèses imposées par des symboles modulaires . Nous montrons que les valeurs normalisées des symboles modulaires, ordonnées selon ce comptage possèdent une répartition gaussienne.
@article{JTNB_2009__21_3_721_0, author = {Petridis, Yiannis N. and Risager, Morten S.}, title = {Hyperbolic lattice-point counting and modular symbols}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {721--734}, publisher = {Universit\'e Bordeaux 1}, volume = {21}, number = {3}, year = {2009}, doi = {10.5802/jtnb.698}, zbl = {1214.11065}, mrnumber = {2605543}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/jtnb.698/} }
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 SP - 721 EP - 734 VL - 21 IS - 3 PB - Université Bordeaux 1 UR - http://archive.numdam.org/articles/10.5802/jtnb.698/ DO - 10.5802/jtnb.698 LA - en ID - JTNB_2009__21_3_721_0 ER -
%0 Journal Article %A Petridis, Yiannis N. %A Risager, Morten S. %T Hyperbolic lattice-point counting and modular symbols %J Journal de théorie des nombres de Bordeaux %D 2009 %P 721-734 %V 21 %N 3 %I Université Bordeaux 1 %U http://archive.numdam.org/articles/10.5802/jtnb.698/ %R 10.5802/jtnb.698 %G en %F JTNB_2009__21_3_721_0
Petridis, Yiannis N.; Risager, Morten S. Hyperbolic lattice-point counting and modular symbols. Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 3, pp. 721-734. doi : 10.5802/jtnb.698. http://archive.numdam.org/articles/10.5802/jtnb.698/
[1] J. Bourgain, A. Gamburd, P. Sarnak, Sieving and expanders. C. R. Math. Acad. Sci. Paris 343 (2006), no. 3, 155–159. | MR
[2] F. Chamizo, Some applications of large sieve in Riemann surfaces. Acta Arith. 77 (1996), no. 4, 315–337. | MR | Zbl
[3] J. Delsarte, Sur le gitter fuchsien. C. R. Acad. Sci. Paris 214 (1942), 147–179. | MR
[4] D. Goldfeld, Zeta functions formed with modular symbols. Automorphic forms, automorphic representations, and arithmetic (Fort Worth, TX, 1996), 111–121, Proc. Sympos. Pure Math., 66, Part 1, Ager. Math. Soc., Providence, RI, 1999. | MR | Zbl
[5] D. Goldfeld, The distribution of modular symbols. Number theory in progress, Vol. 2 (Zakopane-Kościelisko, 1997), 849–865, de Gruyter, Berlin, 1999. | MR | Zbl
[6] A. Good, Local analysis of Selberg’s trace formula. Lecture Notes in Mathematics, 1040. Springer-Verlag, Berlin, 1983. i+128 pp. | MR | Zbl
[7] D. Goldfeld, C. O’Sullivan, Estimating Additive Character Sums for Fuchsian Groups. The Ramanujan Journal 7 (1) (2003), 241–267. | MR | Zbl
[8] H. Huber, Zur analytischen Theorie hyperbolischen Raumformen und Bewegungsgruppen I. Math. Ann. 138 (1959), 1–26; II. Math. Ann. 142 (1960/1961), 385–398; Nachtrag zu II, Math. Ann. 143 (1961), 463—464. | MR | Zbl
[9] H. Iwaniec, Spectral methods of automorphic forms. Second edition. Graduate Studies in Mathematics, 53. American Mathematical Society, Providence, RI; Revista Matemática Iberoamericana, Madrid, 2002. xii+220 pp. | MR | Zbl
[10] A. Kontorovich, Ph.D thesis, Columbia University, 2007.
[11] P. Lax, R. S. Phillips, The asymptotic distribution of lattice points in Euclidean and non-Euclidean space. J. Funct. Anal. 46 (1982) 280–350. | MR | Zbl
[12] M. Loève, Probability theory I. Fourth edition, Springer, New York, 1977. | MR | Zbl
[13] S. J. Patterson, A lattice-point problem in hyperbolic space. Mathematika 22 (1975), no. 1, 81–88. | MR | Zbl
[14] Y. N. Petridis, M. S. Risager, Modular symbols have a normal distribution. Geom. Funct. Anal. 14 (2004), no. 5, 1013–1043. | MR | Zbl
[15] Y. N. Petridis, M. S. Risager, The distribution of values of the Poincaré pairing for hyperbolic Riemann surfaces. J. Reine Angew. Math., 579 (2005), 159–173. | MR | Zbl
[16] R. Phillips, Z. Rudnick, The circle problem in the hyperbolic plane. J. Funct. Anal. 121 (1994), no. 1, 78–116. | MR | Zbl
[17] R. Phillips, P. Sarnak, The spectrum of Fermat curves. Geom. Funct. Anal. 1 (1991), no. 1, 80–146. | MR | Zbl
[18] M. S. Risager, Distribution of modular symbols for compact surfaces. Internat. Math. Res. Notices. 2004, no. 41, 2125–2146. | MR | Zbl
Cited by Sources: