Most recent issue | All issues (1900–Present) | Articles in press | Recently published articles
Prime geodesic theorem in the 3-dimensional hyperbolic space
Authors: Olga Balkanova, Dimitrios Chatzakos, Giacomo Cherubini, Dmitry Frolenkov and Niko Laaksonen
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 11F72; Secondary 11M36, 11L05
DOI: https://doi.org/10.1090/tran/7720
Published electronically: December 28, 2018
Abstract: For a cofinite Kleinian group acting on , we study the prime geodesic theorem on , which asks about the asymptotic behavior of lengths of primitive closed geodesics (prime geodesics) on . Let be the error in the counting of prime geodesics with length at most . For the Picard manifold, , we improve the classical bound of Sarnak, , to . In the process we obtain a mean subconvexity estimate for the Rankin-Selberg -function attached to Maass-Hecke cusp forms. We also investigate the second moment of for a general cofinite group , and we show that it is bounded by .
[1] O. Balkanova and D. Frolenkov, Prime geodesic theorem for the Picard manifold, https://arxiv.org/abs/1804.00275 (2018).
[2] Y. Bonthonneau, Weyl laws for manifolds with hyperbolic cusps, https://arxiv.org/abs/1512.05794 (2015).
Yingchun Cai, Prime geodesic theorem, J. Théor. Nombres Bordeaux 14 (2002), no. 1, 59–72 (English, with English and French summaries). MR 1925990
Giacomo Cherubini and João Guerreiro, Mean square in the prime geodesic theorem, Algebra Number Theory 12 (2018), no. 3, 571–597. MR 3815307, https://doi.org/10.2140/ant.2018.12.571
J. B. Conrey and H. Iwaniec, The cubic moment of central values of automorphic 퐿-functions, Ann. of Math. (2) 151 (2000), no. 3, 1175–1216. MR 1779567, https://doi.org/10.2307/121132
J. Elstrodt, F. Grunewald, and J. Mennicke, Groups acting on hyperbolic space, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. Harmonic analysis and number theory. MR 1483315
I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 7th ed., Elsevier/Academic Press, Amsterdam, 2007. Translated from the Russian; Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger; With one CD-ROM (Windows, Macintosh and UNIX). MR 2360010
Jeffrey Hoffstein and Paul Lockhart, Coefficients of Maass forms and the Siegel zero, Ann. of Math. (2) 140 (1994), no. 1, 161–181. With an appendix by Dorian Goldfeld, Hoffstein and Daniel Lieman. MR 1289494, https://doi.org/10.2307/2118543
Heinz Huber, Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen. II, Math. Ann. 142 (1960/1961), 385–398 (German). MR 0126549, https://doi.org/10.1007/BF01451031
Heinz Huber, Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen. II, Math. Ann. 143 (1961), 463—464 (German). MR 0154980, https://doi.org/10.1007/BF01470758
Henryk Iwaniec, Fourier coefficients of cusp forms and the Riemann zeta-function, Seminar on Number Theory, 1979–1980 (French), Univ. Bordeaux I, Talence, 1980, pp. Exp. No. 18, 36. MR 604215
Henryk Iwaniec, Mean values for Fourier coefficients of cusp forms and sums of Kloosterman sums, Number theory days, 1980 (Exeter, 1980)	London Math. Soc. Lecture Note Ser., vol. 56, Cambridge Univ. Press, Cambridge-New York, 1982, pp. 306–321. MR 697274
Henryk Iwaniec, Prime geodesic theorem, J. Reine Angew. Math. 349 (1984), 136–159. MR 743969, https://doi.org/10.1515/crll.1984.349.136
Henryk Iwaniec, Spectral methods of automorphic forms, 2nd ed., Graduate Studies in Mathematics, vol. 53, American Mathematical Society, Providence, RI; Revista Matemática Iberoamericana, Madrid, 2002. MR 1942691
Shin-ya Koyama, 퐿^{∞}-norms on eigenfunctions for arithmetic hyperbolic 3-manifolds, Duke Math. J. 77 (1995), no. 3, 799–817. MR 1324641, https://doi.org/10.1215/S0012-7094-95-07724-2
Shin-ya Koyama, Prime geodesic theorem for arithmetic compact surfaces, Internat. Math. Res. Notices 8 (1998), 383–388. MR 1628243, https://doi.org/10.1155/S1073792898000257
Shin-ya Koyama, Prime geodesic theorem for the Picard manifold under the mean-Lindelöf hypothesis, Forum Math. 13 (2001), no. 6, 781–793. MR 1861249, https://doi.org/10.1515/form.2001.034
Shin-Ya Koyama, Correction to "퐿^{∞}-norms of eigenfunctions for arithmetic hyperbolic 3-manifolds," Duke Math. J. 77 (1995), 799–817 [ MR1324641], Duke Math. J. 165 (2016), no. 2, 413–415. MR 3457678, https://doi.org/10.1215/00127094-3481590
Xiaoqing Li, Bounds for 퐺퐿(3)×퐺퐿(2) 퐿-functions and 퐺퐿(3) 퐿-functions, Ann. of Math. (2) 173 (2011), no. 1, 301–336. MR 2753605, https://doi.org/10.4007/annals.2011.173.1.8
Wen Zhi Luo and Peter Sarnak, Quantum ergodicity of eigenfunctions on 푃푆퐿₂(퐙)\퐇², Inst. Hautes Études Sci. Publ. Math. 81 (1995), 207–237. MR 1361757
Yoichi Motohashi, A trace formula for the Picard group. I, Proc. Japan Acad. Ser. A Math. Sci. 72 (1996), no. 8, 183–186. MR 1420613
Yoichi Motohashi, Trace formula over the hyperbolic upper half space, Analytic number theory (Kyoto, 1996) London Math. Soc. Lecture Note Ser., vol. 247, Cambridge Univ. Press, Cambridge, 1997, pp. 265–286. MR 1694996, https://doi.org/10.1017/CBO9780511666179.018
Maki Nakasuji, Prime geodesic theorem via the explicit formula of Ψ for hyperbolic 3-manifolds, Proc. Japan Acad. Ser. A Math. Sci. 77 (2001), no. 7, 130–133. MR 1857290
Yiannis N. Petridis and Morten S. Risager, Local average in hyperbolic lattice point counting, with an appendix by Niko Laaksonen, Math. Z. 285 (2017), no. 3-4, 1319–1344. MR 3623751, https://doi.org/10.1007/s00209-016-1749-z
R. S. Phillips and P. Sarnak, The Weyl theorem and the deformation of discrete groups, Comm. Pure Appl. Math. 38 (1985), no. 6, 853–866. MR 812352, https://doi.org/10.1002/cpa.3160380614
P. Sarnak, The arithmetic and geometry of some hyperbolic three-manifolds, Acta Math. 151 (1983), no. 3-4, 253–295. MR 723012, https://doi.org/10.1007/BF02393209
Goro Shimura, On the holomorphy of certain Dirichlet series, Proc. London Math. Soc. (3) 31 (1975), no. 1, 79–98. MR 0382176, https://doi.org/10.1112/plms/s3-31.1.79
K. Soundararajan and Matthew P. Young, The prime geodesic theorem, J. Reine Angew. Math. 676 (2013), 105–120. MR 3028757
Yoshio Tanigawa, Selberg trace formula for Picard groups, Algebraic number theory (Kyoto Internat. Sympos., Res. Inst. Math. Sci., Univ. Kyoto, Kyoto, 1976) Japan Soc. Promotion Sci., Tokyo, 1977, pp. 229–242. MR 0472711
Nigel Watt, Spectral large sieve inequalities for Hecke congruence subgroups of 푆퐿(2,ℤ[핚]), J. Number Theory 140 (2014), 349–424. MR 3181662, https://doi.org/10.1016/j.jnt.2014.01.018
O. Balkanova and D. Frolenkov, Prime geodesic theorem for the Picard manifold, https://arxiv.org/abs/1804.00275 (2018).
Y. Bonthonneau, Weyl laws for manifolds with hyperbolic cusps, https://arxiv.org/abs/1512.05794 (2015).
Yingchun Cai, Prime geodesic theorem, J. Théor. Nombres Bordeaux 14 (2002), no. 1, 59-72 (English, with English and French summaries). MR 1925990
Giacomo Cherubini and João Guerreiro, Mean square in the prime geodesic theorem, Algebra Number Theory 12 (2018), no. 3, 571-597. MR 3815307, https://doi.org/10.2140/ant.2018.12.571
J. B. Conrey and H. Iwaniec, The cubic moment of central values of automorphic -functions, Ann. of Math. (2) 151 (2000), no. 3, 1175-1216. MR 1779567, https://doi.org/10.2307/121132
Jeffrey Hoffstein and Paul Lockhart, Coefficients of Maass forms and the Siegel zero, Ann. of Math. (2) 140 (1994), no. 1, 161-181. With an appendix by Dorian Goldfeld, Hoffstein and Daniel Lieman. MR 1289494, https://doi.org/10.2307/2118543
Heinz Huber, Zur analytischen theorie hyperbolischer raumformen und bewegungsgruppen. II, Math. Ann. 142 (1960/1961), 385-398 (German). MR 0126549, https://doi.org/10.1007/BF01451031
Heinz Huber, Zur analytischen theorie hyperbolischer raumformen und bewegungsgruppen. II, Math. Ann. 143 (1961), 463--464 (German). MR 0154980, https://doi.org/10.1007/BF01470758
Henryk Iwaniec, Fourier coefficients of cusp forms and the Riemann zeta-function, Seminar on Number Theory, 1979-1980 (French), Univ. Bordeaux I, Talence, 1980, pp. Exp. No. 18, 36. MR 604215
Henryk Iwaniec, Mean values for Fourier coefficients of cusp forms and sums of Kloosterman sums, Number theory days, 1980 (Exeter, 1980) London Math. Soc. Lecture Note Ser., vol. 56, Cambridge Univ. Press, Cambridge, England-New York, 1982, pp. 306-321. MR 697274
Henryk Iwaniec, Prime geodesic theorem, J. Reine Angew. Math. 349 (1984), 136-159. MR 743969, https://doi.org/10.1515/crll.1984.349.136
Shin-ya Koyama, -norms on eigenfunctions for arithmetic hyperbolic -manifolds, Duke Math. J. 77 (1995), no. 3, 799-817. MR 1324641, https://doi.org/10.1215/S0012-7094-95-07724-2
Shin-ya Koyama, Prime geodesic theorem for arithmetic compact surfaces, Internat. Math. Res. Notices 8 (1998), 383-388. MR 1628243, https://doi.org/10.1155/S1073792898000257
Shin-ya Koyama, Prime geodesic theorem for the Picard manifold under the mean-Lindelöf hypothesis, Forum Math. 13 (2001), no. 6, 781-793. MR 1861249, https://doi.org/10.1515/form.2001.034
Shin-Ya Koyama, Correction to `` -norms of eigenfunctions for arithmetic hyperbolic 3-manifolds,'' Duke Math. J. 77 (1995), 799-817 [MR1324641], Duke Math. J. 165 (2016), no. 2, 413-415. MR 3457678, https://doi.org/10.1215/00127094-3481590
Xiaoqing Li, Bounds for -functions and -functions, Ann. of Math. (2) 173 (2011), no. 1, 301-336. MR 2753605, https://doi.org/10.4007/annals.2011.173.1.8
Wen Zhi Luo and Peter Sarnak, Quantum ergodicity of eigenfunctions on , Inst. Hautes Études Sci. Publ. Math. 81 (1995), 207-237. MR 1361757
Yoichi Motohashi, A trace formula for the Picard group. I, Proc. Japan Acad. Ser. A Math. Sci. 72 (1996), no. 8, 183-186. MR 1420613
Yoichi Motohashi, Trace formula over the hyperbolic upper half space, Analytic number theory (Kyoto, 1996) London Math. Soc. Lecture Note Ser., vol. 247, Cambridge Univ. Press, Cambridge, England, 1997, pp. 265-286. MR 1694996, https://doi.org/10.1017/CBO9780511666179.018
Maki Nakasuji, Prime geodesic theorem via the explicit formula of for hyperbolic 3-manifolds, Proc. Japan Acad. Ser. A Math. Sci. 77 (2001), no. 7, 130-133. MR 1857290
Yiannis N. Petridis and Morten S. Risager, Local average in hyperbolic lattice point counting, with an appendix by Niko Laaksonen, Math. Z. 285 (2017), no. 3-4, 1319-1344. MR 3623751, https://doi.org/10.1007/s00209-016-1749-z
R. S. Phillips and P. Sarnak, The Weyl theorem and the deformation of discrete groups, Comm. Pure Appl. Math. 38 (1985), no. 6, 853-866. MR 812352, https://doi.org/10.1002/cpa.3160380614
P. Sarnak, The arithmetic and geometry of some hyperbolic three-manifolds, Acta Math. 151 (1983), no. 3-4, 253-295. MR 723012, https://doi.org/10.1007/BF02393209
Goro Shimura, On the holomorphy of certain Dirichlet series, Proc. London Math. Soc. (3) 31 (1975), no. 1, 79-98. MR 0382176, https://doi.org/10.1112/plms/s3-31.1.79
K. Soundararajan and Matthew P. Young, The prime geodesic theorem, J. Reine Angew. Math. 676 (2013), 105-120. MR 3028757
Yoshio Tanigawa, Selberg trace formula for Picard groups, Algebraic number theory (Kyoto Internat. Sympos., Res. Inst. Math. Sci., Univ. Kyoto, Kyoto, 1976) Japan Soc. Promotion Sci., Tokyo, 1977, pp. 229-242. MR 0472711
Nigel Watt, Spectral large sieve inequalities for Hecke congruence subgroups of , J. Number Theory 140 (2014), 349-424. MR 3181662, https://doi.org/10.1016/j.jnt.2014.01.018
Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 11F72, 11M36, 11L05
Retrieve articles in all journals with MSC (2010): 11F72, 11M36, 11L05
Olga Balkanova
Affiliation: Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Chalmers tvärgata 3, 412 96 Gothenburg, Sweden
Email: olgabalkanova@gmail.com
Dimitrios Chatzakos
Affiliation: Université de Lille 1 Sciences et Technologies and Centre Europen pour les Mathmatiques, la Physique et leurs interactions (CEMPI), Cité Scientifique, 59655 Villeneuve d Ascq Cédex, France
Email: Dimitrios.Chatzakos@math.univ-lille1.fr
Giacomo Cherubini
Affiliation: Dipartimento di Matematica, Università degli Studi di Genova, via Dodecaneso 35, 16146 Genoa, Italy
Email: cherubini@dima.unige.it
Dmitry Frolenkov
Affiliation: National Research University Higher School of Economics, Moscow, Russia; and Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina Street, Moscow 119991, Russia
Email: frolenkov@mi.ras.ru
Niko Laaksonen
Affiliation: McGill University, Department of Mathematics and Statistics, Burnside Hall, 805 Sherbrooke Street West, Montreal, Quebec H3A 0B9, Canada
Email: n.laaksonen@ucl.ac.uk
Keywords: Prime geodesic theorem, Selberg trace formula, Kuznetsov trace formula, Kloosterman sums
Received by editor(s): August 22, 2018
Additional Notes: The first author was partially supported by Royal Swedish Academy of Sciences project no. MG2018-0002 and by the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013) with ERC grant agreement no. 615722 MOTMELSUM
The second author would like to thank the School of Mathematics of the University of Bristol and the Mathematics Department of King's College London for their support and hospitality during the academic year 2016-17. During these visits he received funding from an LMS 150th Anniversary Postdoctoral Mobility Grant 2016-17 and the European Unions Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 335141 "Nodal". He is also currently supported by the Labex CEMPI (ANR-11-LABX-0007-01).
The third author was supported by a Leibniz fellowship and thanks the MFO for the excellent working conditions, and by a "Ing. Giorgio Schirillo" postdoctoral grant from the "Istituto Nazionale di Alta Matematica".
The last author would like to thank the Mathematics Department at KTH in Stockholm, where he received funding from KAW 2013.0327. He would also like to thank the Department of Mathematics and Statistics at McGill University and the Centre de Recherches Mathématiques for their hospitality.