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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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. 372 (2019), 5355-5374
MSC (2010): Primary 11F72; Secondary 11M36, 11L05
Published electronically: December 28, 2018
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Abstract: For $ \Gamma $ a cofinite Kleinian group acting on $ \mathbb{H}^3$, we study the prime geodesic theorem on $ M=\Gamma \backslash \mathbb{H}^3$, which asks about the asymptotic behavior of lengths of primitive closed geodesics (prime geodesics) on $ M$. Let $ E_{\Gamma }(X)$ be the error in the counting of prime geodesics with length at most $ \log X$. For the Picard manifold, $ \Gamma =\mathrm {PSL}(2,\mathbb{Z}[i])$, we improve the classical bound of Sarnak, $ E_{\Gamma }(X)=O(X^{5/3+\epsilon })$, to $ E_{\Gamma }(X)=O(X^{13/8+\epsilon })$. In the process we obtain a mean subconvexity estimate for the Rankin-Selberg $ L$-function attached to Maass-Hecke cusp forms. We also investigate the second moment of $ E_{\Gamma }(X)$ for a general cofinite group $ \Gamma $, and we show that it is bounded by $ O(X^{16/5+\epsilon })$.

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Additional Information

Olga Balkanova
Affiliation: Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Chalmers tvärgata 3, 412 96 Gothenburg, Sweden

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

Giacomo Cherubini
Affiliation: Dipartimento di Matematica, Università degli Studi di Genova, via Dodecaneso 35, 16146 Genoa, Italy

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

Niko Laaksonen
Affiliation: McGill University, Department of Mathematics and Statistics, Burnside Hall, 805 Sherbrooke Street West, Montreal, Quebec H3A 0B9, Canada

Keywords: Prime geodesic theorem, Selberg trace formula, Kuznetsov trace formula, Kloosterman sums
Received by editor(s): August 22, 2018
Published electronically: December 28, 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.
Article copyright: © Copyright 2018 American Mathematical Society