AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
Period Functions for Maass Wave Forms and Cohomology
About this Title
R. Bruggeman, Mathematisch Instituut Universiteit Utrecht, 3508 TA Utrecht, Nederland, J. Lewis, Massachusetts Institute of Technology, Cambridge, MA 02139, USA and D. Zagier, Max-Planck-Institut für Mathematik, 53111 Bonn, Deutschland; and Collège de France, 75005 Paris, France
Publication: Memoirs of the American Mathematical Society
Publication Year:
2015; Volume 237, Number 1118
ISBNs: 978-1-4704-1407-8 (print); 978-1-4704-2503-6 (online)
DOI: https://doi.org/10.1090/memo/1118
Published electronically: January 8, 2015
Keywords: Maass form,
period function,
cohomology group,
parabolic cohomology,
principal series,
Petersson scalar product,
cup product
MSC: Primary 11F37, 11F67, 11F75, 22E40
Table of Contents
Chapters
- Introduction
- 1. Eigenfunctions of the hyperbolic Laplace operator
- 2. Maass forms and analytic cohomology: cocompact groups
- 3. Cohomology of infinite cyclic subgroups of $\mathrm {PSL}_2(\mathbb {R})$
- 4. Maass forms and semi-analytic cohomology: groups with cusps
- 5. Maass forms and differentiable cohomology
- 6. Distribution cohomology and Petersson product
- List of notations
Abstract
We construct explicit isomorphisms between spaces of Maass wave forms and cohomology groups for discrete cofinite groups $\Gamma \subset \mathrm {PSL}_2({\mathbb {R}})$.
In the case that $\Gamma$ is the modular group $\mathrm {PSL}_2({\mathbb {Z}})$ this gives a cohomological framework for the results in Period functions for Maass wave forms. I, of J. Lewis and D. Zagier in Ann. Math. 153 (2001), 191–258, where a bijection was given between cuspidal Maass forms and period functions.
We introduce the concepts of mixed parabolic cohomology group and semi-analytic vectors in principal series representation. This enables us to describe cohomology groups isomorphic to spaces of Maass cusp forms, spaces spanned by residues of Eisenstein series, and spaces of all $\Gamma$-invariant eigenfunctions of the Laplace operator.
For spaces of Maass cusp forms we also describe isomorphisms to parabolic cohomology groups with smooth coefficients and standard cohomology groups with distribution coefficients. We use the latter correspondence to relate the Petersson scalar product to the cup product in cohomology.
- Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956
- Roelof W. Bruggeman, Automorphic forms, hyperfunction cohomology, and period functions, J. Reine Angew. Math. 492 (1997), 1–39. MR 1488063, DOI 10.1515/crll.1997.492.1
- R. Bruggeman, Quantum Maass forms, The Conference on $L$-Functions, World Sci. Publ., Hackensack, NJ, 2007, pp. 1–15. MR 2310286
- R. Bruggeman, J. Lewis, and D. Zagier, Function theory related to the group $\textrm {PSL}_2(\Bbb R)$, From Fourier analysis and number theory to Radon transforms and geometry, Dev. Math., vol. 28, Springer, New York, 2013, pp. 107–201. MR 2986956, DOI 10.1007/978-1-4614-4075-8_{7}
- R. W. Bruggeman and T. Mühlenbruch, Eigenfunctions of transfer operators and cohomology, J. Number Theory 129 (2009), no. 1, 158–181. MR 2468476, DOI 10.1016/j.jnt.2008.08.003
- Ulrich Bunke and Martin Olbrich, Gamma-cohomology and the Selberg zeta function, J. Reine Angew. Math. 467 (1995), 199–219. MR 1355929
- Ulrich Bunke and Martin Olbrich, Resolutions of distribution globalizations of Harish-Chandra modules and cohomology, J. Reine Angew. Math. 497 (1998), 47–81. MR 1617426, DOI 10.1515/crll.1998.043
- Anton Deitmar and Joachim Hilgert, Cohomology of arithmetic groups with infinite dimensional coefficient spaces, Doc. Math. 10 (2005), 199–216. MR 2148074
- Anton Deitmar and Joachim Hilgert, A Lewis correspondence for submodular groups, Forum Math. 19 (2007), no. 6, 1075–1099. MR 2367955, DOI 10.1515/FORUM.2007.042
- M. Eichler, Eine Verallgemeinerung der Abelschen Integrale, Math. Z. 67 (1957), 267–298 (German). MR 89928, DOI 10.1007/BF01258863
- Roger Godement, Topologie algébrique et théorie des faisceaux, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1252, Hermann, Paris, 1958 (French). Publ. Math. Univ. Strasbourg. No. 13. MR 0102797
- Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
- Sigurdur Helgason, Topics in harmonic analysis on homogeneous spaces, Progress in Mathematics, vol. 13, Birkhäuser, Boston, Mass., 1981. MR 632696
- Lars Hörmander, An introduction to complex analysis in several variables, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR 0203075
- 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
- Marvin I. Knopp, Some new results on the Eichler cohomology of automorphic forms, Bull. Amer. Math. Soc. 80 (1974), 607–632. MR 344454, DOI 10.1090/S0002-9904-1974-13520-2
- Joseph Lehner, Discontinuous groups and automorphic functions, Mathematical Surveys, No. VIII, American Mathematical Society, Providence, R.I., 1964. MR 0164033
- John B. Lewis, Eigenfunctions on symmetric spaces with distribution-valued boundary forms, J. Functional Analysis 29 (1978), no. 3, 287–307. MR 512246, DOI 10.1016/0022-1236(78)90032-0
- John B. Lewis, Spaces of holomorphic functions equivalent to the even Maass cusp forms, Invent. Math. 127 (1997), no. 2, 271–306. MR 1427619, DOI 10.1007/s002220050120
- J. Lewis and D. Zagier, Period functions and the Selberg zeta function for the modular group, The mathematical beauty of physics (Saclay, 1996) Adv. Ser. Math. Phys., vol. 24, World Sci. Publ., River Edge, NJ, 1997, pp. 83–97. MR 1490850
- J. Lewis and D. Zagier, Period functions for Maass wave forms. I, Ann. of Math. (2) 153 (2001), no. 1, 191–258. MR 1826413, DOI 10.2307/2661374
- H. Maass, Lectures on modular functions of one complex variable, 2nd ed., Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 29, Tata Institute of Fundamental Research, Bombay, 1983. With notes by Sunder Lal. MR 734485
- Ju. I. Manin, Periods of cusp forms, and $p$-adic Hecke series, Mat. Sb. (N.S.) 92(134) (1973), 378–401, 503 (Russian). MR 0345909
- Dieter H. Mayer, The thermodynamic formalism approach to Selberg’s zeta function for $\textrm {PSL}(2,\textbf {Z})$, Bull. Amer. Math. Soc. (N.S.) 25 (1991), no. 1, 55–60. MR 1080004, DOI 10.1090/S0273-0979-1991-16023-4
- M. Möller and A. D. Pohl, Period functions for Hecke triangle groups, and the Selberg zeta function as a Fredholm determinant, Ergodic Theory Dynam. Systems 33 (2013), no. 1, 247–283. MR 3009112, DOI 10.1017/S0143385711000794
- H. Neunhöffer, Über die analytische Fortsetzung von Poincaréreihen, S.-B. Heidelberger Akad. Wiss. Math.-Natur. Kl. (1973), 33–90 (German). MR 0352007
- Anke D. Pohl, Period functions for Maass cusp forms for $\Gamma _0(p)$: a transfer operator approach, Int. Math. Res. Not. IMRN 14 (2013), 3250–3273. MR 3085759, DOI 10.1093/imrn/rns146
- Anke D. Pohl, A dynamical approach to Maass cusp forms, J. Mod. Dyn. 6 (2012), no. 4, 563–596. MR 3008410, DOI 10.3934/jmd.2012.6.563
- A. D. Pohl: Odd and even Maass cusp forms for Hecke triangle groups, and the billiard flow. arXiv:1303.0528
- Henrik Schlichtkrull, Hyperfunctions and harmonic analysis on symmetric spaces, Progress in Mathematics, vol. 49, Birkhäuser Boston, Inc., Boston, MA, 1984. MR 757178
- Goro Shimura, Sur les intégrales attachées aux formes automorphes, J. Math. Soc. Japan 11 (1959), 291–311 (French). MR 120372, DOI 10.2969/jmsj/01140291
- G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746
- D. Zagier: Periods of modular forms, traces of Hecke operators, and multiple zeta values. In Hokei-keishiki to L-kansuu no kenkyuu (= Research on Automorphic Forms and L-Functions), RIMS Kokyuroku 843 (1993), 162–170.
- D. Zagier: New points of view on the Selberg zeta function. Proceedings of the Japanese-German Seminar "Explicit Structures of Modular Forms and Zeta Functions", Ryushi-do (2002), 1-10.
- Don Zagier, Quantum modular forms, Quanta of maths, Clay Math. Proc., vol. 11, Amer. Math. Soc., Providence, RI, 2010, pp. 659–675. MR 2757599