Abstract
In this chapter, we introduce Maass wave/cusp forms of real weight on subgroups of finite index in the full modular group with compatible multiplier systems. After introducing multiplier systems and the unitary automorphic factor and discussing the hyperbolic Laplace operator and Maass operators, we define Maass waveforms. We then use Hecke operators in the special case where the Maass waveform is of weight 0 and trivial multiplier system on the full group. We conclude the chapter by mentioning some spectral theory for the Laplace operator and its Friedrichs extension and discuss briefly Selberg’s conjecture.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
L. Bers, F. John, M. Schechter, Partial Differential Equations. Lectures in Applied Mathematics, vol. 3 (Interscience Publishers, Geneva, 1964). ISBN 978-0-8218-0049-2
A.R. Booker, A. Strömbergsson, A. Venkatesh, Effective computation of Maass cusp forms. Int. Math. Res. Not. 2006, 71281. https://doi.org/10.1155/IMRN/2006/71281
A. Borel, Automorphic Forms on \(SL_2(\mathbb {R})\) (Cambridge University Press, Cambridge, 1997). ISBN 978-0511896064. https://doi.org/10.1017/CBO9780511896064
R.W. Bruggeman, Modular forms of varying weight. I. Math. Z. 190(4), 477–495 (1985). MR808915. http://www.ams.org/mathscinet-getitem?mr=808915; https://doi.org/10.1007/BF01214747
R.W. Bruggeman, Families of Automorphic Forms. Monographs in Mathematics (Birkhäuser Boston Inc., Boston, MA, 1994). ISBN 3-7643-5046-6
D. Bump, Automorphic Forms and Representations. Cambridge Studies in Advanced Mathematics, vol. 55 (Cambridge University Press, Cambridge, 1997). ISBN 0-521-55098-X
D. Bump, J.W. Cogdell, E. de Shalit, D. Gaitsgory, E. Kowalski, S.S. Kudla, An Introduction to the Langlands Program (Birkhäuser Boston, Cambridge, 2004). ISBN 978-0-8176-3211-3. https://doi.org/10.1007/978-0-8176-8226-2
Encyclopedia of Mathematics (2013). http://www.encyclopediaofmath.org/
S.S. Gelbart, An elementary introduction to the Langlands program. Bull. Am. Math. Soc. (N.S.) 10(2), 177–219 (1984)
A. Good, Cusp forms and eigenfunctions of the Laplacian. Math. Ann. 255(4), 523–548 (1984). https://doi.org/10.1007/BF01451932
D.A. Hejhal, The Selberg trace formula and the Riemann zeta function. Duke Math. J. 43(3) (1976), 441–482. https://doi.org/10.1215/S0012-7094-76-04338-6
D.A. Hejhal, The Selberg Trace Formula for PSL(2, R), Vol. 2. Lecture Notes in Mathematics, vol. 1001 (Springer, Berlin, 1983). ISBN 3-540-12323-7
M.N. Huxley, Scattering matrices for congruence subgroups, in Modular Forms (Durham, 1983). Ellis Horwood Series in Mathematics and its Applications: Statistics and Operational Research (Horwood, Chichester, 1984), pp. 141–156. MR803366. http://www.ams.org/mathscinet-getitem?mr=803366
H. Iwaniec, Topics in Classical Automorphic Forms. Graduate Studies in Mathematics, vol. 17 (American Mathematical Society, Providence, RI, 1997). ISBN 0-8218-0777-3. https://doi.org/10.1090/gsm/017
H. Iwaniec, Spectral Methods of Automorphic Forms. Graduate Studies in Mathematics, vol. 53, 2nd edn. (American Mathematical Society, Providence, RI); Revista Matemática Iberoamericana, Madrid, 2002. ISBN 0-8218-3160-7. https://doi.org/10.1090/gsm/053
T. Kato, Perturbation Theory for Linear Operators. Reprint of the 1980 edition. Classics in Mathematics (Springer, Berlin, 1995). ISBN 3-540-58661-X
T. Kubota Elementary Theory of Eisenstein Series (Kodansha Ltd., Tokyo/Wiley, New York, 1973)
S. Lang, SL2(R) (Springer, New York, 1985). Corrected second printing, 1998. ISBN 0-387-96198-4
R.P. Langlands, Problems in the theory of automorphic forms, in Lectures in Modern Analysis and Applications, III. Lecture Notes in Mathematics, vol. 170 (Springer, New York, 1970), pp. 18–61. ISBN 978-3-540-05284-5. https://doi.org/10.1007/BFb0079065
P.D. Lax, R.S. Phillips, Scattering Theory for Automorphic Functions. Annals of Mathematics Studies, vol. 87 (Princeton University Press, Princeton, NJ, 1976). ISBN 0-691-08179-4
The LMFDB Collaboration, The L-Functions and Modular Forms Database (2019). http://www.lmfdb.org. Accessed 4 Apr 2020
W. Luo, Z. Rudnick, P. Sarnak, On Selberg’s eigenvalue conjecture. Geom. Funct. Anal. 5(2), 387–401 (1995). https://doi.org/10.1007/BF01895672
H. Maass, Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen. Math. Ann. 121, 141–183 (1949). https://doi.org/10.1007/BF01329622
H. Maass, Lectures on Modular Functions of One Complex Variable, 2nd edn. (Tata Institute of Fundamental Research, Bombay, 1983). http://www.math.tifr.res.in/~publ/ln/tifr29.pdf
W. Magnus, F. Oberhettinger, R.P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics. 3rd edn. Die Grundlehren der mathematischen Wissenschaften, vol. 52 (Springer, New York, 1966). MR0232968. http://www.ams.org/mathscinet-getitem?mr=232968
Ju.I. Manin, Periods of cusp forms, and p-adic Hecke series. Mat. Sb. (N. Ser.) 92(134), 378–401 (1973). English translation appeared in: Math. USSR-Sb. 21(3), 371–393 (1973). MR345909. http://www.ams.org/mathscinet-getitem?mr=345909
T. Mühlenbruch, Systems of automorphic forms and period functions. Ph.D. Thesis, Utrecht University, 2003
T. Mühlenbruch, W. Raji, Generalized Maass wave forms. Proc. Am. Math. Soc. 141, 1143–1158 (2013)
NIST Digital Library of Mathematical Functions. Version 1.0.8. Release date 25 April 2014. https://dlmf.nist.gov/
L. Pukánszky, The Plancherel formula for the universal covering group of SL(R, 2). Math. Ann. 156, 96–143 (1964)
W. Roelcke, Automorphe formen in der hyperbolishce Ebene, I. Math. Ann. 167, 292–337 (1966)
P. Sarnak, Selberg’s eigenvalue conjecture. Not. Am. Math. Soc. 42, 1272–1277 (1995)
A. Selberg, On the estimation of Fourier coefficients of modular forms, in Theory of Numbers, ed. by A.L. Whiteman. Proceedings of Symposia in Pure Mathematics VIII (American Mathematical Society, Providence, RI, 1965), pp. 1–15. ISBN 978-0-8218-1408-6. MR0182610. http://www.ams.org/mathscinet-getitem?mr=0182610
L.J. Slater, Confluent Hypergeometric Functions (Cambridge University Press, New York, 1960). MR107026. http://www.ams.org/mathscinet-getitem?mr=107026
F. Strömberg, Computational aspects of Maass waveforms. PhD-thesis, Uppsala Universitet, 2005. ISBN 91-506-1794-X. URN: urn:nbn:se:uu:diva-4778: http://urn.kb.se/resolve?urn=urn%3Anbn%3Ase%3Auu%3Adiva-4778. DiVA:diva2:165722: http://uu.diva-portal.org/smash/record.jsf?pid=diva2%3A165722
F. Strömberg, Hecke operators for Maass waveforms on \(PSL(2,{\mathbb {Z}})\) with integer weight and eta multiplier. Int. Math. Res. Not. 2007 (2007). https://doi.org/10.1093/imrn/rnm062
A. Terras, Harmonic Analysis on Symmetric Spaces and Applications, I (Springer, New York, 1985)
A. Terras, Harmonic Analysis and Symmetric Spaces and Applications, II (Springer, Berlin, 1988)
H. Then, Maass cusp forms for large eigenvalues. Math. Comp. 74, 363–381 (2005). https://doi.org/10.1090/S0025-5718-04-01658-8
A. Venkov, Spectral Theory of Automorphic Functions and Its Applications. Mathematics and its Applications (Soviet Series), vol. 51 (Kluwer Academic Publishers Group, Dordrecht, 1990). ISBN 0-7923-0487-X. https://doi.org/10.1007/978-94-009-1892-4
E. Wirtinger, Zur formalen Theorie der Funktionen von mehr komplexen Veränderlichen. Math. Ann. 97, 357–375 (1926, in German). https://doi.org/10.1007/BF01447872
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Mühlenbruch, T., Raji, W. (2020). Maass Wave Forms of Real Weight. In: On the Theory of Maass Wave Forms. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-030-40475-8_3
Download citation
DOI: https://doi.org/10.1007/978-3-030-40475-8_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-40477-2
Online ISBN: 978-3-030-40475-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)