Abstract
We saw in the previous chapter that Maass operators E ± k map Maass cusp forms of weight k to Maass cusp forms of weight k ± 2. Can we exploit this result and use Maass operators to get a whole family of Maass cusp forms by starting with a “reference” Maass waveform of weight k and generate all possible images by repeatedly applying one of the Maass operators?
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References
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, Modular forms of varying weight. III. J. Reine Angew. Math. 371, 144–190 (1986). https://doi.org/10.1515/crll.1986.371.144
R.W. Bruggeman, Families of Automorphic Forms. Monographs in Mathematics (Birkhäuser Boston Inc., Boston, MA, 1994). ISBN 3-7643-5046-6
V.A. Bykovskii, Functional equations for Hecke-Maass series. Funct. Anal. Appl. 34, 23–32, 95 (2000)
I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, 4th edn. (Academic, New York/London, 1965). MR197789. http://www.ams.org/mathscinet-getitem?mr=197789
I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products. (Academic, New York, 1980). ISBN 0-12-294760-6
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
H. Iwanieca, P. Sarnak, The non-vanishing of central values of automorphic L-functions and Landau-Siegel zeros. Israel J. Math. 120, 155–177 (2000)
J. Lewis, D. Zagier, Period functions for Maass wave forms. I. Ann. Math. 153, 191–258 (2001)
S. Liu, Nonvanishing of central L-values of Maass forms. Adv. Math. 332, 403–437 (2018). arXiv:1702.07084v2. https://arxiv.org/abs/1702.07084v2; https://doi.org/10.1016/j.aim.2018.05.017
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
Y. Motohashi, A note on the mean value of the zeta and L-functions. XII. Proc. Jpn. Acad. Ser. A Math. Sci. 78(3), 36–41 (2002)
T. Mühlenbruch, Systems of automorphic forms and period functions. Ph.D. Thesis, Utrecht University, 2003
T. Mühlenbruch, W. Raji, Eichler integrals for Maass cusp forms of half-integral weight. Ill. J. Math. 57(2), 445–475 (2013)
R. Narasimhan, Y. Nievergelt, Complex Analysis in One Variable, 2nd edn. (Birkhäuser Boston Inc., Boston, 2001). ISBN 0-8176-4164-5
H. Rademacher, Topics in Analytic Number Theory (Springer, New York, 1973)
L.J. Slater, Confluent Hypergeometric Functions (Cambridge University Press, New York, 1960). MR107026. http://www.ams.org/mathscinet-getitem?mr=107026
W. Walter, Analysis I. Grundwissen Mathematik, vol. 3 (Springer, New York, 1985)
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Mühlenbruch, T., Raji, W. (2020). Families of Maass Cusp Forms, L-Series, and Eichler Integrals. In: On the Theory of Maass Wave Forms. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-030-40475-8_4
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