Rational functions and modular forms
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Abstract:
There are two elementary methods for constructing modular forms that dominate in literature. One of them uses automorphic Poincaré series and the other one theta functions. We start a third elementary approach to modular forms using rational functions that have certain properties regarding pole distribution and growth. We prove modularity with contour integration methods and Weil’s converse theorem, without using the classical formalism of Eisenstein series and $L$-functions.References
- Anatoli Andrianov, Introduction to Siegel modular forms and Dirichlet series, Universitext, Springer, New York, 2009. MR 2468862, DOI 10.1007/978-0-387-78753-4
- Tom M. Apostol, Modular functions and Dirichlet series in number theory, 2nd ed., Graduate Texts in Mathematics, vol. 41, Springer-Verlag, New York, 1990. MR 1027834, DOI 10.1007/978-1-4612-0999-7
- Bruce C. Berndt and Armin Straub, On a secant Dirichlet series and Eichler integrals of Eisenstein series, Math. Z. 284 (2016), no. 3-4, 827–852. MR 3563256, DOI 10.1007/s00209-016-1675-0
- Bruce C. Berndt and Alexandru Zaharescu, Finite trigonometric sums and class numbers, Math. Ann. 330 (2004), no. 3, 551–575. MR 2099193, DOI 10.1007/s00208-004-0559-5
- Yingkun Li and Michael Neururer, A magnetic modular form, Int. J. Number Theory 15 (2019), no. 5, 907–924. MR 3955840, DOI 10.1142/S1793042119500489
- Fred Diamond and Jerry Shurman, A first course in modular forms, Graduate Texts in Mathematics, vol. 228, Springer-Verlag, New York, 2005. MR 2112196
- Martin Dickson and Michael Neururer, Products of Eisenstein series and Fourier expansions of modular forms at cusps, J. Number Theory 188 (2018), 137–164. MR 3778627, DOI 10.1016/j.jnt.2017.12.013
- J. Franke, A dominated convergence theorem for Eisenstein series, in preparation, 2019.
- J. Franke, Rational functions, Cotangent sums and Eichler integrals, in preparation, 2019.
- Eberhard Freitag, Hilbert modular forms, Springer-Verlag, Berlin, 1990. MR 1050763, DOI 10.1007/978-3-662-02638-0
- E. Freitag, Siegelsche Modulfunktionen, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 254, Springer-Verlag, Berlin, 1983 (German). MR 871067, DOI 10.1007/978-3-642-68649-8
- Özlem Imamoḡlu and Winfried Kohnen, Representations of integers as sums of an even number of squares, Math. Ann. 333 (2005), no. 4, 815–829. MR 2195146, DOI 10.1007/s00208-005-0699-2
- Toshitsune Miyake, Modular forms, Reprint of the first 1989 English edition, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2006. Translated from the 1976 Japanese original by Yoshitaka Maeda. MR 2194815
Additional Information
- J. Franke
- Affiliation: Mathematisches Institut Universität Heidelberg, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany
- MR Author ID: 1263601
- Email: jfranke@mathi.uni-heidelberg.de
- Received by editor(s): June 18, 2019
- Received by editor(s) in revised form: January 20, 2020
- Published electronically: June 30, 2020
- Communicated by: Matthew A. Papanikolas
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 4151-4164
- MSC (2010): Primary 11F11
- DOI: https://doi.org/10.1090/proc/15034
- MathSciNet review: 4135285