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Elliptic Zeta Functions and Equivariant Functions

Published online by Cambridge University Press:  20 November 2018

Abdellah Sebbar  and
Isra Al-Shbeil
Abdellah Sebbar
Affiliation:
Department of Mathematics and Statistics, University of Ottawa, Ottawa ON K1N 6N5, e-mail: asebbar@uottawa.ca
Isra Al-Shbeil
Affiliation:
Department of Mathematics and Statistics, University of Ottawa, Ottawa ON K1N 6N5, e-mail: ialsh010@uottawa.ca
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Abstract

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In this paper we establish a close connection between three notions attached to a modular subgroup, namely, the set of weight two meromorphic modular forms, the set of equivariant functions on the upper half-plane commuting with the action of the modular subgroup, and the set of elliptic zeta functions generalizing the Weierstrass zeta functions. In particular, we show that the equivariant functions can be parameterized by modular objects as well as by elliptic objects.

Keywords

11F1235Q1532L10modular formequivariant functionelliptic zeta function
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 61 , Issue 2 , 01 June 2018 , pp. 376 - 389
Copyright
Copyright © Canadian Mathematical Society 2018

References

[1] Brady, M. M., Meromorphic Solutions ofa System offunctional equations involving the modular group. Proc. Amer. Math. Soc. 30 (1971), 271–277. http://dx.doi.org/10.1090/S0002-9939-1971-0280712-5Google Scholar
[2] Conway, J. H., Understandinggroups like To(N). In: Groups, difference sets, and the Monster (Columbus, OH, 1993), Ohio State Univ. Math. Res. Inst. Publ., 4, de Gruyter, Berlin, 1996, pp. 327–343.Google Scholar
[3] Elbasraoui, A. and Sebbar, A., Equivariant forms: structure and geometry. Canad. Math. Bull. 56 (2013), no. 3, 520–533. http://dx.doi.Org/10.4153/CMB-2O11-195-2Google Scholar
[4] Elbasraoui, A. and Sebbar, A., Rational equivariant forms. Int. J. Number Theory 8 (2012), no. 4, 963–981. http://dx.doi.Org/10.1142/S1 793042112500571Google Scholar
[5] Lang, S., Elliptic functions. Second ed., Graduate Texts in Mathematics, 112, Springer-Verlag, New York, 1987. http://dx.doi.org/10.1007/978-1-4612-4752-4Google Scholar
[6] Saber, H. and Sebbar, A., On the critical points of modular forms. J. Number Theory 132 (2012), no. 8, 1780–1787. http://dx.doi.Org/10.1016/j.jnt.2012.03.004Google Scholar
[7] Saber, H. and Sebbar, A., Equivariant functions and vector-valued modular forms. Int. J. Number Theory 10 (2014), no. 4, 949–954. http://dx.doi.Org/10.1142/S1793042114500092Google Scholar
[8] Sebbar, A. and Sebbar, A., Equivariant functions and integrals of elliptic functions. Geom. Dedicata 160 (2012), 373–414. http://dx.doi.Org/10.1007/s10711-011-9688-7Google Scholar
[9] Shimura, G., Introduction to the arithmetic theory of automorphic functions. Kan Memorial Lectures, 1, Publications of the Mathematical Society of Japan, 11, Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, NJ, 1971.Google Scholar
[10] Tannery, J. and Molk, J., Elements de la theorie des fonctions elliptiques. Gauthier-Villars, Paris, 1990.Google Scholar