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Certain aspects of holomorphic function theory on some genus-zero arithmetic groups

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  01 March 2017

Jay Jorgenson ,
Lejla Smajlović  and
Holger Then
Jay Jorgenson
Affiliation:
Department of Mathematics, The City College of New York, Convent Avenue at 138th Street, New York, NY 10031, USA email jjorgenson@mindspring.com
Lejla Smajlović
Affiliation:
Department of Mathematics, University of Sarajevo, Zmaja od Bosne 35, 71 000 Sarajevo, Bosnia and Herzegovina email lejlas@pmf.unsa.ba
Holger Then
Affiliation:
Department of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, United Kingdom email holger.then@bristol.ac.uk

Abstract

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There are a number of fundamental results in the study of holomorphic function theory associated to the discrete group $\operatorname{PSL}(2,\mathbb{Z})$, including the following statements: the ring of holomorphic modular forms is generated by the holomorphic Eisenstein series of weights four and six, denoted by $E_{4}$ and $E_{6}$; the smallest-weight cusp form $\unicode[STIX]{x1D6E5}$ has weight twelve and can be written as a polynomial in $E_{4}$ and $E_{6}$; and the Hauptmodul $j$ can be written as a multiple of $E_{4}^{3}$ divided by $\unicode[STIX]{x1D6E5}$. The goal of the present article is to seek generalizations of these results to some other genus-zero arithmetic groups $\unicode[STIX]{x1D6E4}_{0}(N)^{+}$ with square-free level $N$, which are related to ‘Monstrous moonshine conjectures’. Certain aspects of our results are generated from extensive computer analysis; as a result, many of the space-consuming results are made available on a publicly accessible web site. However, we do present in this article specific results for certain low-level groups.

MSC classification

Primary: 11F12: Automorphic forms, one variable
Secondary: 11F20: Dedekind eta function, Dedekind sums
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 19 , Issue 2 , 2016 , pp. 360 - 381
Copyright
© The Author(s) 2017 

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