Certain aspects of holomorphic function theory on some genus-zero arithmetic groups. Issue 2 (March 2017)

• Record Type: Journal Article
• Authors: Jorgenson, Jay; Smajlović, Lejla; Then, Holger
• Is Part Of: LMS journal of computation and mathematics. Volume 19:Issue 2(2016)
• Page Start: 360
• Page End: 381
• Publication Date: 2017-03
• Subjects: 11F12 (primary), -- 11F20 (secondary)
• DOI: 10.1112/S1461157016000425
• Languages: English
• ISSNs: 1461-1570

Abstract

Abstract : 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.