### Abstract

There are a number of fundamental results in the study of holomorphic function theory associated to the discrete group
PSL(2,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
E4
and
E6
; the smallest-weight cusp form
Δ
has weight twelve and can be written as a polynomial in
E4
and
E6
; and the Hauptmodul
j
can be written as a multiple of
E34
divided by
Δ
. The goal of the present article is to seek generalizations of these results to some other genus-zero arithmetic groups
Γ0(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.

Original language | English |
---|---|

Pages (from-to) | 360-381 |

Number of pages | 22 |

Journal | LMS Journal of Computation and Mathematics |

Volume | 19 |

Issue number | 2 |

Early online date | 1 Mar 2017 |

DOIs | |

Publication status | Published - 2017 |

### Keywords

- Mathematics - Number Theory

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## Cite this

Jorgenson, J., Smajlović, L., & Then, H. (2017). Certain aspects of holomorphic function theory on some genus zero arithmetic groups.

*LMS Journal of Computation and Mathematics*,*19*(2), 360-381. https://doi.org/10.1112/S1461157016000425