TY - JOUR
T1 - Certain aspects of holomorphic function theory on some genus zero arithmetic groups
AU - Jorgenson, Jay
AU - Smajlović, Lejla
AU - Then, Holger
PY - 2017
Y1 - 2017
N2 - 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.
AB - 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.
KW - Mathematics - Number Theory
U2 - 10.1112/S1461157016000425
DO - 10.1112/S1461157016000425
M3 - Article
VL - 19
SP - 360
EP - 381
JO - LMS Journal of Computation and Mathematics
JF - LMS Journal of Computation and Mathematics
SN - 1461-1570
IS - 2
ER -