Modular equations for congruence subgroups of genus zero

2019 ◽  
Vol 51 (1) ◽  
pp. 187-204
Author(s):  
Bumkyu Cho
Keyword(s):  
Modular Equations ◽  
Genus Zero ◽  
Congruence Subgroups

Modular equations and the genus zero property of moonshine functions

1997 ◽  
Vol 129 (3) ◽  
pp. 413-443 ◽  
Author(s):  
C.J. Cummins ◽  
T. Gannon
Keyword(s):  
Modular Equations ◽  
Genus Zero

scholarly journals On Conjugacy Classes of Congruence Subgroups of PSL(2, R)

2009 ◽  
Vol 12 ◽  
pp. 264-274 ◽  
Author(s):  
C. J. Cummins
Keyword(s):  
Congruence Subgroup ◽  
Principal Congruence ◽  
Conjugacy Classes ◽  
Genus Zero ◽  
Congruence Subgroups ◽  
Principal Congruence Subgroup ◽  
Genus One

AbstractLet G be a subgroup of PSL(2, R) which is commensurable with PSL(2, Z). We say that G is a congruence subgroup of PSL(2, R) if G contains a principal congruence subgroup /overline Γ(N) for some N. An algorithm is given for determining whether two congruence subgroups are conjugate in PSL(2, R). This algorithm is used to determine the PSL(2, R) conjugacy classes of congruence subgroups of genus-zero and genus-one. The results are given in a table.

scholarly journals Proof of a conjecture of Ramanujan

1967 ◽  
Vol 8 (1) ◽  
pp. 14-32 ◽  
Author(s):  
A. O. L. Atkin
Keyword(s):  
Fourier Coefficient ◽  
Modular Equations ◽  
Genus Zero ◽  
Modular Invariant ◽  
Image Position

We writeandso that p(n) is the number of unrestricted partitions of n. Ramanujan [1] conjectured in 1919 that if q = 5, 7, or 11, and 24m ≡ 1 (mod qn), then p(m) ≡ 0 (mod qn). He proved his conecture for n = 1 and 2†, but it was not until 1938 that Watson [4] proved the conjecture for q = 5 and all n, and a suitably modified form for q = 7 and all n. (Chowla [5] had previously observed that the conjecture failed for q = 7 and n = 3.) Watson's method of modular equations, while theoretically available for the case q = 11, does not seem to be so in practice even with the help of present-day computers. Lehner [6, 7] has developed an essentially different method, which, while not as powerful as Watson's in the cases where Γ0(q) has genus zero, is applicable in principle to all primes q without prohibitive calculation. In particular he proved the conjecture for q = 11 and n = 3 in [7]. Here I shall prove the conjecture for q = 11 and all n, following Lehner's approach rather than Watson's. I also prove the analogous and essentially simpler result for c(m), the Fourier coefficient‡ of Klein's modular invariant j (τ) as

scholarly journals Fundamental domains for genus-zero and genus-one congruence subgroups

2010 ◽  
Vol 13 ◽  
pp. 222-245
Author(s):  
C. J. Cummins
Keyword(s):  
Fundamental Domain ◽  
Congruence Subgroup ◽  
Discrete Subgroup ◽  
Magma Source ◽  
Genus Zero ◽  
Congruence Subgroups ◽  
Fundamental Domains ◽  
Finite Set ◽  
Data Files ◽  
Genus One

AbstractIn this paper, we compute Ford fundamental domains for all genus-zero and genus-one congruence subgroups. This is a continuation of previous work, which found all such groups, including ones that are not subgroups ofPSL(2,ℤ). To compute these fundamental domains, an algorithm is given that takes the following as its input: a positive square-free integerf, which determines a maximal discrete subgroup Γ0(f)+ofSL(2,ℝ); a decision procedure to determine whether a given element of Γ0(f)+is in a subgroupG; and the index ofGin Γ0(f)+. The output consists of: a fundamental domain forG, a finite set of bounding isometric circles; the cycles of the vertices of this fundamental domain; and a set of generators ofG. The algorithm avoids the use of floating-point approximations. It applies, in principle, to any group commensurable with the modular group. Included as appendices are: MAGMA source code implementing the algorithm; data files, computed in a previous paper, which are used as input to compute the fundamental domains; the data computed by the algorithm for each of the congruence subgroups of genus zero and genus one; and an example, which computes the fundamental domain of a non-congruence subgroup.

scholarly journals On Rademacher's conjecture: congruence subgroups of genus zero of the modular group

2004 ◽  
Vol 277 (1) ◽  
pp. 408-428 ◽  
Author(s):  
Kok Seng Chua ◽  
Mong Lung Lang ◽  
Yifan Yang
Keyword(s):  
Modular Group ◽  
Genus Zero ◽  
Congruence Subgroups

Modular Equations and Discrete, Genus-Zero Subgroups of SL(2, ℝ) Containing Γ(N)

2002 ◽  
Vol 45 (1) ◽  
pp. 36-45 ◽  
Author(s):  
C. J. Cummins
Keyword(s):  
Discrete Subgroup ◽  
Modular Equations ◽  
Genus Zero ◽  
Modular Polynomials ◽  
Automorphic Functions

AbstractLet G be a discrete subgroup of SL(2, ℝ) which contains Γ(N) for some N. If the genus of X(G) is zero, then there is a unique normalised generator of the field of G-automorphic functions which is known as a normalised Hauptmodul. This paper gives a characterisation of normalised Hauptmoduls as formal q series using modular polynomials.

Determination of Hauptmoduls and Construction of Abelian Extensions of Quadratic Number Fields

2007 ◽  
Vol 50 (3) ◽  
pp. 334-346
Author(s):  
Hung-Jen Chiang-Hsieh ◽  
Yifan Yang
Keyword(s):  
Number Fields ◽  
Cyclic Extension ◽  
Quadratic Number ◽  
Modular Functions ◽  
Genus Zero ◽  
Congruence Subgroups ◽  
Abelian Extensions ◽  
Quadratic Number Fields

AbstractWe obtain Hauptmoduls of genus zero congruence subgroups of the type (p) := Γ0(p) + wp, where p is a prime and wp is the Atkin–Lehner involution. We then use the Hauptmoduls, along with modular functions on Γ1(p) to construct families of cyclic extensions of quadratic number fields. Further examples of cyclic extension of bi-quadratic and tri-quadratic number fields are also given.

On the hard-hexagon model and the theory of modular functions

1988 ◽  
Vol 325 (1588) ◽  
pp. 643-702 ◽  
Keyword(s):  
Exact Solution ◽  
Order Parameter ◽  
Modular Group ◽  
Lattice Gas ◽  
Modular Equations ◽  
Modular Functions ◽  
Congruence Subgroups ◽  
Geometrical Properties ◽  
Mathematical Properties ◽  
Hard Hexagon Model

The mathematical properties of the exact solution of the hard-hexagon lattice gas model are investigated by using the Klein-Fricke theory of modular functions. In particular, it is shown that the order-parameter R and the reciprocal activity z' for the model can be expressed in terms of hauptmoduls that are associated with certain congruence subgroups of the full modular group Known modular equations are then used to prove that R (z') is an algebraic function of A connection is established between the singular points of this function and the geometrical properties of the icosahedron .

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