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.

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 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.

On lifting Fuchsian Groups

1980 ◽  
Vol 87 (1) ◽  
pp. 61-67
Author(s):  
A. H. M. Hoare
Keyword(s):  
Hyperbolic Plane ◽  
Discrete Subgroup ◽  
Universal Covering ◽  
Fuchsian Groups ◽  
Universal Covering Group ◽  
Genus Zero ◽  
Canonical Map ◽  
Covering Group ◽  
Compact Case ◽  
Fundamental Polygon

Let Γ be a discrete subgroup of G = PSL(2,ℝ) and p be the canonical map from the universal covering group on to PSL(2,ℝ). By a continuity argument on a fundamental polygon for Γ acting on the hyperbolic plane 2, Milnor (4) obtained a presentation for p−1(Γ) whenever 2/Γ is compact and of genus zero. Using Teichmüller theory and a double Reidemeister-Schreier process, Macbeath(2) showed that the general compact case can be deduced from Milnor's result. We give here a method of obtaining a presentation which uses only the geometry of a Dirichlet region and which applies equally to the non-compact case.

scholarly journals Ramanujan’s function k(τ)=r(τ)r 2(2τ) and its modularity

2020 ◽  
Vol 18 (1) ◽  
pp. 1727-1741
Author(s):  
Yoonjin Lee ◽  
Yoon Kyung Park
Keyword(s):  
Continued Fraction ◽  
Quadratic Field ◽  
Singular Values ◽  
Modular Function ◽  
Imaginary Quadratic Field ◽  
Modular Equations ◽  
Positive Rational Number ◽  
Congruence Relations ◽  
Symmetry Relations ◽  
Ray Class Field

Abstract We study the modularity of Ramanujan’s function k ( τ ) = r ( τ ) r 2 ( 2 τ ) k(\tau )=r(\tau ){r}^{2}(2\tau ) , where r ( τ ) r(\tau ) is the Rogers-Ramanujan continued fraction. We first find the modular equation of k ( τ ) k(\tau ) of “an” level, and we obtain some symmetry relations and some congruence relations which are satisfied by the modular equations; these relations are quite useful for reduction of the computation cost for finding the modular equations. We also show that for some τ \tau in an imaginary quadratic field, the value k ( τ ) k(\tau ) generates the ray class field over an imaginary quadratic field modulo 10; this is because the function k is a generator of the field of the modular function on Γ 1 ( 10 ) {{\mathrm{\Gamma}}}_{1}(10) . Furthermore, we suggest a rather optimal way of evaluating the singular values of k ( τ ) k(\tau ) using the modular equations in the following two ways: one is that if j ( τ ) j(\tau ) is the elliptic modular function, then one can explicitly evaluate the value k ( τ ) k(\tau ) , and the other is that once the value k ( τ ) k(\tau ) is given, we can obtain the value k ( r τ ) k(r\tau ) for any positive rational number r immediately.

scholarly journals Pseudo almost automorphic solutions of class r in α-norm under the light of measure theory

2020 ◽  
Vol 7 (1) ◽  
pp. 81-101
Author(s):  
Issa Zabsonre ◽  
Djendode Mbainadji
Keyword(s):  
Phase Space ◽  
Spectral Decomposition ◽  
Measure Theory ◽  
New Approach ◽  
Almost Automorphic ◽  
Almost Automorphic Functions ◽  
Pseudo Almost Automorphic ◽  
Automorphic Functions

AbstractUsing the spectral decomposition of the phase space developed in Adimy and co-authors, we present a new approach to study weighted pseudo almost automorphic functions in the α-norm using the measure theory.

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