Modular equations and the genus zero property of moonshine functions

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

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

Canadian Mathematical Bulletin ◽

10.4153/cmb-2002-004-6 ◽

2002 ◽

Vol 45 (1) ◽

pp. 36-45 ◽

Cited By ~ 1

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.

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Some modular equations analogous to Ramanujan’s identities

Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas ◽

10.1007/s13398-021-01002-w ◽

2021 ◽

Vol 115 (2) ◽

Author(s):

H. M. Srivastava ◽

B. R. Srivatsa Kumar ◽

R. Narendra

Keyword(s):

Modular Equations

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

Open Mathematics ◽

10.1515/math-2020-0105 ◽

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.

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