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Ramanujan’s modular equations and Atkin-Lehner involutions. (English) Zbl 0922.11040

This paper concerns a set of identities of which the following is the simplest:

-Q+1 Q=P+8 P,

where

P=η 3 (τ)η 3 (3τ) η 3 (2τ)η 3 (6τ)andQ=η 6 (τ)η 6 (6τ) η 6 (2τ)η 6 (3τ)·

Here η is the Dedekind η-function. The paper offers about a dozen identities of this shape, together with a unified method of proof, given in §4: “Let σ be a modular function with invariance group A such that the genus g(A * )0. If there exists a group G such that m:=[G:A]<, g(G * )=0, and AG, then we always have an identity of the form

i=1 m σ| g i =n(f) d(f),

where G= i=1 m g i A, n(x) and d(x) are both polynomials in x, and f generates the function field of G * . To determine d(f), we first set :=polesof i=1 m σ| g i {polesoff}. Since by assumption, f is a bijection from G{}, we conclude that

d(x)= p (x-f(p)) e p ,

where

e p =-orderof( i=1 n σ| g i )atp orderof(f-f(p))atp·

Note that d(x) is defined to be 1 if =φ. The polynomial n(x) can then be determined by comparing the Fourier expansions of d(f) i=1 m σ| g i and f at .”

In practice, G is an extension of a level N congruence subgroup by an Atkin-Lehner involution N e =aebcNde, where eN. This new method of proof yields some modular equations beyond those of Ramanujan.

The final section lists “all genus 0 discrete groups Γ, Γ 0 (N)ΓΓ 0 (N)+, where the Γ 0 (N)+ is generated by Γ 0 (N) together with all its Atkin-Lehner involutions”.


MSC:
11F11Holomorphic modular forms of integral weight
11F03Modular and automorphic functions
11F20Dedekind eta function, Dedekind sums
References:
[1]B. C. Berndt,Ramanujan Notebooks Part III, Springer-Verlag, New York, 1991.
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[11]S. Ramanujan,Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957.