Further properties of a function of Ogg and Ligozat
Abstract
Certain identities of Ramanujan may be succinctly expressed in terms of the rational function wN(g) = wN(f) - 1/wN(f) on the modular curve X0(N), where f is a certain modular unit on the Nebentypus cover X(N) introduced by Ogg and Ligozat for N prime congruent to 1 (mod 4) and wN is the Fricke involution. These correspond to levels N = 5, 13, where the genus of X0(N) is zero. In this paper we produce some analogs of these identities for each wN(g) such that X0(N) has genus 1, 2, and also for each h = g + wN(g) such that the Atkin-Lehner quotient X0+(N) has genus 1, 2. We also found that if n is the degree of the field of definition F of the non-trivial zeros of the latter, then the degree of the normal closure of F over Q is the n-th solution of Singmaster's Problem.
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