Further properties of a function of Ogg and Ligozat

Abstract

Certain identities of Ramanujan may be succinctly expressed in terms of the rational function w_N(g) = w_N(f) - 1/w_N(f) on the modular curve X_0(N), where f is a certain modular unit on the Nebentypus cover X_χ(N) introduced by Ogg and Ligozat for N prime congruent to 1 (mod 4) and w_N is the Fricke involution. These correspond to levels N = 5, 13, where the genus of X_0(N) is zero. In this paper we produce some analogs of these identities for each w_N(g) such that X_0(N) has genus 1, 2, and also for each h = g + w_N(g) such that the Atkin-Lehner quotient X_0+(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.