Further Properties of a Function of Ogg and Ligozat

Certain identities of Ramanujan may be succinctly expressed in terms of the rational function ğχ = f̆χ − 1 f̆χ on the modular curve X0(N), where f̆χ = wNfχ and fχ is a certain modular unit on the Nebentypus cover Xχ(N) introduced by Ogg and Ligozat for prime N ≡ 1 (mod 4) and wN is the Fricke involution. These correspond to levels N = 5, 13, where the genus gN of X0(N) is zero. In this paper we study a slightly more general kind of relations for each ğχ such that X0(N) has genus gN = 1, 2, and also for each hχ = gχ + ğχ such that the Atkin-Lehner quotient X 0 (N) has genus g + N = 1, 2. It turns out 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.