Schneider-Siegel theorem for a family of values of a harmonic weak Maass form at Hecke orbits

Abstract

Let j(z) be the modular j-invariant function. Let tau be an algebraic number in the complex upper half plane H. It was proved by Schneider and Siegel that if tau is not a CM point, i.e., [Q(tau) : Q] not equal 2, then j(tau) is transcendental. Let f be a harmonic weak Maass form of weight 0 on Gamma(0)(N). In this paper, we consider an extension of the results of Schneider and Siegel to a family of values of f on Hecke orbits of tau. For a positive integer m, let T-m denote the m-th Hecke operator. Suppose that the coefficients of the principal part of f at the cusp i infinity are algebraic, and that f has its poles only at cusps equivalent to i infinity. We prove, under a mild assumption on f, that, for any fixed tau, if N is a prime such that N >= 23 and N is not an element of (23, 29, 31, 41, 47, 59, 71}, then f(T-m.tau) are transcendental for infinitely many positive integers m prime to N.