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

Abstract

AbstractLet {j(z)} be the modular j-invariant function. Let τ be an algebraic number in the complex upper half plane {\mathbb{H}}. It was proved by Schneider and Siegel that if τ is not a CM point, i.e., {[\mathbb{Q}(\tau):\mathbb{Q}]\neq 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 τ. 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\infty} are algebraic, and that f has its poles only at cusps equivalent to {i\infty}. We prove, under a mild assumption on f, that, for any fixed τ, if N is a prime such that {N\geq 23} and {N\notin\{23,29,31,41,47,59,71\}}, then {f(T_{m}.\tau)} are transcendental for infinitely many positive integers m prime to N.