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

Dohoon Choi, Subong Lim

Research output: Contribution to journalArticle

Abstract

Let j(z) be the modular j-invariant function. Let τ be an algebraic number in the complex upper half plane H. It was proved by Schneider and Siegel that if τ is not a CM point, i.e., [Q(τ) : Q] = 2, then j(τ) is transcendental. Let f be a harmonic weak Maass form of weight 0 on ?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 Tm denote the m-th Hecke operator. Suppose that the coefficients of the principal part of f at the cusp i∞ are algebraic, and that f has its poles only at cusps equivalent to i∞. We prove, under a mild assumption on f , that, for any fixed τ, if N is a prime such that N ≥ 23 and N ? {23, 29, 31, 41, 47, 59, 71}, then f(Tm.τ) are transcendental for infinitely many positive integers m prime to N.

Original languageEnglish
JournalForum Mathematicum
DOIs
Publication statusAccepted/In press - 2019 Jan 1

Keywords

  • CM point
  • Harmonic weak Maass form
  • meromorphic differential

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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