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 journalArticlepeer-review

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 Γ0(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
Pages (from-to)139-150
Number of pages12
JournalForum Mathematicum
Volume32
Issue number1
DOIs
Publication statusPublished - 2020 Jan 1

Bibliographical note

Publisher Copyright:
© 2020 De Gruyter. All rights reserved.

Keywords

  • CM point
  • Harmonic weak Maass form
  • meromorphic differential

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics

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