TY - JOUR

T1 - Schneider-Siegel theorem for a family of values of a harmonic weak Maass form at Hecke orbits

AU - Choi, Dohoon

AU - Lim, Subong

N1 - Publisher Copyright:
© 2019 Walter de Gruyter GmbH, Berlin/Boston.

PY - 2019

Y1 - 2019

N2 - 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.

AB - 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.

KW - CM point

KW - Harmonic weak Maass form

KW - meromorphic differential

UR - http://www.scopus.com/inward/record.url?scp=85072600654&partnerID=8YFLogxK

U2 - 10.1515/forum-2018-0295

DO - 10.1515/forum-2018-0295

M3 - Article

AN - SCOPUS:85072600654

JO - Forum Mathematicum

JF - Forum Mathematicum

SN - 0933-7741

ER -