## 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 Γ_{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∞ 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(T_{m}.τ) are transcendental for infinitely many positive integers m prime to N.

Original language | English |
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Pages (from-to) | 139-150 |

Number of pages | 12 |

Journal | Forum Mathematicum |

Volume | 32 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2020 Jan 1 |

## Keywords

- CM point
- Harmonic weak Maass form
- meromorphic differential

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics