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

  • Dohoon Choi und Subong Lim ORCID logo EMAIL logo
Veröffentlicht/Copyright: 13. September 2019
Forum Mathematicum
Aus der Zeitschrift Forum Mathematicum Band 32 Heft 1

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.

Keywords: Harmonic weak Maass form; CM point; meromorphic differential
MSC 2010: 11F03; 11F25

Communicated by Jan Bruinier

Acknowledgements

The authors appreciate the referee for helpful comments.

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Received: 2018-12-04
Revised: 2019-05-26
Published Online: 2019-09-13
Published in Print: 2020-01-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. On the distribution of zeros of derivatives of the Riemann ξ-function
  3. Representations of constant socle rank for the Kronecker algebra
  4. Abstract bivariant Cuntz semigroups II
  5. Curves on Segre threefolds
  6. 𝒩(p, q, s)-type spaces in the unit ball of ℂn(II): Carleson measure and its application
  7. On a class of critical Robin problems
  8. On the description of multidimensional normal Hausdorff operators on Lebesgue spaces
  9. Spectral asymptotics for Krein–Feller operators with respect to 𝑉-variable Cantor measures
  10. Schneider–Siegel theorem for a family of values of a harmonic weak Maass form at Hecke orbits
  11. Modified energy method and applications for the well-posedness for the higher-order Benjamin–Ono equation and the higher-order intermediate long wave equation
  12. Some remarks on products of sets in the Heisenberg group and in the affine group
  13. Codimension growth of central polynomials of Lie algebras
  14. Unramified Whittaker functions for certain Brylinski–Deligne covering groups
  15. Tilting classes over commutative rings
https://doi.org/10.1515/forum-2018-0295
Schlagwörter für diesen Artikel
Harmonic weak Maass form; CM point; meromorphic differential

Artikel in diesem Heft

  1. Frontmatter
  2. On the distribution of zeros of derivatives of the Riemann ξ-function
  3. Representations of constant socle rank for the Kronecker algebra
  4. Abstract bivariant Cuntz semigroups II
  5. Curves on Segre threefolds
  6. 𝒩(p, q, s)-type spaces in the unit ball of ℂn(II): Carleson measure and its application
  7. On a class of critical Robin problems
  8. On the description of multidimensional normal Hausdorff operators on Lebesgue spaces
  9. Spectral asymptotics for Krein–Feller operators with respect to 𝑉-variable Cantor measures
  10. Schneider–Siegel theorem for a family of values of a harmonic weak Maass form at Hecke orbits
  11. Modified energy method and applications for the well-posedness for the higher-order Benjamin–Ono equation and the higher-order intermediate long wave equation
  12. Some remarks on products of sets in the Heisenberg group and in the affine group
  13. Codimension growth of central polynomials of Lie algebras
  14. Unramified Whittaker functions for certain Brylinski–Deligne covering groups
  15. Tilting classes over commutative rings
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