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Integrality properties of the CM-values of certain weak Maass forms

  • Eric Larson EMAIL logo and Larry Rolen
Published/Copyright: March 1, 2013
Forum Mathematicum
From the journal Forum Mathematicum Volume 27 Issue 2

Abstract

In a recent paper, Bruinier and Ono prove that the coefficients of certain weight -1/2 harmonic Maass forms are traces of singular moduli for weak Maass forms. In particular, for the partition function p(n), they prove that

p(n)=124n-1·∑Pp(αQ),

where Pp is a weak Maass form and αQ ranges over a finite set of discriminant -24n+1 CM points. Moreover, they show that 6·(24n-1)·Pp(αQ) is always an algebraic integer, and they conjecture that (24n-1)·Pp(αQ) is always an algebraic integer. Here we prove a general theorem which implies this conjecture as a corollary.

Keywords: Singular moduli; partition function; Maass forms
MSC: 11F37; 11P99

Funding source: NSF

Award Identifier / Grant number: Emory 2011 REU

The authors are grateful for the support of the NSF in funding the Emory 2011 REU. The authors would like to thank our advisor Ken Ono for his guidance, useful conversations, improving the quality of exposition of this article, and hosting the REU.

Received: 2012-8-3
Revised: 2012-12-2
Published Online: 2013-3-1
Published in Print: 2015-3-1

© 2015 by De Gruyter

Articles in the same Issue

  1. Frontmatter
  2. Near abelian profinite groups
  3. On Calabi–Yau threefolds associated to a web of quadrics
  4. On the partial fraction decomposition of the restricted partition generating function
  5. Projectively flat Finsler manifolds with infinite dimensional holonomy
  6. Weighted version of Carleson measure and multilinear Fourier multiplier
  7. Injective objects and retracts of Fraïssé limits
  8. On hypersurfaces containing projective varieties
  9. Additive reducts of algebraically closed valued fields
  10. On the subconvexity problem for GL(3) × GL(2) L-functions
  11. An algebraic property of Hecke operators and two indefinite theta series
  12. Sum formula of multiple Hurwitz-zeta values
  13. Classification of homogeneous holomorphic hermitian principal bundles over G/K
  14. Integrality properties of the CM-values of certain weak Maass forms
  15. Heat kernel for fractional diffusion operators with perturbations
  16. New class of multiple weights and new weighted inequalities for multilinear operators
  17. Balanced Hermitian geometry on 6-dimensional nilmanifolds
  18. Extensions of tame algebras and finite group schemes of domestic representation type
  19. Weighted estimates for multilinear Fourier multipliers
  20. Cluster tilting vs. weak cluster tilting in Dynkin type A infinity
  21. From ring epimorphisms to universal localisations
  22. Brauer algebras of type B
  23. Existence problems for the p-Laplacian
  24. Averaging operators over nondegenerate quadratic surfaces in finite fields
  25. Lawson homology for abelian varieties
https://doi.org/10.1515/forum-2012-0112
Keywords for this article
Singular moduli; partition function; Maass forms

Articles in the same Issue

  1. Frontmatter
  2. Near abelian profinite groups
  3. On Calabi–Yau threefolds associated to a web of quadrics
  4. On the partial fraction decomposition of the restricted partition generating function
  5. Projectively flat Finsler manifolds with infinite dimensional holonomy
  6. Weighted version of Carleson measure and multilinear Fourier multiplier
  7. Injective objects and retracts of Fraïssé limits
  8. On hypersurfaces containing projective varieties
  9. Additive reducts of algebraically closed valued fields
  10. On the subconvexity problem for GL(3) × GL(2) L-functions
  11. An algebraic property of Hecke operators and two indefinite theta series
  12. Sum formula of multiple Hurwitz-zeta values
  13. Classification of homogeneous holomorphic hermitian principal bundles over G/K
  14. Integrality properties of the CM-values of certain weak Maass forms
  15. Heat kernel for fractional diffusion operators with perturbations
  16. New class of multiple weights and new weighted inequalities for multilinear operators
  17. Balanced Hermitian geometry on 6-dimensional nilmanifolds
  18. Extensions of tame algebras and finite group schemes of domestic representation type
  19. Weighted estimates for multilinear Fourier multipliers
  20. Cluster tilting vs. weak cluster tilting in Dynkin type A infinity
  21. From ring epimorphisms to universal localisations
  22. Brauer algebras of type B
  23. Existence problems for the p-Laplacian
  24. Averaging operators over nondegenerate quadratic surfaces in finite fields
  25. Lawson homology for abelian varieties
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