Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-22T05:12:26.523Z Has data issue: false hasContentIssue false
  • English
  • Français

Fourier Coefficients of Vector-valued Modular Forms of Dimension 2

Published online by Cambridge University Press:  20 November 2018

Cameron Franc  and
Geoffrey Mason
Cameron Franc
Affiliation:
Department of Mathematics, University of California, Santa Cruz e-mail: [email protected]@ucsc.edu
Geoffrey Mason
Affiliation:
Department of Mathematics, University of California, Santa Cruz e-mail: [email protected]@ucsc.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove the following theorem. Suppose that $F\,=\,\left( {{f}_{1}},\,{{f}_{2}} \right)$ is a 2-dimensional, vector-valued modular form on $\text{S}{{\text{L}}_{2}}\left( \mathbb{Z} \right)$ whose component functions ${{f}_{1}}$, ${{f}_{2}}$ have rational Fourier coefficients with bounded denominators. Then ${{f}_{1}}$ and ${{f}_{2}}$ are classical modular forms on a congruence subgroup of the modular group.

Keywords

11F4111G99vector-valued modular formmodular groupbounded denominators
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 57 , Issue 3 , 01 September 2014 , pp. 485 - 494
Copyright
Copyright © Canadian Mathematical Society 2014

References

[1] Anderson, Greg and Moore, Greg, Rationality in conformal field theory. Comm. Math. Phys. 117 (1988), 441–450. http://dx.doi.org/10.1007/BF01223375 Google Scholar
[2] Bantay, Peter and Gannon, Terry, Vector-valued modular functions for the modular group and the hypergeometric equation. Commun. Number Theory Phys. 1 (2007), 651–680. http://dx.doi.org/10.4310/CNTP.2007.v1.n4.a2 Google Scholar
[3] Kaneko, Masanobu and Koike, Masao, On modular forms arising from a differential equation of hypergeometric type. Ramanujan J. 7 (2003), 145–164. http://dx.doi.org/10.1023/A:1026291027692 Google Scholar
[4] Kaneko, Masanobu and Zagier, Don, Supersingular j-invariants, hypergeometric series, and Atkin’s orthogonal polynomials. In: Computational perspectives on number theory (Chicago, IL, 1995), AMS/IP Stud. Adv. Math. 7, Amer. Math. Soc., Providence, RI, 1998, 97–126.Google Scholar
[5] Marks, Christopher and Mason, Geoffrey, Structure of the module of vector-valued modular forms. J. London Math. Soc. (2) 82 (2010), 32–48. http://dx.doi.org/10.1112/jlms/jdq020 Google Scholar
[6] Mason, Geoffrey, 2-dimensional vector-valued modular forms. Ramanujan J. 17 (2008), 405–427. http://dx.doi.org/10.1007/s11139-007-9054-4 Google Scholar
[7] On the Fourier coefficients of 2-dimensional vector-valued modular forms. Proc. Amer. Math. Soc. 140 (2012), 1921–1930. http://dx.doi.org/10.1090/S0002-9939-2011-11098-0 Google Scholar
[8] Tsutsumi, Hiroyuki, Modular differential equations of second order with regular singularities at elliptic points for SL2(Z). Proc. Amer. Math. Soc. 134 (2006), 931–941. http://dx.doi.org/10.1090/S0002-9939-05-08115-3 Google Scholar