Usage
  • 27 views
  • 25 downloads

Unbounded Denominators for Hypergeometric Functions and Modular Forms

  • Author / Creator
    Bernstein, Tobias
  • Atkin and Swinnerton-Dyer conjectured a simple characterization of those Fuchsian groups whose modular forms have integral Fourier coefficient. It has a natural and far-reaching generalization, which we will call the vASD conjecture, to vector-valued modular forms. We confirm vASD conjecture for all 1-dimensional multipliers of $\Gamma(2)$, and set the stage to test it for higher dimensions for $\Gamma(2)$ and other Fuchsian groups. In order to do so, we investigate the similar question for hypergeometric functions, namely when the denominators of its coefficients are unbounded. We do this using $p$-adic methods, checking when the coefficients are $p$-adically unbounded for a given $p$. We generalize the results of \cite{base} for the standard hypergeometric function $_2F_1$ to the generalized hypergeometric function $_nF_{n-1}$ with rational parameters. In particular, we provide a necessary and sufficient condition for a given prime $p$, applicable to all but finitely many primes, which determines when its coefficients are $p$-adically unbounded; these are equivalent but different to the conditions found earlier by Dwork and by Christol. Also, we show that the results from \cite{base} concerning when the density of unbounded primes is 0 or 1 respectively extend to the case of $_nF_{n-1}$, and strengthen each slightly. We additionally show that the structure of the set of unbounded primes from the $_2F_1$ case extends to the $_nF_{n-1}$ case. We end with a discussion of modular forms and a brief overview of how the work on hypergeometric functions will apply to the vASD conjecture.

  • Subjects / Keywords
  • Graduation date
    Spring 2019
  • Type of Item
    Thesis
  • Degree
    Master of Science
  • DOI
    https://doi.org/10.7939/r3-gj2r-gp17
  • License
    Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.