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
  • Degree
    Master of Science
  • DOI
  • License
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