Cameron Franc, Terry Gannon, Geoffrey Mason
Published 2017-08-14Version 1
We study the question of when the coefficients of a hypergeometric series are p-adically unbounded for a given rational prime p. Our first main result is a necessary and sufficient criterion (applicable to all but finitely many primes) for determining when the coefficients of a hypergeometric series with rational parameters are p-adically unbounded. This criterion is then used to show that the set of unbounded primes for a given series is, up to a finite discrepancy, a finite union of primes in arithmetic progressions. This set can be computed explicitly. We characterize when the density of the set of unbounded primes is 0, and when it is 1. Finally, we discuss the connection between this work and the unbounded denominators conjecture concerning Fourier coefficients of modular forms.
Modular Forms and Certain ${}_2F_1(1)$ Hypergeometric Series
On the $p$-integrality of $A$-hypergeometric series
On integrality properties of hypergeometric series
![QR code for arXiv:1708.04213 [math.NT]](http://oss.arxitics.com/images/favicon.svg)