On unbounded denominators and hypergeometric series

We study the question of when the coefficients of a hypergeometric series are p-adically unbounded for a given rational prime p. Our first 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 (equivalent but different conditions were found earlier by Dwork in [12] and Christol in [9]). We then show that the set of unbounded primes for a given series is, up to a finite discrepancy, a finite union of the set of primes in certain arithmetic progressions and we explain how this set can be computed. We characterize when the density of the set of unbounded primes is 0 (a similar result is found in [9]), and when it is 1. Finally, we discuss the connection between this work and the unbounded denominators conjecture for Fourier coefficients of modular forms.