Cameron Franc

Bounded primes for hypergeometric series

A power series with rational coefficients is said to be bounded
at a rational prime p if its coefficients are bounded in the p-adic
topology. The question of when a hypergeometric series with rational
parameters is bounded at a prime p was studied by Dwork and Christol in
the 1970s and 1980s. Recently we revisited this question in joint work
with Terry Gannon and Geoff Mason. We showed that the set of bounded
primes for a fixed hypergeometric series has a Dirichlet density. In fact,
with finitely many exceptions, this set is a union of sets of primes in
arithmetic progressions. Recently (in joint work with a class of
undergraduates at the University of Saskatchewan) we found an efficient
formula for computing this density, and we have used this formula to
explore the generic global behaviour of the density of bounded primes for
hypergeometric series. Unsurprisingly, the density of bounded primes
appears to be quite small in general. In line with this we have
established an upper bound on the density of bounded primes for certain
specialisations of hypergeometric parameters. In this talk we will report
on these results.