### Bounded primes for hypergeometric series

Heilbronn Number Theory Seminar

9th May 2018, 4:00 pm – 5:00 pm

Howard House, 4th Floor Seminar Room

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.

## Comments are closed.