On algebraicity and special values of hypergeometric functions

Hypergeometric functions enjoy the following remarkable property: while they are usually transcendental in characteristic zero, their reductions modulo p, when they exist, are always algebraic.

In this talk, I will investigate, for certain families $h(x)$ of 2F1, the variation of the Galois groups of $h(x) \bmod p$ when $p$ varies. If time permits, I will also explain how algebraic equations can help to assign an arithmetic-relevant value to $h(1)$ and compare it with its actual value when $h$ is viewed as a complex function.