Galois Groups of D-finite Series modulo Primes

A power series with rational number coefficients is called D-finite if it satisfies a linear differential equation with polynomial coefficients over the rationals. Taking the reduction of such a D-finite power series modulo a prime number, one often obtains an algebraic power series. For example, for diagonals of multivariate rational functions this was observed by Furstenberg in 1967, and for hypergeometric functions this can be deduced from results by Christol, and was made precise in recent work by Vargas-Montoya. In these cases, one can consider the (usual, algebraic) Galois group of the reduction over the field of rational functions.

In this talk, I will showcase many examples of D-finite series of different nature for which we are (almost) able to compute said Galois groups for all prime numbers for which they are defined. I will then collect these observations to raise questions on the general behavior of these groups: Is there some uniformity of the Galois groups of the reductions of a given D-finite series with respect to the prime number, and does there exist an object in characteristic zero governing their behavior, like the differential Galois group of the minimal equation for the series? Much like the Grothendieck p-curvature conjecture, we are looking for a local-global principle, but we are concerned with only a single solution of a differential equation, instead of a full basis.

This talk is based on joint work with X. Caruso and D. Vargas-Montoya.