Seminar

Consider a vector bundle over P^1_K equipped with a meromorphic connection, for some number field K. When this data arises from the de Rham realization of some family of motives (i.e., as a Gauss-Manin connection), for all but finitely many primes p this connection admits a p-adic analytic Frobenius structure. A priori, these structures are defined in incompatible categories and so have nothing to do with each other. The purpose of the talk is to speculate on the extent to which these structures "commute" in some meaningful way, and to give a piece of hard evidence (from the theory of hypergeometric differential equations) that we find very suggestive.