Hasse invariants and mod p solutions of A-hypergeometric systems

Igusa noted that the Hasse invariant of the Legendre family of elliptic curves over a finite field of odd characteristic is a solution mod p of a Gaussian hypergeometric equation. We show that any family of exponential sums over a finite field has a Hasse invariant which is a sum of products of mod p solutions of A-hypergeometric systems.