Hasse invariants and mod $p$ solutions of $A$-hypergeometric systems

TL;DR

This paper demonstrates that Hasse invariants of exponential sum families over finite fields can be expressed as sums of products of mod p solutions of A-hypergeometric systems, linking algebraic invariants to hypergeometric solutions.

Contribution

It establishes a general connection between Hasse invariants and solutions of A-hypergeometric systems for exponential sums over finite fields.

Findings

Hasse invariants can be written as sums of products of mod p solutions.

The approach generalizes Igusa's observation for elliptic curves.

Provides a new perspective on the algebraic structure of exponential sums.

Abstract

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.