Hypergeometric Integrals Modulo $p$ and Hasse--Witt Matrices

Abstract

We consider the KZ differential equations over $\mathbb{C}$ in the case, when the hypergeometric solutions are one-dimensional integrals. We also consider the same differential equations over a finite field ${\mathbb{F}}_p$. We study the space of polynomial solutions of these differential equations over ${\mathbb{F}}_p$, constructed in a previous work by V. Schechtman and the second author. Using Hasse-Witt matrices we identify the space of these polynomial solutions over ${\mathbb{F}}_p$ with the space dual to a certain subspace of regular differentials on an associated curve. We also relate these polynomial solutions over ${\mathbb{F}}_p$ and the hypergeometric solutions over $\mathbb{C}$.

Latex, 36 pages, v2: tex presentation is improved, v3: Introduction is edited