Hypergeometric Integrals Modulo p and Hasse–Witt Matrices

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 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 \({\mathbb {C}}\).