Eigenvectors and Eigenvalues of $p$-Curvature Operators for KZ--type Differential Connections

A KZ-type connection in characteristic $p$ has $p$-curvature operators. These operators are commuting endomorphisms of the connection. We present a Bethe-ansatz-type construction of their eigenvectors and eigenvalues. The eigenvector is constructed as the value of a characteristic $p$ hypergeometric integral and extends to a flat eigensection over the Frobenius neighborhood of the corresponding Bethe point.