Eigenvectors of $p$-Curvature for Geometric Difference Equations

Abstract

A qKZ-type additive discrete flat connection in characteristic $p$ has $p$-curvature operators. They are commuting automorphisms of the connection. We present a Bethe-ansatz-type construction of eigensections and eigenvalues of the $p$-curvature operators. An eigensection of $p$-curvature operators is a discrete hypergeometric sum over the finite lattice ${\mathbb{Z}}^r/p{\mathbb{Z}}^r$. Thus, the $p$-curvature eigensections are constructed by a finite, discrete analogue of a hypergeometric integral.

Latex, 31 pages