Congruences for Hasse-Witt matrices and solutions of $p$-adic KZ equations

Abstract

We prove general Dwork-type congruences for Hasse–Witt matrices attached to tuples of Laurent polynomials.We apply this result to establishing arithmetic and p-adic analytic properties of functions originating from polynomial solutions modulo $p^s$ of Knizhnik–Zamolodchikov (KZ) equations, the solutions which come as coefficients of master polynomials and whose coefficients are integers. As an application we show that the $p$-adic KZ connection associated with the family of hyperelliptic curves $y^2 = (t - z_1) \dotsc (t - z_{2g+1})$ has an invariant subbundle of rank $g$. Notice that the corresponding complex KZ connection has no nontrivial subbundles due to the irreducibility of its monodromy representation.