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

TL;DR

This paper establishes Dwork-type congruences for Hasse--Witt matrices linked to Laurent polynomials and explores their implications for the arithmetic and $p$-adic properties of solutions to KZ equations, revealing invariant subbundles in specific hyperelliptic cases.

Contribution

It introduces new Dwork-type congruences for Hasse--Witt matrices and applies them to analyze $p$-adic properties of polynomial solutions of KZ equations, uncovering invariant subbundles.

Findings

Proved general Dwork-type congruences for Hasse--Witt matrices.

Established $p$-adic analytic properties of solutions to KZ equations.

Identified invariant subbundles in $p$-adic KZ connections for hyperelliptic curves.

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, 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 $y^2=(t-z_1)\dots (t-z_{2g+1})$ 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.