Dwork-type congruences and $p$-adic KZ connection

TL;DR

This paper investigates the $p$-adic KZ connection for a family of algebraic curves, revealing an invariant subbundle structure absent in the complex case, through novel Dwork-type congruences for Hasse--Witt matrices.

Contribution

It introduces new Dwork-type congruences that establish the existence of invariant subbundles in the $p$-adic KZ connection, contrasting with the irreducibility in the complex setting.

Findings

The $p$-adic KZ connection has an invariant subbundle of rank $g$.

The complex KZ connection has no nontrivial proper subbundles.

New Dwork--type congruences are developed for Hasse--Witt matrices.

Abstract

We show that the $p$-adic KZ connection associated with the family of curves $y^q=(t-z_1)\dots (t-z_{qg+1})$ has an invariant subbundle of rank $g$, while the corresponding complex KZ connection has no nontrivial proper subbundles due to the irreducibility of its monodromy representation. The construction of the invariant subbundle is based on new Dwork--type congruences for associated Hasse--Witt matrices.