On $p$-adic solutions to KZ equations, ordinary crystals, and $p^s$-hypergeometric solutions

TL;DR

This paper studies the $p$-adic solutions to KZ equations associated with hyperelliptic curves, revealing the structure of their flat sections and hypergeometric solutions in the context of ordinary crystals and reductions modulo powers of $p$.

Contribution

It establishes the relationship between flat sections of the KZ connection and $p^s$-hypergeometric solutions, and describes the structure of the unit root subcrystal over $Z_p$ and its reductions.

Findings

The Gauss-Manin connection has an ordinary $F$-crystal structure.

All local flat sections annihilate the unit root subcrystal.

The space of flat sections is a free $Z_p$-module of rank $g$.

Abstract

We consider the KZ connection associated with a family of hyperelliptic curves of genus $g$ over the ring of $p$-adic integers $\mathbb{Z}_p$. Then the dual connection is the Gauss-Manin connection of that family. We observe that the Gauss-Manin connection has an ordinary $F$-crystal structure and its unit root subcrystal is of rank $g$. We prove that all local flat sections of the KZ connection annihilate the unite root subcrystal, and the space of all local flat sections of the KZ connection is a free $\mathbb{Z}_p$-module of rank $g$. We also consider the reduction modulo $p^s$ of the unit root subcrystal for any $s\geq 1$. We prove that its annihilator is generated by the so-called $p^s$-hypergeometric flat sections of the KZ connection. In particular, that means that the reduction modulo $p^s$ of an arbitrary local flat section of the KZ connection over $\mathbb{Z}_p$ is a linear combination of the $p^s$-hypergeometric flat sections.