On p-adic solutions to KZ equations, ordinary crystals, and pˢ-hypergeometric solutions

We consider the KZ connection associated with a family of hyperelliptic curves of genus g over the ring of p-adic integers Zₚ. 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 Zₚ-module of rank g. We also consider the reduction modulo pˢ of the unit root subcrystal for any s≥ 1. We prove that its annihilator is generated by the so-called pˢ-hypergeometric flat sections of the KZ connection. In particular, that means that the reduction modulo pˢ of an arbitrary local flat section of the KZ connection over Zₚ is a linear combination of the pˢ-hypergeometric flat sections.