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

2024/06/27 by Varchenko, Alexander, Vologodsky, Vadim · 1 citation
#Algebraic Geometry (math.AG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2406.19318

Abstract

We consider the KZ connection associated with a family of hyperelliptic curves of genus g over the ring of p-adic integers 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 ℤ_p-module of rank g. We also consider the reduction modulo p^s of the unit root subcrystal for any s≥ 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 ℤ_p is a linear combination of the p^s-hypergeometric flat sections.

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• p-curvature operators and Satake-type phenomenon for \fraksl₂ KZ equations with κ=± 2

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