Ghosts and congruences for $p^s$-approximations of hypergeometric periods

TL;DR

This paper establishes Dwork-type congruences for Laurent polynomial constant terms and explores their implications for hypergeometric and KZ equations, revealing new p-adic properties and structural differences between p-adic and complex solutions.

Contribution

It introduces general Dwork-type congruences for Laurent polynomial constant terms and applies them to analyze p-adic properties of hypergeometric and KZ solutions.

Findings

Proves Dwork-type congruences for Laurent polynomial constant terms

Shows p-adic KZ connection has an invariant line subbundle

Demonstrates differences in subbundle structures between p-adic and complex cases

Abstract

We prove general Dwork-type congruences for constant terms 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 hypergeometric and KZ equations, solutions which come as coefficients of master polynomials and whose coefficients are integers. As an application we show that the simplest example of a $p$-adic KZ connection has an invariant line subbundle while its complex analog has no nontrivial subbundles due to the irreducibility of the monodromy group.