Dynamical and qKZ equations modulo $p^s$, an example

TL;DR

This paper studies a specific system of dynamical and qKZ equations related to elliptic integrals, solving them modulo prime powers and analyzing their p-adic limits to understand invariant line bundles.

Contribution

It provides an explicit example of solving dynamical and qKZ equations modulo prime powers and explores the p-adic limit behavior of solutions and associated line bundles.

Findings

Solutions modulo $p^n$ are explicitly constructed.

p-adic limits of solutions define invariant line bundles.

Line bundles are invariant under both dynamical and qKZ connections.

Abstract

We consider an example of the joint system of dynamical differential equations and qKZ difference equations with parameters corresponding to equations for elliptic integrals. We solve this system of equations modulo any power $p^n$ of a prime integer $p$. We show that the $p$-adic limit of these solutions as $n\to\infty$ determines a sequence of line bundles, each of which is invariant with respect to the corresponding dynamical connection, and that sequence of line bundles is invariant with respect to the corresponding qKZ difference connection.