Finding All Solutions of qKZ Equations in Characteristic $p$

Evgeny Mukhin^a, Alexander Varchenko<sup>b

a</sup> Department of Mathematical Sciences, Indiana University Indianapolis, 402 North Blackford St, Indianapolis, IN 46202-3216, USA
^b Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA

Abstract: In [J. Lond. Math. Soc. 109 (2024), e12884, 22 pages], the difference qKZ equations were considered modulo a prime number $p$ and a family of polynomial solutions of the qKZ equations modulo $p$ was constructed by an elementary procedure as suitable $p$-approximations of the hypergeometric integrals. In this paper, we study in detail the first family of nontrivial examples of the qKZ equations in characteristic $p$. We describe all solutions of these qKZ equations in characteristic $p$ by demonstrating that they all stem from the $p$-hypergeometric solutions. We also prove a Lagrangian property (called the orthogonality property) of the subbundle of the qKZ bundle spanned by the $p$-hypergeometric sections. This paper extends the results of [arXiv:2405.05159] on the differential KZ equations to the difference qKZ equations.

Keywords: qKZ equations, $p$-hypergeometric solutions, orthogonality relations, $p$-curvature.

MSC: 11D79, 12H25, 32G34, 33C05, 33E30

Received: September 23, 2025; in final form December 16, 2025; Published online January 2, 2026

Language: English

DOI: 10.3842/SIGMA.2026.001

ArXiv: 2503.06326