SIGMA 22 (2026), 001, 19 pages      arXiv:2503.06326      https://doi.org/10.3842/SIGMA.2026.001

Finding All Solutions of qKZ Equations in Characteristic $p$

Evgeny Mukhin ^a and Alexander Varchenko ^b
a) 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

Received September 23, 2025, in final form December 16, 2025; Published online January 02, 2026

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.

Key words: qKZ equations; $p$-hypergeometric solutions; orthogonality relations; $p$-curvature.

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