Hyperelliptic integrals modulo $p$ and Cartier–Manin matrices

Alexander Varchenko (Department of Mathematics, University of North Carolina, Chapel Hill, N.C., U.S.A.; and Faculty of Mathematics and Mechanics, Lomonosov Moscow State University, Moscow, Russia)

Abstract

The hypergeometric solutions of the KZ equations were constructed almost 30 years ago. The polynomial solutions of the KZ equations over the finite field $\mathbb{F}_p$ with a prime number $p$ of elements were constructed only recently. In this paper we consider an example of the KZ equations whose hypergeometric solutions are given by hyperelliptic integrals of genus $g$. It is known that in this case the total $2g$-dimensional space of holomorphic (multivalued) solutions is given by the hyperelliptic integrals. We show that the recent construction of the polynomial solutions over the field $\mathbb{F}_p$ in this case gives only a $g$-dimensional space of solutions, that is, a “half” of what the complex analytic construction gives. We also show that all the constructed polynomial solutions over the field $\mathbb{F}_p$ can be obtained by reduction modulo $p$ of a single distinguished hypergeometric solution. The corresponding formulas involve the entries of the Cartier–Manin matrix of the hyperelliptic curve.

That situation is analogous to an example of the elliptic integral considered in the classical Y.I. Manin’s paper [6] in 1961.

Keywords

KZ equations, hyperelliptic integrals, Cartier–Manin matrix, reduction to characteristic $p$.

2010 Mathematics Subject Classification

Primary 13A35. Secondary 32G20, 33C60.

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The author was supported in part by NSF grants DMS-1665239 and DMS-1954266.

Received 8 June 2018

Accepted 9 September 2020

Published 11 November 2020