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Journal Article

Hyperelliptic integrals modulo p and Cartier-Manin matrices


Varčenko,  Aleksandr N.
Max Planck Institute for Mathematics, Max Planck Society;

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Varčenko, A. N. (2020). Hyperelliptic integrals modulo p and Cartier-Manin matrices. Pure and Applied Mathematics Quarterly, 16(3), 315-336. doi:10.4310/PAMQ.2020.v16.n3.a1.

Cite as: https://hdl.handle.net/21.11116/0000-0008-0D0A-0
The hypergeometric solutions of the KZ equations were constructed almost 30
years ago. The polynomial solutions of the KZ equations over the finite field $F_p$ with a prime number $p$ of elements were constructed recently. In this paper we consider the 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 solutions is given by the hyperelliptic integrals. We show that the recent construction of the polynomial solutions over the field $F_p$ 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 $F_p$ 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 the example of the elliptic integral considered in the classical Y.I. Manin's paper in 1961.