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#### Hyperelliptic integrals modulo p and Cartier-Manin matrices

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https://dx.doi.org/10.4310/PAMQ.2020.v16.n3.a1

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1806.03289.pdf

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##### Citation

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

##### 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 $F_p$ 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.

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.