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.

中文翻译：

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超椭圆积分模 $p$ 和 Cartier-Manin 矩阵

KZ 方程的超几何解是在近 30 年前构建的。最近构建了具有素数$p$元素的有限域$F_p$上的KZ方程的多项式解。在本文中，我们考虑 KZ 方程的例子，其超几何解由 $g$ 属的超椭圆积分给出。众所周知，在这种情况下，全纯解的总 $2g$ 维空间由超椭圆积分给出。我们表明，在这种情况下，$F_p$ 域上多项式解的最新构造仅给出了一个 $g$ 维的解空间，即复杂解析构造给出的“一半”。我们还表明，所有在域 $F_p$ 上构造的多项式解都可以通过对单个可区分的超几何解的减模 $p$ 来获得。相应的公式涉及到超椭圆曲线的 Cartier-Manin 矩阵的项。这种情况类似于 1961 年 YI Manin 的经典论文中考虑的椭圆积分的例子。