Hyperelliptic integrals modulo $p$ and Cartier-Manin matrices

The hypergeometric solutions of the KZ differential equations were constructed almost 30 years ago. The polynomial solutions of the KZ equations over the finite field $F_p$ with a prime number $p$ elements were constructed recently. I will 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. It turns out that the recent construction of the polynomial solutions over the field $F_p$ in this case gives a $g$-dimensional space of solutions, that is, a "half" of what the complex analytic construction gives. It also turns out 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.
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