  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.
