  We consider the Gauss-Manin differential equations for hypergeometric
integrals associated with a family of weighted arrangements of hyperplanes
moving parallelly to themselves. We reduce these equations modulo a prime
integer $p$ and construct polynomial solutions of the new differential
equations as $p$-analogs of the initial hypergeometric integrals.
  In some cases we interpret the $p$-analogs of the hypergeometric integrals as
sums over points of hypersurfaces defined over the finite field $F_p$. That
interpretation is similar to the interpretation by Yu.I. Manin in [Ma] of the
number of point on an elliptic curve depending on a parameter as a solution of
a classical hypergeometric differential equation.
  We discuss the associated Bethe ansatz.
