  It is known that solutions of the KZ equations can be written in the form of
multidimensional hypergeometric integrals. In 2017 in a joint paper of the
author with V. Schechtman the construction of hypergeometric solutions was
modified, and solutions of the KZ equations modulo a prime number $p$ were
constructed. These solutions modulo $p$, called the $p$-hypergeometric
solutions, are polynomials with integer coefficients. A general problem is to
determine the number of independent $p$-hypergeometric solutions and understand
the meaning of that number.
  In this paper we consider the KZ equations associated with the space of
singular vectors of weight $n-2r$ in the tensor power $W^{\otimes n}$ of the
vector representation of $\frak{sl}_2$. In this case, the hypergeometric
solutions of the KZ equations are given by $r$-dimensional hypergeometric
integrals. We consider the module of the corresponding $p$-hypergeometric
solutions, determine its rank, and show that the rank equals the dimension of
the space of suitable square integrable differential $r$-forms.
