On the number of p-hypergeometric solutions of KZ equations

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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n-2r$$\end{document} in the tensor power W⊗n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W^{\otimes n}$$\end{document} of the vector representation of sl2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {sl}_2$$\end{document}. 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 is equal to the dimension of the space of suitable square integrable differential r-forms.