Determinant of ${\mathbb{F}}_p$-hypergeometric solutions under ample reduction

Abstract

We consider the KZ differential equations over $\mathbb{C}$ in the case, when the hypergeometric solutions are one-dimensional integrals. We also consider the same differential equations over a finite field ${\mathbb{F}}_p$. We study the polynomial solutions of these differential equations over ${\mathbb{F}}_p$, constructed in a previous work joint with V.\,Schechtman and called the ${\mathbb{F}}_p$-hypergeometric solutions. The dimension of the space of ${\mathbb{F}}_p$-hypergeometric solutions depends on the prime number $p$. We say that the KZ equations have ample reduction for a prime $p$, if the dimension of the space of ${\mathbb{F}}_p$-hypergeometric solutions is maximal possible, that is, equal to the dimension of the space of solutions of the corresponding KZ equations over $\mathbb{C}$. Under the assumption of ample reduction, we prove a determinant formula for the matrix of coordinates of basis ${\mathbb{F}}_p$-hypergeometric solutions. The formula is analogous to the corresponding formula for the determinant of the matrix of coordinates of basis complex hypergeometric solutions, in which binomials $(z_i-z_j{)}^{M_i+M_j}$ are replaced with $(z_i-z_j{)}^{M_i+M_j-p}$ and the Euler gamma function $\Gamma (x)$ is replaced with a suitable ${\mathbb{F}}_p$-analog ${\Gamma }_{{\mathbb{F}}_p}(x)$ defined on ${\mathbb{F}}_p$.

Latex, 22 pages, v2: references added