Determinant of 𝔽_{𝕡}-hypergeometric solutions under ample reduction

Abstract

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