Notes on $2D$ ${\mathbb{F}}_p$-Selberg integrals

We prove a two-dimensional ${\mathbb{F}}_p$-Selberg integral formula, in which the two-dimensional ${\mathbb{F}}_p$-Selberg integral $\bar{S}(a,b,c;l_1,l_2)$ depends on positive integer parameters $a,b,c$, $l_1,l_2$ and is an element of the finite field ${\mathbb{F}}_p$ with odd prime number $p$ of elements. The formula is motivated by the analogy between multidimensional hypergeometric solutions of the KZ equations and polynomial solutions of the same equations reduced modulo $p$.