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 $\overline S(a,b,c;l_1,l_2)$ depends on positive integer parameters $a,b,c$, $l_1,l_2$ and is an element of the finite field $\mathbb{F}_p$ with an 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 $p$.