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 \(p\) .