Appell-Lauricella hypergeometric functions over finite fields and algebraic varieties

TL;DR

This paper establishes finite field analogues of Appell-Lauricella hypergeometric functions, connecting algebraic geometry and finite field point counting through hypersurfaces with group actions.

Contribution

It introduces finite field versions of these hypergeometric functions and relates them to the rational points on specific hypersurfaces with symmetry.

Findings

Finite field analogues of Appell-Lauricella functions are constructed.

Rational point counts on hypersurfaces are expressed via these functions.

Connections between algebraic varieties and hypergeometric functions over finite fields are demonstrated.

Abstract

We prove finite field analogues of integral representations of Appell- Lauricella hypergeometric functions in many variables. We consider certain hypersurfaces having a group action and compute the numbers of rational points associated with characters of the group, which will be expressed in terms of Appell-Lauricella functions over finite fields.