A Cubic Transformation Formula for Appell-Lauricella Hypergeometric Functions over Finite Fields

TL;DR

This paper introduces finite-field versions of Appell-Lauricella hypergeometric functions, establishes their geometric connections, and proves a finite-field analogue of a classical cubic transformation, advancing understanding of hypergeometric functions over finite fields.

Contribution

It defines multivariable finite-field hypergeometric functions, proves a cubic transformation conjecture, and links these functions to point counts on generalized Picard curves.

Findings

Established a finite-field analogue of Koike and Shiga's cubic transformation.

Constructed formulas for counting points on generalized Picard curves over finite fields.

Derived transformation and reduction formulas for finite-field Appell-Lauricella functions.

Abstract

We define a finite-field version of Appell-Lauricella hypergeometric functions built from period functions in several variables, paralleling the development by Fuselier, et. al in the single variable case. We develop geometric connections between these functions and the family of generalized Picard curves. In our main result, we use finite-field Appell-Lauricella functions to establish a finite-field analogue of Koike and Shiga's cubic transformation for the Appell hypergeometric function $F_1$, proving a conjecture of Ling Long. We use our multivariable period functions to construct formulas for the number of $\mathbb{F}_p$-points on the generalized Picard curves. We also give some transformation and reduction formulas for the period functions, and consequently for the finite-field Appell-Lauricella functions.