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

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.