Appell–Lauricella Hypergeometric Functions over Finite Fields, and a New Cubic Transformation Formula

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. (Hypergeometric functions over finite fields, arXiv:1510.02575v2) 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 (Koike and Shiga, J. Number Theory 124:123–141, 2007) for the Appell hypergeometric function F1, proving a conjecture of Ling Long. We also prove a finite field analogue of Gauss’ quadratic arithmetic geometric mean. We use our multivariable period functions to construct formulas for the number of \(\mathbb {F}_p\)-points on the generalized Picard curves. Lastly, we give some transformation and reduction formulas for the period functions, and consequently for the finite-field Appell–Lauricella functions.