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 recent development by Fuselier, Long, Ramakrishna, and the last two authors 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 F1, proving a conjecture of Long. We use our multivariable period functions to construct formulas for the number of Fp-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. © 2018, SpringerNature.