Hypergeometric Functions over Finite Fields

We discuss recent work of the authors in which we study the translation of classical hypergeometric transformation and evaluation formulas to the finite field setting.Our approach is motivated by the desire for both an algorithmic type approach that closely parallels the classical case, and an approach that aligns with geometry. In light of these objectives, we focus on period functions in our construction which makes point counting on the corresponding varieties as straightforward as possible.We are also motivated by previous work joint with Deines, Fuselier, Long, and Tu in which we study generalized Legendre curves using periods to determine a condition for when the endomorphism algebra of the primitive part of the associated Jacobian variety contains a quaternion algebra over ℚ $${\mathbb {Q}}$$ . In most cases this involves computing Galois representations attached to the Jacobian varieties using Greene’s finite field hypergeometric functions.