Hypergeometric properties of genus 3 generalized Legendre curves

Abstract

Abstract. Inspired by a result of Manin, we study the relationship between certain period integrals and the trace of Frobenius of genus 3 generalized Legendre curves. We show that both of these properties can be computed in terms of "matching" classical and finite field hypergeometric functions, a phenomenon that has also been observed in elliptic curves and many higher dimensional varieties.

Evaluation of Certain Hypergeometric Functions over Finite Fields

“…Key ingredients in their approach are some transformation formulas of hypergeometric functions over finite fields, such as a finite field analogue of Clausen's theorem. In recent years, Frechette, Fuselier, Goodson, Lennon, Ono, Papanikolas, and Salerno [10,11,12,13,15,22,23,26] investigated connections between hypergeometric functions over finite fields and elliptic curves, algebraic varieties, and Hecke eigenforms. Very recently, McCarthy-Papanikolas [24] linked the hypergeometric functions to Siegel modular forms.…”