A Picard family of curves and hypergeometric functions over finite fields I

TL;DR

This paper derives an explicit formula for the trace of Frobenius for a specific family of algebraic curves over finite fields, expressed via finite field hypergeometric functions, advancing understanding in arithmetic geometry.

Contribution

It introduces a new expression for Frobenius traces of Picard family curves using finite field hypergeometric functions, linking algebraic geometry and special functions.

Findings

Explicit Frobenius trace formula for the family of curves

Connection established between algebraic curves and hypergeometric functions

Provides tools for further research in finite field arithmetic geometry

Abstract

We give an expression for the trace of Frobenius for the family of curves \[ y^3 = x (x-1)(x-\lambda)(x-\mu) \] over finite fields in terms of finite field hypergeometric functions.