The Green-Tao theorem for affine curves over F_q

TL;DR

This paper extends Green-Tao type results to the coordinate rings of affine curves over finite fields, using a carefully chosen polynomial subring and the Riemann-Roch theorem to establish the existence of long arithmetic progressions of prime elements.

Contribution

It introduces a novel approach for affine curve coordinate rings over finite fields, generalizing Green-Tao theorems to this new setting.

Findings

Established Green-Tao type theorems for affine curve coordinate rings over finite fields.

Developed a method to select polynomial subrings with desirable properties.

Utilized Riemann-Roch theorem to support the proof structure.

Abstract

Green and Tao famously proved in a 2008 paper that there are arithmetic progressions of prime numbers of arbitrary lengths. Soon after, analogous statements were proved by Tao for the ring of Gaussian integers and by L\^e for the polynomial rings over finite fields. In 2020 this was extented to orders of arbitrary number fields by Kai-Mimura-Munemasa-Seki-Yoshino. We settle the case of the coordinate rings of affine curves over finite fields. The main contribution of this paper is subtle choice of a polynomial subring of the given ring which plays the role of $\mathbb Z$ in the number field case. This choice and the proof of its pleasant properties eventually depend on the Riemann-Roch formula.