  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.
