Linear patterns of prime elements in number fields

Abstract

We prove a number field analogue of the Green--Tao--Ziegler theorem on simultaneous prime values of degree 1 polynomials whose linear parts are pairwise linearly independent. Applications of our results include a Hasse principle of rational points for certain fibrations $X\to \mathbb{P}^1$ over number fields $K$ which had only been available over $\mathbb Q $ by Harpaz--Skorobogatov--Wittenberg, and construction of elliptic curves having some specified ranks due to Koymans--Pagano and Zywina. This latter family of results led to a negative answer to a generalized Hilbert Tenth Problem.