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. This can be used to prove a Hasse principle result for certain fibrations $X\to \mathbb{P}^1$ over a number field $K$ extending a result of Harpaz--Skorobogatov--Wittenberg which was only available over $\mathbb Q $. The main technical content is the proof that the von Mangoldt function $\Lambda _K$ of a number field $K$ is well approximated by its Cramer/Siegel models in the Gowers norm sense. Via the inverse theory of the Gowers norm, this is achieved by showing that the difference of $\Lambda _K$ and its model is asymptotically orthogonal to nilsequences. To prove the asymptotic orthogonality, we use Mitsui's Prime Element Theorem as the base case and proceed by upgrading Green--Tao's type I/II sum computation to the general number field. Other applications of our results include the negative resolution of Hilbert's Tenth Problem over all number rings by Koymans--Pagano.