Linear patterns of prime elements in number fields

TL;DR

This paper establishes a number field analogue of a key theorem on prime values of linear polynomials, with applications to rational points, elliptic curves, and the Hilbert Tenth Problem over number fields.

Contribution

It introduces a new theorem extending Green--Tao--Ziegler results to number fields, enabling advances in rational points and elliptic curve constructions.

Findings

Proved a number field analogue of the Green--Tao--Ziegler theorem.

Applied results to Hasse principle for fibrations over number fields.

Constructed elliptic curves with specified ranks, impacting the generalized Hilbert Tenth Problem.

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.