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Linear patterns of prime elements in number fields

Wataru Kai

TL;DR

The paper establishes a number field analogue of the Green–Tao–Ziegler theorem for simultaneous prime values of affine-linear forms with independent linear parts, by showing the von Mangoldt function $\\Lambda_K$ is well approximated by Cramér and Siegel models in the Gowers $U^{s+1}$-norm and exploiting inverse Gowers theory to prove orthogonality to nilsequences. The authors develop a robust framework combining a $W$-trick, Mitsui’s prime-element base, a Vaughan-type decomposition over ideals, and a factorization theorem to reduce to equidistributed nilsequence settings. They deduce a GTZ-type asymptotic formula for primes represented by affine-linear forms over number fields, including positivity criteria, and apply this to a Hasse principle for certain fibrations and to results on Hilbert’s Tenth problem over number rings. The work also extends prior Q-based results to arbitrary number fields, incorporating Siegel zeros and localized rings $\\mathcal{O}_K[S^{-1}]$ in a unified analytic approach with nilsequence techniques. Overall, the paper advances the interface between analytic number theory, additive combinatorics, and arithmetic geometry in the number-field setting, with significant consequences for rational points and undecidability questions.

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 $Λ_K$ 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 $Λ_K$ 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.

Linear patterns of prime elements in number fields

TL;DR

The paper establishes a number field analogue of the Green–Tao–Ziegler theorem for simultaneous prime values of affine-linear forms with independent linear parts, by showing the von Mangoldt function is well approximated by Cramér and Siegel models in the Gowers -norm and exploiting inverse Gowers theory to prove orthogonality to nilsequences. The authors develop a robust framework combining a -trick, Mitsui’s prime-element base, a Vaughan-type decomposition over ideals, and a factorization theorem to reduce to equidistributed nilsequence settings. They deduce a GTZ-type asymptotic formula for primes represented by affine-linear forms over number fields, including positivity criteria, and apply this to a Hasse principle for certain fibrations and to results on Hilbert’s Tenth problem over number rings. The work also extends prior Q-based results to arbitrary number fields, incorporating Siegel zeros and localized rings in a unified analytic approach with nilsequence techniques. Overall, the paper advances the interface between analytic number theory, additive combinatorics, and arithmetic geometry in the number-field setting, with significant consequences for rational points and undecidability questions.

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 over a number field extending a result of Harpaz--Skorobogatov--Wittenberg which was only available over . The main technical content is the proof that the von Mangoldt function of a number field 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 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.
Paper Structure (65 sections, 68 theorems, 529 equations)

This paper contains 65 sections, 68 theorems, 529 equations.

Table of Contents

  1. Introduction
  2. Applications
  3. Hasse principle for rational points on varieties over number fields
  4. Hilbert's Tenth Problem over number rings
  5. Notation and conventions
  6. General notation
  7. Cubes in Euclidean spaces
  8. Ideals
  9. Outline of proof
  10. The von Neumann theorem
  11. Siegel model
  12. Gowers Inverse Theorem
  13. Factorization theorem
  14. Mitsui's Prime Element Theorem
  15. Vaughan decomposition
  16. ...and 50 more sections

Key Result

Theorem 1.1

Let $\psi _1,\dots ,\psi _t\in \mathcal{O}_K [X_1,\dots ,X_d]$ be polynomials of degree $1$ such that their linear parts $\dot\psi _i$ are pairwise linearly independent over $K$. Then we have an asymptotic formula as $N\to +\infty$: Here, $C_{\psi _1,\dots ,\psi _t }$ is a constant depending on $\psi _1,\dots ,\psi _t$, which is positive if (and only if) the following condition is satisfied: (An

Theorems & Definitions (157)

  • Theorem 1.1: Theorem \ref{['thm:prime-values']} and Remark \ref{['rem:product-beta']}
  • Theorem 1.2: Harpaz--Skorobogatov--Wittenberg Harpaz-Skorobogatov-Wittenberg
  • Definition 2.1: Gowers norm
  • Theorem 2.2: Theorem \ref{['thm:Mangoldt-Cramer-close']}
  • Proposition 3.1: existence of norm-length compatible bases
  • proof
  • Definition 3.2: norm-length compatible bases
  • Proposition 3.3: change of basis matrices
  • proof
  • Proposition 3.4: normalized norm function
  • ...and 147 more