2023/06/29 by Kai, Wataru · 2 citations
#Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2306.16983
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→ ℙ1 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 ΛK 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 ΛK 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.