On the square-free representation function of a norm form and nilsequences

TL;DR

This paper proves that the square-free restriction of a norm form's representation function does not correlate with nilsequences, enabling applications in rational points research via the fibration method.

Contribution

It establishes a non-correlation result between square-free norm form representation functions and nilsequences, advancing understanding in number theory and algebraic geometry.

Findings

Square-free representation function does not correlate with nilsequences

Application to rational points via the fibration method

Supports recent work on rational points by Harpaz and Wittenberg

Abstract

We show that the restriction to square-free numbers of the representation function attached to a norm form does not correlate with nilsequences. By combining this result with previous work of Browning and the author, we obtain an application that is used in recent work of Harpaz and Wittenberg on the fibration method for rational points.