A motivic circle method

TL;DR

This paper adapts the circle method to the Grothendieck ring of varieties, enabling direct approximation of moduli space classes and enhancing geometric understanding without point counting.

Contribution

It introduces a novel version of the circle method within the Grothendieck ring framework, expanding tools for studying moduli spaces of rational curves.

Findings

Approximate classes of moduli spaces directly in the Grothendieck ring

Provides new insights into the geometry of moduli spaces

Circumvents traditional point counting methods

Abstract

The circle method has been successfully used over the last century to study rational points on hypersurfaces. More recently, a version of the method over function fields, combined with spreading out techniques, has led to a range of results about moduli spaces of rational curves on hypersurfaces. In this paper a version of the circle method is implemented in the setting of the Grothendieck ring of varieties. This allows us to approximate the classes of these moduli spaces directly, without relying on point counting, and leads to a deeper understanding of their geometry.