Motivic configurations on the line

TL;DR

This paper introduces a new operation on motivic homotopy classes of endomorphisms of the projective line, depending on rational point configurations, and provides an algebraic formula for its image under the unstable degree map.

Contribution

It defines a configuration-dependent operation in the motivic setting and derives an explicit algebraic formula for its image under the unstable degree map.

Findings

Operation depends on configuration via discriminant

Local-to-global formula for unstable degree

Generalization from local to global endomorphisms

Abstract

For each configuration of rational points on the affine line, we define an operation on the group of unstable A1 motivic homotopy classes of endomorphisms of the projective line. We also derive an algebraic formula for the image of such an operation under Cazanave and Morel's unstable degree map, which is valued in an extension of the Grothendieck--Witt group. In contrast to the topological setting, these operations depend on the choice of configuration of points via a discriminant. We prove this by first showing a local-to-global formula for the global unstable degree as a modified sum of local terms. We then use an anabelian argument to generalize from the case of local degrees of a global rational function to the case of an arbitrary collection of endomorphisms of the projective line.