A Motivic Grothendieck-Teichm\"uller Group

TL;DR

This paper establishes the Beilinson-Soulé vanishing conjecture for motives related to genus 0 moduli spaces, proving these motives are mixed Tate and defining a motivic Grothendieck-Teichmüller group via Tannakian formalism.

Contribution

It proves the vanishing conjecture for these motives, shows they are mixed Tate, and constructs a motivic Grothendieck-Teichmüller group using Tannakian categories and functoriality.

Findings

Motives over moduli spaces are mixed Tate.

The Beilinson-Soulé vanishing conjecture holds for these motives.

A motivic Grothendieck-Teichmüller group is defined.

Abstract

This paper proves the Beilinson-Soul{\'e} vanishing conjecture for motives attached to the moduli spaces of curves of genus 0 with n marked points. As part of the proof, it is also proved that these motives are mixed Tate. As a consequence of Levine's work, one obtains then well defined categories of mixed Tate motives over the moduli spaces of curves . It is shown that morphisms between moduli spaces forgetting marked points and embedding as boundary components induce functors between those categories and how tangential bases points fit in these functorialities. Tannakian formalism attaches groups to these categories and morphisms reflecting the functorialities leading to the definition of a motivic Grothendieck-Teichm{\"u}ller group. Proofs of the above properties rely on the geometry of the tower of the moduli spaces . This allows us to treat the general case of motives over Spec(Z) with integral coefficients working in Spitzweck's category of motives. From there, passing to Q-coefficients we deal with the classical tannakian formalism and explain how working over Spec(Q) allows a more concrete description of the tannakian group.