A motivic Grothendieck–Teichmüller group

Abstract

We prove the Beilinson–Soulé vanishing conjecture for motives attached to the moduli spaces ${\mathcal{ℳ}}_{0,n}$ of curves of genus $0$ with $n$ marked points. As part of the proof, we also show that these motives are mixed Tate. As a consequence of Levine’s work, we thus obtain a well-defined category of mixed Tate motives over the moduli space of curves ${\mathcal{ℳ}}_{0,n}$. We furthermore show that the morphisms between the moduli spaces ${\mathcal{ℳ}}_{0,n}$ obtained by forgetting marked points and by embedding boundary components induce functors between the associated categories of mixed Tate motives. Finally, we explain how tangential base points fit into these functorialities.

The categories we construct are Tannakian, and therefore have attached Tannakian fundamental groups, connected by morphisms induced by those between the categories. This system of groups and morphisms leads to the definition of a motivic Grothendieck–Teichmüller group.

The proofs of the above results rely on the geometry of the tower of the moduli spaces ${\mathcal{ℳ}}_{0,n}$. This allows us to treat the general case of motives over $Spec\left(\mathbb{Z}\right)$ with coefficients in $\mathbb{Z}$, working in Spitzweck’s category of motives. From there, passing to $\mathbb{Q}$ coefficients, we deal with the classical Tannakian formalism and explain how working over $Spec\left(\mathbb{Q}\right)$ yields a more concrete description of the Tannakian groups.