We prove the Beilinson–Soulé vanishing conjecture for motives attached to the moduli spaces
of curves
of genus
with
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
.
We furthermore show that the morphisms between the moduli spaces
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
.
This allows us to treat the general case of motives over
with
coefficients in
,
working in Spitzweck’s category of motives. From there, passing to
coefficients, we deal with the classical Tannakian formalism and explain how working
over
yields a more concrete description of the Tannakian groups.
