We prove that the category of mixed Tate motives over Z is spanned by the motivic fundamental group of P 1 minus three points. We prove a conjecture by M. Hoffman which states that every multiple zeta value is a Q-linear combination of ζ(n1, . . . , nr), where ni ∈ {2, 3}.

Aʙʀ. -We prove a conjecture due to Goncharov and Manin which states that the periods of the moduli spaces M0,n of Riemann spheres with n marked points are multiple zeta values. We do this by introducing a differential algebra of multiple polylogarithms on M0,n and proving that it is closed under the operation of taking primitives. The main idea is to apply a version of Stokes' formula iteratively to reduce each period integral to multiple zeta values.We also give a geometric interpretation of the double shuffle relations, by showing that they are two extreme cases of general product formulae for periods which arise by considering natural maps between moduli spaces.R. -Nous démontrons une conjecture de Goncharov et Manin qui prédit que les périodes des espaces de modules M0,n des courbes de genre 0 avec n points marqués sont des valeurs zêta multiples. Nous introduisons une algèbre différentielle de fonctions polylogarithmes multiples sur M0,n dans laquelle il existe des primitives. L'idée principale est d'appliquer une version de la formule de Stokes récursivement pour réduire chaque intégrale de périodes à une combinaison linéaire de valeurs zêta multiples.Nous donnons également une interprétation géométrique des double relations de mélange pour les valeurs zêta multiples. En considérant des applications naturelles entre les espaces des modules, on déduit des formules de produit générales entre leurs périodes. Les doubles relations de mélange s'obtiennent comme deux cas particuliers de cette construction.

Abstract. We introduce a new method for computing massless Feynman integrals analytically in parametric form. An analysis of the method yields a criterion for a primitive Feynman graph G to evaluate to multiple zeta values. The criterion depends only on the topology of G, and can be checked algorithmically. As a corollary, we reprove the result, due to Bierenbaum and Weinzierl, that the massless 2-loop 2-point function is expressible in terms of multiple zeta values, and generalize this to the 3, 4, and 5-loop cases. We find that the coefficients in the Taylor expansion of planar graphs in this range evaluate to multiple zeta values, but the non-planar graphs with crossing number 1 may evaluate to multiple sums with 6 th roots of unity. Our method fails for the five loop graphs with crossing number 2 obtained by breaking open the bipartite graph K 3,4 at one edge.

The values at 1 of single-valued multiple polylogarithms span a certain subalgebra of multiple zeta values. The properties of this algebra are studied from the point of view of motivic periods.