Multiple zeta value cycles in low weight

In a recent work, the author has constructed two families of algebraic cycles in Bloch cycle algebra over the prjective line minus 3 points that are expected to correspond to multiple polylogarithms in one variable and have a good specialization at 1 related to multiple zeta values. This is a short presentation, by the way of toy examples in low weight (5), of this contruc- tion and could serve as an introduction to the general setting. Working in low weight also makes it possible to push ("by hand") the construction further. In particular, we will not only detail the construction of the cycle but we will also associate to these cycles explicit elements in the bar construction over the cycle algebra and make as explicit as possible the "bottow-left" coefficient of the Hodge realization periods matrix. That is, in a few relevant cases we will associated to each cycles an integral showing how the specialization at 1 is related to multiple zeta values. We will be particularly interested in a new weight 3 example .Comment: revised version

EXPLICIT ASSOCIATOR RELATIONS FOR MULTIPLE ZETA VALUES

Associators were introduced by Drinfel’d in [Dri91] as a monodromy representation of a Knizhnik-Zamolodchikov equation. Associators can be briefly described as formal series in two non-commutative variables satisfying three equations. These three equations yield a large number of algebraic relations between the coefficients of the series, a situation which is particularly interesting in the case of the original Drinfel’d associator, whose coefficients are multiple zetas values. In the first part of this paper, we work out these algebraic relations among multiple zeta values by direct use of the defining relations of associators. While well-known for the first two relations, the algebraic relations we obtain for the third (pentagonal) relation, which are algorithmically explicit although we do not have a closed formula, do not seem to have been previously written down. The second part of the paper shows that if one has an explicit basis for the bar-construction of the moduli space M0,5 of genus zero Riemann surfaces with 5 marked points at one’s disposal, then the task of writing down the algebraic relations corresponding to the pentagon relation becomes significantly easier and more economical compared to the direct calculation above. We discuss the explicit basis described by Brown and Gangl, which is dual to the basis of the enveloping algebra of the braids Lie algebra UB5. In order to write down the relation between multiple zeta values, we then remark that it is enough to write down the relations associated to elements that generate the bar construction as an algebra. This corresponds to looking at the bar construction modulo shuffle, which is dual to the Lie algebra of 5-strand braids. We write down, in the appendix, the associated algebraic relations between multiple zeta values in weights 2 and 3

A relative basis for mixed Tate motives over the projective line minus three points

International audienceIn a previous work, the author have built two families of distinguished algebraic cycles in Bloch-Kriz cubical cycle complex over the projective line minus three points. The goal of this paper is to show how these cycles induce well-defined elements in the \HH^0 of the bar construction of the cycle complex and thus generated comodules over this \HH^0, that is a mixed Tate motives as in Bloch and Kriz construction. In addition, it is shown that out of the two families only ones is needed at the bar construction level. As a consequence, the author obtains that one of the family gives a basis of the tannakian coLie coalgebra of mixed Tate motives over \ps relatively to the tannakian coLie coalgebra of mixed Tate motives over \Sp(\Q). This in turns provides a new formula for Goncharov motivic coproduct, which arise explicitly as the coaction dual to Ihara action by special derivations

A MOTIVIC GROTHENDIECK-TEICHMÜLLER GROUP

International audienceThis paper proves the Beilinson-Soulé 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-Teichmüller 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

Équations fonctionnelles du dilogarithme

International audienceThis paper proves a " new " family of functional equations (Eqn) for Rogers dilogarithm. These equations rely on the combinatorics of dihedral coordinates on moduli spaces of curves of genus 0, M 0,n. For n = 4 we find back the duality relation while n = 5 gives back the 5 terms relation. It is then proved that the whole family reduces to the 5 terms relation. In the author's knownledge, it is the first time that an infinite family of functional equations for the dilogarithm with an increasing number of variables (n − 3 for (Eqn)) is reduced to the 5 terms relation.Cet article démontre une " nouvelle " famille d'équations fonctionnelles (Eq_n) (n≥ 4) satisfaites par le dilogarithme de Rogers. Ces équations fonctionnelles reflètent la combinatoire des coordonnées diédrales des espaces de modules de courbes de genres 0. Pour n = 4, on retrouve la relation de dualité et pour n = 5, la relation à 5 termes du dilogarithme. Dans une seconde partie, on démontre que la famille (Eq_n) se réduit à la relation à 5 termes. C'est, à la connaissance de l'auteur, la première fois qu'une famille infinie d'équations fonctionnelles du dilogarithme ayant un nombre croissant de variables (n − 3 pour (Eq_n)) se réduit à la relation à 5 termes

Motifs de Tate mixtes et éclatements à la MacPherson-Procesi ; Une application aux valeurs zêta multiples motiviques

In this thesis, we study the close links between multiple zeta values and the geometry of moduli spaces of curves in genius zero. It is shown how the properties of forgetful maps between moduli spaces of curves lead to the double shuffle relations for multiple zeta values MZVs (shuffle and stuffle). The main result of this work shows that these double shuffle relations hold for the motivic multiple zeta values attached to the moduli space of curves defined by Goncharov and Manin. First we show how those double shuffle relations are linked to the geometry of the moduli spaces of curves in genus 0. The next step, after a review on framed mixed motives, is to obtain shuffle relations for for the framed mixed motives defined by Goncharov and Manin, which are attached to both multiple zeta values and moduli spaces of curves. The last chapter of my thesis is devoted to the problem of the motivic stuffle. There, we adapt a theorem from Y. Hu about successive blow-ups to the situation of mixed Tate motives and then build a family of varieties. After some considerations on intersections of specific hypersurfaces in the affine and on the mixed Hodge structure of some relative cohomology groups, this family makes it possible to construct a new version of motivic multiple zeta values. Using the geometry of this family of varieties and these new motivic multiple zeta values it is easy to deduced some motivic stuffle relations for the new motivic multiple zeta values which lead, by comparison with the moduli spaces of curve, to motivic stuffle relations for the motivic multiple zeta values defined by Goncharov and Manin.Dans cette thèse, on étudie liens étroits qui existent entre les valeurs zêta multiples et la géométrie des espaces de modules de courbes en genre 0. En particulier, on y montre comment les deux produits de mélanges (shuffle et stuffle) des valeurs zêta multiples reflètent le comportement de certaines applications d'oubli entre espaces de modules courbes. Un des objectifs de mon travail a été de comprendre comment ces produits de mélange existent dans le cadre des motifs de Tate mixtes attachés aux espaces de module de courbes. On rappellera, dans un premier temps, les définitions et les propriétés des deux produits de mélange. Ensuite, on fera le lien avec la géométrie des espaces de modules de courbes. Puis, après quelques rappels sur les motifs encadrés, on montrera comment effectuer le passage aux motifs de Tate mixtes pour le produit shuffle dans le cadre des valeurs zêta multiples motiviques de Goncharov et Manin. Enfin, le dernier chapitre est consacré au stuffle motivique. Après avoir adapté un théorème de Y. Hu sur les successions d'éclatements à la situation des motifs de Tate mixtes, on construira une famille de variétés. À partir de là, on définira une nouvelles versions des valeurs zêta multiples motiviques. Pour parvenir à cette construction, on étudiera, entre autres, l'intersection d'hypersurfaces particulières et la structure de Hodge mixte de certains groupes de cohomologie relative. On obtient alors une forme de relation stuffle pour les motifs de Tate mixtes encadrés ces nouvelles valeur zêta motiviques dont on déduit les relations de stuffle pour les MZV motiviques de Goncharov et Manin