arXiv subjects

Ismaël Soudères

Publications and source records attributed to Ismaël Soudères.

6 recordsLinked to original sources

arXiv2017

Functional equations for Rogers dilogarithm

This 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.

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arXiv2016

Cycle complex over the projective line minus three points : toward multiple zeta values cycles

In this paper, the author constructs a family of algebraic cycles in Bloch's cubical cycle complex over the projective line minus three points which are expected to correspond to multiple polylogarithms in one variable. Elements in this family are in particular equidimensional over the projective line minus three points. In weight greater or equal to $2$, they are naturaly extended as equidimensional cycle over the affine line. This allows to consider their fibers at the point 1 and this is one of the main differences with Gangl, Goncharov and Levin work where generic arguments are imposed for cycles corresponding to multiple polylogarithms in many variables. Considering the fiber at 1 make it possible to think of these cycles as corresponding multiple zeta values. After the introduction, the author recalls some properties of Bloch's cycle complex, presents the strategy and enlightens the difficulties on a few examples. Then a large section is devoted to the combinatorial situation which is related to the combinatoric of trivalent trees and to a differential on trees already introduced by Gangl Goncharov and Levin. In the last section, two families of cycles are constructed as solution to a "differential system" in Bloch cycle complex. One of this families contains only cycles with empty fiber at 0 and should correspond to multiple polylogarithms while the other contains only cycles empty at 1. The use of two such families is required in order to work with equidimimensional cycles and to insure the admissibility condition.

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arXiv2014

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

In 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.

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arXiv2014

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 .

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arXiv2011

Explicit associator relations for multiple zeta values

Associators were introduced by Drinfel'd in as a monodromy representation of a KZ equation. Associators can be briefly described as formal series in two non-commutative variables satisfying three quations. 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 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 also discuss the explicit basis described by Brown and Gangl, which is dual to the basis of the enveloping algebra of the braids Lie algebra.

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arXiv2008

Motivic double shuffle

The goal of this article is to give an elementary proof of the double shuffle relations directly for the Goncharov and Manin motivic multiple zeta values. The shuffle relation is straightforward, but for the stuffle we use a modification of a method first introduced by P. Cartier for the purpose of proving stuffle for the real multiple zeta values via integrals and blow-up sequences.

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