arXiv subjects

Samuel Baumard

Publications and source records attributed to Samuel Baumard.

3 recordsLinked to original sources

arXiv2015

Relations dans l'algèbre de Lie fondamentale des motifs elliptiques mixtes

Hain and Matsumoto constructed a category of universal mixed elliptic motives and described the fundamental Lie algebra of this category, relating it to a certain graded and filtered Lie algebra E. In an unpublished paper, Aaron Pollack proved a result on relations in a certain quotient of E, and gave several examples of actual relations in small weight that naturally lead to a conjecture about the existence of such relations in all weights. In this article, we prove this conjecture in depth 3, establishing the existence in all weights of relations of the type noted in Pollack's examples. ----- Richard Hain et Makoto Matsumoto ont construit une catégorie de motifs elliptiques mixtes universels et décrit l'algèbre de Lie fondamentale de cette catégorie en la reliant à une certaine algèbre de Lie graduée et filtrée E. Dans un article non publié, Aaron Pollack a démontré un résultat sur les relations dans un certain quotient de E, et donné plusieurs exemples de relations en petit poids qui conduisent à une conjecture sur l'existence de telles relations en tout poids. Dans le présent article, nous prouvons cette conjecture en profondeur 3, établissant l'existence en tout poids de relations du type décrit dans les exemples de Pollack.

math.AG↗

arXiv2015

On the derivation representation of the fundamental Lie algebra of mixed elliptic motives

Richard Hain and Makoto Matsumoto constructed a category of universal mixed elliptic motives, and described the fundamental Lie algebra of this category: it is a semi-direct product of the fundamental Lie algebra Lie$\,π_1(MTM)$ of the category of mixed Tate motives over ${\bf Z}$ with a filtered and graded Lie algebra $u$. This Lie algebra, and in particular the subspace $u$, admits a representation as derivations of the free Lie algebra on two generators. In this paper we study the image $E$ of this representation of $u$, starting from some results by Aaron Pollack, who determined all the relations in a certain filtered quotient of $E$, and gave several examples of relations in low weights in $E$ that are connected to period polynomials of cusp forms on $SL_2({\bf Z})$. Pollack's examples lead to a conjecture on the existence of such relations in all depths and all weights, that we state in this article and prove in depth 3 in all weights. The proof follows quite naturally from Ecalle's theory moulds, to which we give a brief introduction. We prove two useful general theorems on moulds in the appendices.

math.QA↗

arXiv2011

Period polynomial relations between double zeta values

The even weight period polynomial relations in the double shuffle Lie algebra $\mathfrak{ds}$ were discovered by Ihara, and completely classified by the second author by relating them to restricted even period polynomials associated to cusp forms on $\mathrm{SL}_2(\mathbb{Z})$. In an article published in the same year, Gangl, Kaneko and Zagier displayed certain linear combinations of odd-component double zeta values which are equal to scalar multiples of simple zeta values in even weight, and also related them to restricted even period polynomials. In this paper, we relate the two sets of relations, showing how they can be deduced from each other by duality.

math.NT↗