Multiple polylogarithm values at roots of unity

Jianqiang Zhao

doi:10.1016/j.crma.2008.09.011

Number Theory

Multiple polylogarithm values at roots of unity

Presented by: Pierre Deligne

Jianqiang Zhao ¹

¹ Department of Mathematics, Eckerd College, St. Petersburg, FL 33711, USA

Comptes Rendus. Mathématique, Volume 346 (2008) no. 19-20, pp. 1029-1032.

Abstract
Résumé

For any positive integer N let $μ_{N}$ be the group of the Nth roots of unity. In this note we shall study the $Q$ -linear relations among the values of multiple polylogarithms evaluated at $μ_{N}$ . We show that the standard relations considered by Racinet do not provide all the possible relations in the following cases: (i) level $N = 4$ , weight $w = 3$ or 4, and (ii) $w = 2$ , $7 < N < 50$ , and N is a power of 2 or 3, or N has at least two prime factors. We further find some (presumably all) of the missing relations in (i) by using the octahedral symmetry of $P^{1} - ({0, \infty} \cup μ_{4})$ . We also prove some other results when $N = p$ or $N = p^{2}$ (p prime ⩾5) by using the motivic fundamental group of $P^{1} - ({0, \infty} \cup μ_{N})$ .

Soient N un entier positif et $μ_{N}$ le groupe des racines N-ièmes de l'unité. Nous étudions les relations $Q$ -linéaires entre les valeurs de polylogarithmes multiples évalués en ces racines de l'unité. Nous montrons que les relations standard considérées par Racinet ne fournissent pas toutes les relations dans les cas suivants : (i) $N = 4$ , poids $w = 3$ ou 4, et (ii) $w = 2$ , $7 < N < 50$ , et N est une puissance de 2 ou 3, ou N a au moins deux facteurs premiers. Dans le cas (i), nous trouvons des (sans doute, toutes les) relations manquantes à l'aide de la symétrie octaédrale de $P^{1} - ({0, \infty} \cup μ_{4})$ . Utilisant le groupe fondamental motivique de $P^{1} - ({0, \infty} \cup μ_{N})$ , nous obtenons des résultats additionnels quand $N = p$ ou $N = p^{2}$ (p premier ⩾5).

Received: 2008-08-26
Accepted: 2008-09-02
Published online: 2008-10-01

DOI: 10.1016/j.crma.2008.09.011

Author's affiliations:

Jianqiang Zhao ¹

¹ Department of Mathematics, Eckerd College, St. Petersburg, FL 33711, USA

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     author = {Jianqiang Zhao},
     title = {Multiple polylogarithm values at roots of unity},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1029--1032},
     publisher = {Elsevier},
     volume = {346},
     number = {19-20},
     year = {2008},
     doi = {10.1016/j.crma.2008.09.011},
     language = {en},
}

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Jianqiang Zhao. Multiple polylogarithm values at roots of unity. Comptes Rendus. Mathématique, Volume 346 (2008) no. 19-20, pp. 1029-1032. doi : 10.1016/j.crma.2008.09.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.09.011/

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