Zeta-like Multizeta Values for higher genus curves

Lara Rodríguez, José Alejandro; Thakur, Dinesh S.

Journal de théorie des nombres de Bordeaux, Tome 33 (2021) no. 2, pp. 553-581.

Nous démontrons certaines relations (et en conjecturons d’autres) entre les valeurs des multizêtas pour les corps de fonctions de genre positif et de nombre de classes $1$, en nous concentrant sur les valeurs de type zêta, à savoir celles dont le rapport à la valeur zêta de même poids est rationnel (ou, conjecturalement de manière équivalente, algébrique). Ce sont les premières relations connues entre multizêtas dont les coefficients ne sont pas dans un corps premier. Nous semblons avoir une famille universelle. Nous constatons également que, de manière intéressante, le mécanisme selon lequel les relations fonctionnent est assez différent du cas du corps des fractions rationnelles, ce qui soulève des questions intéressantes sur l’interprétation motivique attendue en genre supérieur.

We prove some (and conjecture more) relations between the multizeta values for positive genus function fields of class number one, focusing on the zeta-like values, namely those whose ratio with the zeta value of the same weight is rational (or conjecturally equivalently, algebraic). These are the first known relations between multizetas, which are not with prime field coefficients. We seem to have one universal family. We also find that, interestingly, the mechanism with which the relations work is quite different from the rational function field case, raising interesting questions about the expected motivic interpretation in higher genus.

Classification : 11M32, 11G09, 11G30
Mots clés : t-motives, periods, mixed Tate motives

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