Zeta-like Multizeta Values for higher genus curves

José Alejandro Lara Rodríguez; Dinesh S. Thakur

doi:10.5802/jtnb.1169

José Alejandro Lara Rodríguez ¹ ; Dinesh S. Thakur ²

¹ Facultad de Matemáticas Universidad Autónoma de Yucatán Periférico Norte, Tab. 13615 Mérida, Yucatán., México
² Department of Mathematics University of Rochester Rochester, NY 14627, USA

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

Abstract (VO)
Résumé

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.

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.

Reçu le : 2020-06-26
Révisé le : 2021-03-18
Accepté le : 2021-04-19
Publié le : 2021-09-10

DOI : 10.5802/jtnb.1169

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

Affiliations des auteurs :

José Alejandro Lara Rodríguez ¹ ; Dinesh S. Thakur ²

¹ Facultad de Matemáticas Universidad Autónoma de Yucatán Periférico Norte, Tab. 13615 Mérida, Yucatán., México
² Department of Mathematics University of Rochester Rochester, NY 14627, USA

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Jos\'e Alejandro Lara Rodr{\'\i}guez and Dinesh S. Thakur},
     title = {Zeta-like {Multizeta} {Values} for higher genus curves},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {553--581},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {33},
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     doi = {10.5802/jtnb.1169},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1169/}
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José Alejandro Lara Rodríguez; Dinesh S. Thakur. Zeta-like Multizeta Values for higher genus curves. Journal de théorie des nombres de Bordeaux, Tome 33 (2021) no. 2, pp. 553-581. doi : 10.5802/jtnb.1169. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1169/

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