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 , 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.
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Keywords: t-motives, periods, mixed Tate motives
@article{JTNB_2021__33_2_553_0, 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}, number = {2}, year = {2021}, doi = {10.5802/jtnb.1169}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1169/} }
TY - JOUR AU - José Alejandro Lara Rodríguez AU - Dinesh S. Thakur TI - Zeta-like Multizeta Values for higher genus curves JO - Journal de théorie des nombres de Bordeaux PY - 2021 SP - 553 EP - 581 VL - 33 IS - 2 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1169/ DO - 10.5802/jtnb.1169 LA - en ID - JTNB_2021__33_2_553_0 ER -
%0 Journal Article %A José Alejandro Lara Rodríguez %A Dinesh S. Thakur %T Zeta-like Multizeta Values for higher genus curves %J Journal de théorie des nombres de Bordeaux %D 2021 %P 553-581 %V 33 %N 2 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1169/ %R 10.5802/jtnb.1169 %G en %F JTNB_2021__33_2_553_0
José Alejandro Lara Rodríguez; Dinesh S. Thakur. Zeta-like Multizeta Values for higher genus curves. Journal de théorie des nombres de Bordeaux, Volume 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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