Zeta-like Multizeta Values for higher genus curves
Journal de Théorie des Nombres de Bordeaux, Volume 33 (2021) no. 2, pp. 553-581.

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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1169
Classification: 11M32,  11G09,  11G30
Keywords: t-motives, periods, mixed Tate motives
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
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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, Volume 33 (2021) no. 2, pp. 553-581. doi : 10.5802/jtnb.1169. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1169/

[1] Greg W. Anderson t-motives, Duke Math. J., Volume 53 (1985), pp. 457-502 | Zbl: 0679.14001

[2] Greg W. Anderson Rank one elliptic A-modules and A-harmonic series, Duke Math. J., Volume 73 (1994), pp. 491-542 | MR: 1262925 | Zbl: 0807.11032

[3] Greg W. Anderson Log-algebraicity of twisted A-harmonic series and special values of L-series in characteristic p, J. Number Theory, Volume 60 (1996), pp. 165-209 | Article | MR: 1405732 | Zbl: 0868.11031

[4] Greg W. Anderson; Dinesh S. Thakur Tensor powers of the Carlitz module and zeta values, Ann. Math., Volume 132 (1990) no. 1, pp. 159-191 | Article | MR: 1059938 | Zbl: 0713.11082

[5] Greg W. Anderson; Dinesh S. Thakur Multizeta values for 𝔽 q [t], their period interpretation and relations between them, Int. Math. Res. Not., Volume 2009 (2009) no. 11, pp. 2038-2055 | MR: 2507109 | Zbl: 1183.11052

[6] Bruno Anglès; Federico Pellarin; F. Tavares Ribeiro; F. Demeslay Arithmetic of positive characteristic L-series values in Tate algebras, Compos. Math., Volume 152 (2016) no. 1, pp. 1-61 | Article | MR: 3453387 | Zbl: 1336.11042

[7] Gebhard Böckle The distribution of the zeros of the Goss zeta-function for A=𝔽 2 [x,y]/(y 2 +y+x 3 +x+1), Math. Z., Volume 275 (2013) no. 3-4, pp. 835-861 | MR: 3127039 | Zbl: 1282.11115

[8] Leonard Carlitz On certain functions connected with polynomials in a Galois field, Duke Math. J., Volume 1 (1935), pp. 137-168 | MR: 1545872 | Zbl: 0012.04904

[9] Chieh-Yu Chang; Matthew A. Papanikolas; Jing Yu An effective criterion for Eulerian multizeta values in positive characteristic, J. Eur. Math. Soc., Volume 21 (2019) no. 2, pp. 405-440 | Article | MR: 3896206 | Zbl: 1417.11139

[10] Huei-Jeng Chen Anderson–Thakur polynomials and multizeta values in finite characteristic, Asian J. Math., Volume 21 (2017) no. 6, pp. 1135-1152 | Article | MR: 3778122 | Zbl: 1401.11124

[11] Christophe Debry Towards a class number formula for Drinfeld modules (2016) (Ph. D. Thesis)

[12] Jiangxue Fang Special L-values of abelian t-modules, J. Number Theory, Volume 147 (2015), pp. 300-325 | Article | MR: 3276327 | Zbl: 1360.11074

[13] David Goss Basic structures of Function Field Arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 35, Springer, 1996 | MR: 1423131 | Zbl: 0874.11004

[14] Nathan Green; Matthew A. Papanikolas Special L-values and Shtuka functions for Drinfeld modules on elliptic curves, Res. Math. Sci., Volume 5 (2018) no. 1, 4, 47 pages | MR: 3756176 | Zbl: 1441.11150

[15] José Alejandro Lara Rodríguez; Dinesh S. Thakur Zeta-like multizeta values for 𝔽 q [t], Indian J. Pure Appl. Math., Volume 45 (2014) no. 5, pp. 787-801 | MR: 3286087 | Zbl: 1365.11104

[16] Maxim Mornev Shtuka cohomology and special values of Goss L-functions (2018) (Ph. D. Thesis)

[17] Lenny Taelman A Dirichlet unit theorem for Drinfeld modules, Math. Ann., Volume 348 (2010) no. 4, pp. 899-907 | Article | MR: 2721645 | Zbl: 1217.11062

[18] Dinesh S. Thakur Drinfeld modules and arithmetic in function fields, Int. Math. Res. Not., Volume 1992 (1992) no. 9, pp. 185-197 | Article | MR: 1185833 | Zbl: 0756.11015

[19] Dinesh S. Thakur Shtukas and Jacobi sums, Invent. Math., Volume 111 (1993) no. 3, pp. 557-570 | Article | MR: 1202135 | Zbl: 0770.11032

[20] Dinesh S. Thakur Function Field Arithmetic, World Scientific, 2004 | Zbl: 1061.11001

[21] Dinesh S. Thakur Multizeta in function field arithmetic, t-motives: Hodge structures, transcendence and other motivic aspects (EMS Series of Congress Reports), European Mathematical Society, 2009, pp. 441-452 | Zbl: 1441.11222

[22] Dinesh S. Thakur Relations between multizeta values for 𝔽 q [t], Int. Math. Res. Not., Volume 2009 (2009) no. 12, pp. 2318-2346 | Article | MR: 2511913 | Zbl: 1189.11032

[23] Dinesh S. Thakur Shuffle Relations for Function Field Multizeta Values, Int. Math. Res. Not., Volume 2010 (2010) no. 11, pp. 1973-1980 | MR: 2646351 | Zbl: 1198.11077

[24] Dinesh S. Thakur Arithmetic of Gamma, Zeta and Multizeta values for Function Fields, Arithmetic geometry over global function fields (Advanced Courses in Mathematics - CRM Barcelona), Birkhäuser, 2014 | Zbl: 1396.11128

[25] Dinesh S. Thakur Multizeta values for function fields: A survey, J. Théor. Nombres Bordeaux, Volume 29 (2017) no. 3, pp. 997-1023 | Article | Numdam | MR: 3745257 | Zbl: 1430.11119

[26] George Todd Conjectural characterization for 𝔽 q (t)-linear relations between multizeta values, J. Number Theory, Volume 187 (2018), pp. 264-287 | Article | MR: 3766911 | Zbl: 1401.11125

[27] Jianqiang Zhao Multiple zeta functions, multiple polylogarithms and their special values, Series on Number Theory and Its Applications, 12, World Scientific, 2016 | MR: 3469645 | Zbl: 1367.11002

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