On finite Carlitz multiple polylogarithms

Chieh-Yu Chang; Yoshinori Mishiba

doi:10.5802/jtnb.1011

Chieh-Yu Chang ¹; Yoshinori Mishiba ²

¹ Department of Mathematics National Tsing Hua University Hsinchu City 30042, Taiwan R.O.C.
² Department of Life, Environment and Materials Science Fukuoka Institute of Technology, Japan

Journal de théorie des nombres de Bordeaux, Volume 29 (2017) no. 3, pp. 1049-1058.

Abstract
Résumé

In this paper, we define finite Carlitz multiple polylogarithms and show that every finite multiple zeta value over the rational function field $𝔽_{q} (θ)$ is an $𝔽_{q} (θ)$ -linear combination of finite Carlitz multiple polylogarithms at integral points. It is completely compatible with the formula for Thakur MZV’s established in [6].

Dans cet article nous définissons la notion de polylogarithme multiple fini de Carlitz et montrons que chaque valeur de zêta multiple finie définie sûr un corps de fonctions rationnelles $𝔽_{q} (θ)$ est une combinaison linéaire des valeurs des polylogarithmes multiples finis de Carlitz evalués en des points entiers. Cela est complètement compatible avec la formule des MZVs de Thakur établie dans [6].

Received: 2016-11-07
Revised: 2017-05-08
Accepted: 2017-05-11
Published online: 2017-12-12

DOI: 10.5802/jtnb.1011

Classification: 11R58, 11M38
Keywords: Finite Carlitz multiple polylogarithms, finite multiple zeta values, Anderson–Thakur polynomials

Author's affiliations:

Chieh-Yu Chang ¹; Yoshinori Mishiba ²

¹ Department of Mathematics National Tsing Hua University Hsinchu City 30042, Taiwan R.O.C.
² Department of Life, Environment and Materials Science Fukuoka Institute of Technology, Japan

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Chieh-Yu Chang and Yoshinori Mishiba},
     title = {On finite {Carlitz} multiple polylogarithms},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {1049--1058},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {29},
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     year = {2017},
     doi = {10.5802/jtnb.1011},
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Chieh-Yu Chang; Yoshinori Mishiba. On finite Carlitz multiple polylogarithms. Journal de théorie des nombres de Bordeaux, Volume 29 (2017) no. 3, pp. 1049-1058. doi : 10.5802/jtnb.1011. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1011/

References
Cited by

[1] Greg W. Anderson; W. Dale Brownawell; Matthew A. Papanikolas Determination of the algebraic relations among special $Γ$ -values in positive characteristic, Ann. Math., Volume 160 (2004) no. 1, pp. 237-313 | DOI | Zbl

[2] 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 | DOI | Zbl

[3] 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 | Zbl

[4] Bruno Anglès; Tuan Ngo Dac; Floric Tavares Ribeiro Exceptional zeros of $L$ -series and Bernoulli-Carlitz numbers (2015) (https://arxiv.org/abs/1511.06209v2)

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

[6] Chieh-Yu Chang Linear independence of monomials of multizeta values in positive characteristic, Compos. Math., Volume 150 (2014) no. 11, pp. 1789-1808 | DOI | Zbl

[7] Chieh-Yu Chang Linear relations among double zeta values in positive characteristic, Camb. J. Math., Volume 4 (2016) no. 3, pp. 289-331 | DOI | Zbl

[8] Chieh-Yu Chang; Yoshinori Mishiba On multiple polylogarithms in characteristic $p$ : $v$ -adic vanishing versus $\infty$ -adic Eulerianness (2017) (https://arxiv.org/abs/1511.03490, to appear in Int. Math. Res. Not.)

[9] Chieh-Yu Chang; Matthew A. Papanikolas Algebraic independence of periods and logarithms of Drinfeld modules, J. Am. Math. Soc., Volume 25 (2012) no. 1, pp. 123-150 | DOI | Zbl

[10] Chieh-Yu Chang; Jing Yu Determination of algebraic relations among special zeta values in positive characteristic, Adv. Math., Volume 216 (2007) no. 1, pp. 321-345 | DOI | Zbl

[11] Huei-Jeng Chen On shuffle of double zeta values over $𝔽_{q} [t]$ , J. Number Theory, Volume 148 (2015), pp. 153-163 | DOI | Zbl

[12] David Goss Basic structures of function field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3, 35, Springer, 1996, xiii+422 pages | Zbl

[13] Masanobu Kaneko; Don Zagier Finite multiple zeta values (in preparation)

[14] Yoshinori Mishiba On algebraic independence of certain multizeta values in characteristic $p$ , J. Number Theory, Volume 173 (2017), pp. 512-528 | DOI | Zbl

[15] Matthew A. Papanikolas Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms, Invent. Math., Volume 171 (2008) no. 1, pp. 123-174 | DOI | Zbl

[16] Federico Pellarin; Rudolph Perkins On twisted $A$ -harmonic series and Carlitz finite zeta values (2016) (https://arxiv.org/abs/1512.05953)

[17] Dinesh S. Thakur Multizeta in function field arithmetic To appear in the proceedings of the 2009 Banff workshop (published by European Mathematical Society)

[18] Dinesh S. Thakur Function Field Arithmetic, World Scientific, 2004, xv+388 pages | Zbl

[19] Dinesh S. Thakur Power sums with applications to multizeta and zeta zero distribution for $𝔽_{q} [t]$ , Finite Fields Appl., Volume 15 (2009) no. 4, pp. 534-552 | DOI | Zbl

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

[21] George Todd Linear relations between multizeta values, University of Arizona (USA) (2015) (Ph. D. Thesis)

[22] Michel Waldschmidt Multiple polylogarithms: an introduction., Number theory and discrete mathematics (Chandigarh, 2000) (Trends in Mathematics) (2002), pp. 1-12 | Zbl

[23] Jing Yu Transcendence and special zeta values in characteristic $p$ , Ann. Math., Volume 134 (1991) no. 1, pp. 1-23 | DOI | Zbl

[24] Jing Yu Analytic homomorphisms into Drinfeld modules, Ann. Math., Volume 145 (1997) no. 2, pp. 215-233 | DOI | Zbl

[25] Jianqiang Zhao Multiple zeta functions, multiple polylogarithms and their special values, Series on Number Theory and Its Applications, 12, World Scientific, 2016, xxi+595 pages | Zbl

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