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, Tome 29 (2017) no. 3, pp. 1049-1058.

Résumé
Abstract

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].

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].

Reçu le : 2016-11-07
Révisé le : 2017-05-08
Accepté le : 2017-05-11
Publié le : 2017-12-12

DOI : 10.5802/jtnb.1011

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

Affiliations des auteurs :

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

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{JTNB_2017__29_3_1049_0,
     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},
     number = {3},
     year = {2017},
     doi = {10.5802/jtnb.1011},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1011/}
}

TY  - JOUR
AU  - Chieh-Yu Chang
AU  - Yoshinori Mishiba
TI  - On finite Carlitz multiple polylogarithms
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2017
SP  - 1049
EP  - 1058
VL  - 29
IS  - 3
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1011/
DO  - 10.5802/jtnb.1011
LA  - en
ID  - JTNB_2017__29_3_1049_0
ER  -

%0 Journal Article
%A Chieh-Yu Chang
%A Yoshinori Mishiba
%T On finite Carlitz multiple polylogarithms
%J Journal de théorie des nombres de Bordeaux
%D 2017
%P 1049-1058
%V 29
%N 3
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1011/
%R 10.5802/jtnb.1011
%G en
%F JTNB_2017__29_3_1049_0

Chieh-Yu Chang; Yoshinori Mishiba. On finite Carlitz multiple polylogarithms. Journal de théorie des nombres de Bordeaux, Tome 29 (2017) no. 3, pp. 1049-1058. doi : 10.5802/jtnb.1011. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1011/

Bibliographie
Cité par

[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

Cité par Sources :