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Chieh-Yu Chang; Yoshinori Mishiba
On finite Carlitz multiple polylogarithms
Journal de théorie des nombres de Bordeaux, 29 no. 3 (2017), p. 1049-1058
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Class. Math.: 11R58, 11M38
Keywords: Finite Carlitz multiple polylogarithms, finite multiple zeta values, Anderson–Thakur polynomials

Résumé - Abstract

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


[1] Greg W. Anderson, W. Dale Brownawell & Matthew A. Papanikolas, Determination of the algebraic relations among special $\Gamma $-values in positive characteristic, Ann. Math. 160 (2004), p. 237-313
[2] Greg W. Anderson & Dinesh S. Thakur, Tensor powers of the Carlitz module and zeta values, Ann. Math. 132 (1990), p. 159-191
[3] Greg W. Anderson & Dinesh S. Thakur, Multizeta values for $\mathbb{F}_{q}[t]$, their period interpretation, and relations between them, Int. Math. Res. Not. 2009 (2009), p. 2038-2055
[4] Bruno Anglès, Tuan Ngo Dac & Floric Tavares Ribeiro, “Exceptional zeros of $L$-series and Bernoulli-Carlitz numbers”,, 2015
[5] Leonard Carlitz, On certain functions connected with polynomials in a Galois field, Duke Math. J. 1 (1935), p. 137-168
[6] Chieh-Yu Chang, Linear independence of monomials of multizeta values in positive characteristic, Compos. Math. 150 (2014), p. 1789-1808
[7] Chieh-Yu Chang, Linear relations among double zeta values in positive characteristic, Camb. J. Math. 4 (2016), p. 289-331
[8] Chieh-Yu Chang & Yoshinori Mishiba, “On multiple polylogarithms in characteristic $p$: $v$-adic vanishing versus $\infty $-adic Eulerianness”,, to appear in Int. Math. Res. Not., 2017
[9] Chieh-Yu Chang & Matthew A. Papanikolas, Algebraic independence of periods and logarithms of Drinfeld modules, J. Am. Math. Soc. 25 (2012), p. 123-150
[10] Chieh-Yu Chang & Jing Yu, Determination of algebraic relations among special zeta values in positive characteristic, Adv. Math. 216 (2007), p. 321-345
[11] Huei-Jeng Chen, On shuffle of double zeta values over $\mathbb{F}_q[t]$, J. Number Theory 148 (2015), p. 153-163
[12] David Goss, Basic structures of function field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3 35, Springer, 1996
[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 173 (2017), p. 512-528
[15] Matthew A. Papanikolas, Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms, Invent. Math. 171 (2008), p. 123-174
[16] Federico Pellarin & Rudolph Perkins, “On twisted $A$-harmonic series and Carlitz finite zeta values”,, 2016
[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
[19] Dinesh S. Thakur, Power sums with applications to multizeta and zeta zero distribution for $\mathbb{F}_q[t]$, Finite Fields Appl. 15 (2009), p. 534-552
[20] Dinesh S. Thakur, Shuffle Relations for Function Field Multizeta Values, Int. Math. Res. Not. 2010 (2010), p. 1973-1980
[21] George Todd, Linear relations between multizeta values, Ph. D. Thesis, University of Arizona (USA), 2015
[22] Michel Waldschmidt, Multiple polylogarithms: an introduction., in Number theory and discrete mathematics (Chandigarh, 2000), Trends in Mathematics, Birkhäuser, 2002, p. 1-12
[23] Jing Yu, Transcendence and special zeta values in characteristic $p$, Ann. Math. 134 (1991), p. 1-23
[24] Jing Yu, Analytic homomorphisms into Drinfeld modules, Ann. Math. 145 (1997), p. 215-233
[25] Jianqiang Zhao, Multiple zeta functions, multiple polylogarithms and their special values, Series on Number Theory and Its Applications 12, World Scientific, 2016