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Dinesh S. Thakur
Multizeta values for function fields: A survey
Journal de théorie des nombres de Bordeaux, 29 no. 3 (2017), p. 997-1023, doi: 10.5802/jtnb.1009
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Class. Math.: 11M32, 11G09
Keywords: t-motives, periods, shuffle relations, polylogarithms, mixed Tate motives

Résumé - Abstract

We give a survey of the recent developments in the understanding of the multizeta values for function fields.


[1] Greg W. Anderson, $t$-motives, Duke Math. J. 53 (1986), p. 457-502 Article
[2] Greg W. Anderson, Rank one elliptic $A$-modules and $A$-harmonic series, Duke Math. J. 73 (1994), p. 491-542 Article
[3] Greg W. Anderson, Log-algebraicity of twisted $A$-harmonic series and special values of $L$-series in characteristic $p$, J. Number Theory 60 (1996), p. 165-209 Article
[4] Greg W. Anderson, Digit patterns and the formal additive group, Isr. J. Math. 161 (2007), p. 125-139 Article
[5] 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 Article
[6] Greg W. Anderson & Dinesh S. Thakur, “Ihara power series for $\mathbb{F}_q[t]$”, preprint
[7] Greg W. Anderson & Dinesh S. Thakur, Tensor powers of the Carlitz module and zeta values, Ann. Math. 132 (1990), p. 159-191 Article
[8] 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
[9] Bruno Anglès, Tuan Ngo Dac & Floric Tavares Ribeiro, “Special functions and twisted $L$-series”,, 2016
[10] Bruno Anglès, Federico Pellarin & Floric Tavares Ribeiro, Arithmetic of positive characteristic $L$-series values in Tate algebras, Compos. Math. 152 (2016), p. 1-61 Article
[11] Leonard Carlitz, On certain functions connected with polynomials in a Galois field, Duke Math. J. 1 (1935), p. 137-168 Article
[12] Chieh-Yu Chang, Linear independence of monomials of multizeta values in positive characteristic, Compos. Math. 150 (2014), p. 1789-1808 Article
[13] Chieh-Yu Chang, Linear relations among double zeta values in positive characteristic, Camb. J. Math. 4 (2016), p. 289-331 Article
[14] Chieh-Yu Chang & Yoshinori Mishiba, “On multiple polylogarithms in chracteristic $p$: $v$-adic vanishing versus $\infty $-adic Eulerian”, preprint
[15] Chieh-Yu Chang, Matthew A. Papanikolas & Jing Yu, “An effective criterion for Eulerian multizeta values in positive characteristic”,, to appear in J. Eur. Math. Soc. (JEMS), 2015
[16] Chieh-Yu Chang & Jing Yu, Determination of algebraic relations among special zeta values in positive characteristic, Adv. Math. 216 (2007), p. 321-345 Article
[17] Steven Charlton, $\zeta (\lbrace \lbrace 2\rbrace ^m, 1, \lbrace 2\rbrace ^m, 3\rbrace ^n, \lbrace 2\rbrace ^m)/\pi ^{4n+2m(2n+1)}$ is rational, J. Number Theory 148 (2015), p. 463-477 Article
[18] Huei-Jeng Chen, “Anderson-Thakur polynomials and multizeta values in finite characteristic”, preprint
[19] Huei-Jeng Chen, On shuffle of double zeta values over $\mathbb{F}_q[t]$, J. Number Theory 148 (2015), p. 153-163 Article
[20] Christophe Debry, Towards a class number formula for Drinfeld modules, Ph. D. Thesis, U Amsterdam (The Netherlands), 2016
[21] Leonhard Euler, Meditationes circa singulare serierum genus, Novi Comm. Acad. Sci. Petropol. 20 (1776), p. 140-186
[22] Jiangxue Fang, Special $L$-values of abelian $t$-modules, J. Number Theory 147 (2015), p. 300-325 Article
[23] David Goss, Basic structures of function field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3 35, Springer, 1996
[24] Yasutaka Ihara, Braids, Galois groups and some arithmetic functions, in Proceedings of the international congress of mathematicians (Kyoto, 1990), Math. Soc. Japan, 1991, p. 99-120
[25] Yen-Liang Kuan & Yi-Hsuan Lin, Criterion for deciding zeta-like multizeta values in positive characteristic, Exp. Math. 25 (2016), p. 246-256 Article
[26] Vincent Lafforgue, Valeurs spéciales des fonction $L$ en caractéristique $p$, J. Number Theory 129 (2009), p. 2600-2634 Article
[27] José Alejandro Lara Rodríguez, “Some conjectures and results about multizeta values for $\mathbb{F}_q[t]$” 2009, Master’s thesis for The Autonomous University of Yucatan (Mexico)
[28] José Alejandro Lara Rodríguez, Some conjectures and results about multizeta values for $\mathbb{F}_q[t]$, J. Number Theory 130 (2010), p. 1013-1023 Article
[29] José Alejandro Lara Rodríguez, Relations between multizeta values in characteristic $p$, J. Number Theory 131 (2011), p. 2081-2099 Article
[30] José Alejandro Lara Rodríguez & Dinesh S. Thakur, Zeta-like multizeta values for $\mathbb{F}_q[t]$, Indian J. Pure Appl. Math. 45 (2014), p. 787-801 Article
[31] José Alejandro Lara Rodríguez & Dinesh S. Thakur, Multiplicative relations between coefficients of logarithmic derivatives of $\mathbb{F}_q$-linear functions and applications, J. Algebra Appl. 14 (2015) Article
[32] José Alejandro Lara Rodríguez & Dinesh S. Thakur, Multizeta shuffle relations for function fields with non-rational infinite place, Finite Fields Appl. 37 (2016), p. 344-356 Article
[33] Riad Masri, Multiple zeta values over global function fields, Proc. Symp. Pure Math. 75 (2006), p. 157-75 Article
[34] Yoshinori Mishiba, “On algebraic independence of certain multizeta values in characteristic $p$”,, to appear in J. Number Theory, 2014
[35] Yoshinori Mishiba, $p$-th power relations and Euler-Carlitz relations among multizeta values, RIMS Kôkyûroku Bessatsu B53 (2015), p. 13-29
[36] Matthew A. Papanikolas, “Log-Algebraicity on Tensor Powers of the Carlitz Module and Special Values of Goss L-Functions”, work in progress
[37] Matthew A. Papanikolas, Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms, Invent. Math. 171 (2008), p. 123-174 Article
[38] Federico Pellarin, Values of certain $L$-series in positive characteristic, Ann. Math. 176 (2012), p. 2055-2093 Article
[39] Federico Pellarin, A note on multiple zeta values in Tate algebras, Riv. Mat. Univ. Parma 7 (2016), p. 71-100
[40] Federico Pellarin, A sum shuffle formula for zeta values in Tate algebras, J. Théor. Nombres Bordx 29 (2017)
[41] Federico Pellarin & Rudolph Perkins, “On twisted $A$-harmonic series and Carlitz finite zeta values”,, 2016
[42] Lenny Taelman, A Dirichlet unit theorem for Drinfeld modules, Math. Ann. 348 (2010), p. 899-907 Article
[43] Lenny Taelman, Special L-values of Drinfeld modules, Ann. Math. 175 (2012), p. 369-391 Article
[44] Dinesh S. Thakur, “Multizeta in function field arithmetic”, To appear in the proceedings of the 2009 Banff workshop (published by European Mathematical Society)
[45] Dinesh S. Thakur, Shuffle relations for function field multizeta values, Int. Math. Res. Not. (1973–1980)
[46] Dinesh S. Thakur, Drinfeld modules and arithmetic in function fields, Int. Math. Res. Not. 1992 (1992), p. 185-197 Article
[47] Dinesh S. Thakur, Function Field Arithmetic, World Scientific, 2004
[48] 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 Article
[49] Dinesh S. Thakur, Relations between multizeta values for $\mathbb{F}_q[t]$, Int. Math. Res. Not. 2009 (2009), p. 2318-2346 Article
[50] Dinesh S. Thakur, Arithmetic of Gamma, Zeta and Multizeta values for Function Fields, Arithmetic geometry over Global Function Fields, Advanced courses in Mathematics, Birkhäuser, 2014, p. 195–279 Article
[51] George Todd, Linear relations between multizeta values, Ph. D. Thesis, University of Arizona (USA), 2015
[52] L. I. Wade, Certain quantities transcendental over $GF(p^n, x)$, Duke Math. J. 8 (1941), p. 701-720 Article
[53] Jing Yu, Transcendence and special zeta values in characteristic $p$, Ann. Math. 134 (1991), p. 1-23 Article
[54] Jing Yu, Analytic homomorphisms into Drinfeld modules, Ann. Math. 145 (1997), p. 215-233 Article
[55] Jianqiang Zhao, Multiple zeta functions, multiple polylogarithms and their special values, Series on Number Theory and Its Applications 12, World Scientific, 2016