Nous donnons une vue d’ensemble des développements récents concernant la compréhension des valeurs multi-zêta pour les corps de fonctions.
We give a survey of the recent developments in the understanding of the multizeta values for function fields.
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/jtnb.1009
Mots-clés : t-motives, periods, shuffle relations, polylogarithms, mixed Tate motives
@article{JTNB_2017__29_3_997_0, author = {Thakur, Dinesh S.}, title = {Multizeta values for function fields: {A} survey}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {997--1023}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {29}, number = {3}, year = {2017}, doi = {10.5802/jtnb.1009}, language = {en}, url = {https://www.numdam.org/articles/10.5802/jtnb.1009/} }
TY - JOUR AU - Thakur, Dinesh S. TI - Multizeta values for function fields: A survey JO - Journal de théorie des nombres de Bordeaux PY - 2017 SP - 997 EP - 1023 VL - 29 IS - 3 PB - Société Arithmétique de Bordeaux UR - https://www.numdam.org/articles/10.5802/jtnb.1009/ DO - 10.5802/jtnb.1009 LA - en ID - JTNB_2017__29_3_997_0 ER -
%0 Journal Article %A Thakur, Dinesh S. %T Multizeta values for function fields: A survey %J Journal de théorie des nombres de Bordeaux %D 2017 %P 997-1023 %V 29 %N 3 %I Société Arithmétique de Bordeaux %U https://www.numdam.org/articles/10.5802/jtnb.1009/ %R 10.5802/jtnb.1009 %G en %F JTNB_2017__29_3_997_0
Thakur, Dinesh S. Multizeta values for function fields: A survey. Journal de théorie des nombres de Bordeaux, Tome 29 (2017) no. 3, pp. 997-1023. doi : 10.5802/jtnb.1009. https://www.numdam.org/articles/10.5802/jtnb.1009/
[1]
[2] Rank one elliptic
[3] Log-algebraicity of twisted
[4] Digit patterns and the formal additive group, Isr. J. Math., Volume 161 (2007), pp. 125-139 | DOI | Zbl
[5] Determination of the algebraic relations among special
[6] Ihara power series for
[7] Tensor powers of the Carlitz module and zeta values, Ann. Math., Volume 132 (1990) no. 1, pp. 159-191 | DOI | Zbl
[8] Multizeta values for
[9] Special functions and twisted
[10] Arithmetic of positive characteristic
[11] On certain functions connected with polynomials in a Galois field, Duke Math. J., Volume 1 (1935), pp. 137-168 | DOI | Zbl
[12] Linear independence of monomials of multizeta values in positive characteristic, Compos. Math., Volume 150 (2014) no. 11, pp. 1789-1808 | DOI | Zbl
[13] Linear relations among double zeta values in positive characteristic, Camb. J. Math., Volume 4 (2016) no. 3, pp. 289-331 | DOI | Zbl
[14] On multiple polylogarithms in chracteristic
[15] An effective criterion for Eulerian multizeta values in positive characteristic (2015) https://arxiv.org/abs/1411.0124, to appear in J. Eur. Math. Soc. (JEMS)
[16] Determination of algebraic relations among special zeta values in positive characteristic, Adv. Math., Volume 216 (2007) no. 1, pp. 321-345 | DOI | Zbl
[17]
[18] Anderson-Thakur polynomials and multizeta values in finite characteristic (preprint)
[19] On shuffle of double zeta values over
[20] Towards a class number formula for Drinfeld modules, U Amsterdam (The Netherlands) (2016) (Ph. D. Thesis)
[21] Meditationes circa singulare serierum genus, Novi Comm. Acad. Sci. Petropol., Volume 20 (1776), pp. 140-186 reprinted in Opera Omnia, ser. I 15 (1927), p. 217-267
[22] Special
[23] Basic structures of function field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3, 35, Springer, 1996, xiii+422 pages | Zbl
[24] Braids, Galois groups and some arithmetic functions, Proceedings of the international congress of mathematicians (Kyoto, 1990), Math. Soc. Japan (1991), pp. 99-120 | Zbl
[25] Criterion for deciding zeta-like multizeta values in positive characteristic, Exp. Math., Volume 25 (2016) no. 3, pp. 246-256 | DOI | Zbl
[26] Valeurs spéciales des fonction
[27] Some conjectures and results about multizeta values for
[28] Some conjectures and results about multizeta values for
[29] Relations between multizeta values in characteristic
[30] Zeta-like multizeta values for
[31] Multiplicative relations between coefficients of logarithmic derivatives of
[32] Multizeta shuffle relations for function fields with non-rational infinite place, Finite Fields Appl., Volume 37 (2016), pp. 344-356 | DOI | Zbl
[33] Multiple zeta values over global function fields, Proc. Symp. Pure Math., Volume 75 (2006), p. 157-75 | DOI | Zbl
[34] On algebraic independence of certain multizeta values in characteristic
[35]
[36] Log-Algebraicity on Tensor Powers of the Carlitz Module and Special Values of Goss L-Functions (work in progress)
[37] Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms, Invent. Math., Volume 171 (2008) no. 1, pp. 123-174 | DOI | Zbl
[38] Values of certain
[39] A note on multiple zeta values in Tate algebras, Riv. Mat. Univ. Parma, Volume 7 (2016) no. 1, pp. 71-100 | Zbl
[40] A sum shuffle formula for zeta values in Tate algebras, J. Théor. Nombres Bordx, Volume 29 (2017) no. 3
[41] On twisted
[42] A Dirichlet unit theorem for Drinfeld modules, Math. Ann., Volume 348 (2010) no. 4, pp. 899-907 | DOI | Zbl
[43] Special L-values of Drinfeld modules, Ann. Math., Volume 175 (2012) no. 1, pp. 369-391 | DOI | Zbl
[44] Multizeta in function field arithmetic To appear in the proceedings of the 2009 Banff workshop (published by European Mathematical Society)
[45] Shuffle relations for function field multizeta values, Int. Math. Res. Not. (1973–1980) no. 2010, 11 pages | Zbl
[46] Drinfeld modules and arithmetic in function fields, Int. Math. Res. Not., Volume 1992 (1992) no. 9, pp. 185-197 | DOI | Zbl
[47] Function Field Arithmetic, World Scientific, 2004, xv+388 pages | Zbl
[48] Power sums with applications to multizeta and zeta zero distribution for
[49] Relations between multizeta values for
[50] Arithmetic of Gamma, Zeta and Multizeta values for Function Fields, Arithmetic geometry over Global Function Fields (Advanced courses in Mathematics), Birkhäuser, 2014, pp. 195-279 | DOI
[51] Linear relations between multizeta values, University of Arizona (USA) (2015) (Ph. D. Thesis)
[52] Certain quantities transcendental over
[53] Transcendence and special zeta values in characteristic
[54] Analytic homomorphisms into Drinfeld modules, Ann. Math., Volume 145 (1997) no. 2, pp. 215-233 | DOI | Zbl
[55] Multiple zeta functions, multiple polylogarithms and their special values, Series on Number Theory and Its Applications, 12, World Scientific, 2016, xxi+595 pages | Zbl
- Note on the Linear Independence of Alternating Multiple Zeta Values in Positive Characteristic, Acta Mathematica Vietnamica, Volume 49 (2024) no. 3, p. 485 | DOI:10.1007/s40306-024-00554-4
- Colored multizeta values in positive characteristic, Forum Mathematicum (2024) | DOI:10.1515/forum-2023-0226
- Zagier–Hoffman’s Conjectures in Positive Characteristic, Forum of Mathematics, Pi, Volume 12 (2024) | DOI:10.1017/fmp.2024.18
- Universal families of Eulerian multiple zeta values in positive characteristic, Advances in Mathematics, Volume 422 (2023), p. 109003 | DOI:10.1016/j.aim.2023.109003
- On Thakur’s basis conjecture for multiple zeta values in positive characteristic, Forum of Mathematics, Pi, Volume 11 (2023) | DOI:10.1017/fmp.2023.26
- Algebraic relations among Goss’s zeta values on elliptic curves, Forum of Mathematics, Sigma, Volume 11 (2023) | DOI:10.1017/fms.2023.94
- On Log-Algebraic Identities for Anderson t-Modules and Characteristic p Multiple Zeta Values, International Mathematics Research Notices, Volume 2023 (2023) no. 16, p. 13687 | DOI:10.1093/imrn/rnac141
- Vanishing of multizeta values over at negative integers, Canadian Mathematical Bulletin, Volume 65 (2022) no. 1, p. 9 | DOI:10.4153/s0008439521000035
- Trivial multiple zeta values in Tate algebras, International Mathematics Research Notices, Volume 2022 (2022) no. 18, p. 14319 | DOI:10.1093/imrn/rnab104
- On multi-poly-Bernoulli–Carlitz numbers, Journal of Number Theory, Volume 232 (2022), p. 406 | DOI:10.1016/j.jnt.2018.11.003
- Zeta-like multiple zeta values in positive characteristic, Mathematische Zeitschrift, Volume 301 (2022) no. 2, p. 2037 | DOI:10.1007/s00209-022-02970-4
- Zeta-like Multizeta Values for higher genus curves, Journal de Théorie des Nombres de Bordeaux, Volume 33 (2021) no. 2, p. 553 | DOI:10.5802/jtnb.1169
Cité par 12 documents. Sources : Crossref