Geodesic
On Thakur’s basis conjecture for multiple zeta values in positive characteristic
Forum of Mathematics, Pi, Tome 11 (2023)

Voir la notice de l'article provenant de la source Cambridge University Press

In this paper, we study multiple zeta values (abbreviated as MZV’s) over function fields in positive characteristic. Our main result is to prove Thakur’s basis conjecture, which plays the analogue of Hoffman’s basis conjecture for real MZV’s. As a consequence, we derive Todd’s dimension conjecture, which is the analogue of Zagier’s dimension conjecture for classical real MZV’s. 
@article{10_1017_fmp_2023_26,
     author = {Chieh-Yu Chang and Yen-Tsung Chen and Yoshinori Mishiba},
     title = {On {Thakur{\textquoteright}s} basis conjecture for multiple zeta values in positive characteristic},
     journal = {Forum of Mathematics, Pi},
     publisher = {mathdoc},
     volume = {11},
     year = {2023},
     doi = {10.1017/fmp.2023.26},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fmp.2023.26/}
}
TY  - JOUR
AU  - Chieh-Yu Chang
AU  - Yen-Tsung Chen
AU  - Yoshinori Mishiba
TI  - On Thakur’s basis conjecture for multiple zeta values in positive characteristic
JO  - Forum of Mathematics, Pi
PY  - 2023
VL  - 11
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1017/fmp.2023.26/
DO  - 10.1017/fmp.2023.26
LA  - en
ID  - 10_1017_fmp_2023_26
ER  - 
%0 Journal Article
%A Chieh-Yu Chang
%A Yen-Tsung Chen
%A Yoshinori Mishiba
%T On Thakur’s basis conjecture for multiple zeta values in positive characteristic
%J Forum of Mathematics, Pi
%D 2023
%V 11
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1017/fmp.2023.26/
%R 10.1017/fmp.2023.26
%G en
%F 10_1017_fmp_2023_26
Chieh-Yu Chang; Yen-Tsung Chen; Yoshinori Mishiba. On Thakur’s basis conjecture for multiple zeta values in positive characteristic. Forum of Mathematics, Pi, Tome 11 (2023). doi: 10.1017/fmp.2023.26

[1] Anderson, G. W., ‘t-motives’, Duke Math. J. 53(2) (1986), 457–502.Google Scholar | DOI

[2] Anderson, G. W., Brownawell, W. D. and Papanikolas, M. A., ‘Determination of the algebraic relations among special -values in positive characteristic’, Ann. of Math. (2) 160(1) (2004), 237–313.Google Scholar | DOI

[3] Anderson, G. W. and Thakur, D. S., ‘Tensor powers of the Carlitz module and zeta values’, Ann. of Math. (2) 132(1) (1990), 159–191.Google Scholar | DOI

[4] Anderson, G. W. and Thakur, D. S., ‘Multizeta values for F [t], their period interpretation, and relations between them’, Int. Math. Res. Not. IMRN (11) (2009), 2038–2055.Google Scholar

[5] André, Y., Une Introduction aux Motifs (Motifs Purs, Motifs Mixtes, Périodes), Panoramas et Synthéses vol. 17 (Société Mathématique de France, Paris, 2004).Google Scholar

[6] Brown, F., ‘Mixed Tate motives over Z’, Ann. of Math. (2) 175(2) (2012), 949–976.Google Scholar | DOI

[7] Burgos Gil, J. I. and Fresán, J., ‘Multiple zeta values: from numbers to motives’, to appear in Clay Mathematics Proceedings.Google Scholar

[8] Carlitz, L., ‘On certain functions connected with polynomials in a Galois field’, Duke Math. J. 1(2) (1935), 137–168.Google Scholar | DOI

[9] Chang, C.-Y., ‘Linear independence of monomials of multizeta values in positive characteristic’, Compositio Math. 150 (2014), 1789–1808.Google Scholar | DOI

[10] Chang, C.-Y., ‘Linear relations among double zeta values in positive characteristic’, Camb. J. Math. 4(3) (2016), 89–331.Google Scholar | DOI

[11] Chang, C.-Y., Chen, Y.-T. and Mishiba, Y., ‘Algebra structure on multiple zeta values in positive characteristic’, Camb. J. Math. 10(4) (2022), 743–783.Google Scholar | DOI

[12] Chen, Y.-T. and Harada, R., ‘On lower bounds of the dimension of multizeta values in positive characteristic’, Doc. Math. 26 (2021), 537–559.Google Scholar | DOI

[13] Chang, C.-Y. and Mishiba, Y., ‘On a conjecture of Furusho over function fields’, Invent. Math. 223 (2021), 49–102.Google Scholar | DOI

[14] Chang, C.-Y., Papanikolas, M. A. and Yu, J., ‘An effective criterion for Eulerian multizeta values in positive characteristic’, J. Eur. Math. Soc. (JEMS) 21(2) (2019), 405–440.Google Scholar | DOI

[15] Chen, H.-J., ‘On shuffle of double zeta values over F[t]’, J. Number Theory 148 (2015), 153–163.Google Scholar | DOI

[16] Deligne, P. and Goncharov, A. B., ‘Groupes fondamentaux motiviques de Tate mixte’, Ann. Sci. École Norm. Sup. (4) 38(1) (2005), 1–56.Google Scholar | DOI

[17] Gangl, H., Kazeko, M. and Zagier, D., ‘Double zeta values and modular forms’, in Automorphic Forms and Zeta Functions (World Sci. Publ., Hackensack, NJ, 2006), 71–106.Google Scholar | DOI

[18] Goncharov, A. B., ‘Multiple polylogarithms and mixed Tate motives’, .Google Scholar | arXiv

[19] Goss, D., ‘L-adic zeta functions, v-series, and measures for function fields. With an addendum’, Invent. Math. 55(2) (1979), 107–119.Google Scholar | DOI

[20] Goss, D., Basic Structures of Function Field Arithmetic (Springer-Verlag, Berlin, 1996).Google Scholar | DOI

[21] Hoffman, M., ‘The algebra of multiple harmonic series’, J. Algebra 194 (1997), 477–495.Google Scholar | DOI

[22] Im, B. H., Kim, H., Le, K. N., Ngo Dac, T. and Pham, L. H., ‘Zagier-Hoffman’s conjectures in positive characteristic’, .Google Scholar | arXiv

[23] Ihara, K., Kaneko, M. and Zagier, D., ‘Derivation and double shuffle relations for multiple zeta values’, Compositio Math. 142 (2006), 307–338.Google Scholar | DOI

[24] Kuan, Y.-L. and Lin, Y.-H., ‘Criterion for deciding zeta-like multizeta values in positive characteristic’, Exp. Math. 25(3) (2016), 246–256.Google Scholar | DOI

[25] Lara Rodríguez, J. A. and Thakur, D. S., ‘Zeta-like multizeta values for F[t]’, Indian J. Pure Appl. Math. 45(5) (2014), 785–798.Google Scholar

[26] Ngo Dac, T., ‘On Zagier-Hoffman’s conjectures in positive characteristic’, Ann. of Math. (2) 194(1) (2021), 361–392.Google Scholar | DOI

[27] Papanikolas, M. A., ‘Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms’, Invent. Math. 171(1) (2008), 123–174.Google Scholar | DOI

[28] Racinet, G., ‘Doubles mélanges des polylogarithmes multiples aux racines de l’unité’, Publ. Math. Inst. Hautes Études Sci. 95 (2002), 185–231.Google Scholar | DOI

[29] Terasoma, T., ‘Mixed Tate motives and multiple zeta values’, Invent. Math. 149(2) (2002), 339–369.Google Scholar | DOI

[30] Todd, G., ‘A conjectural characterization for F(t)-linear relations between multizeta values’, J. Number Theory 187, 264–287 (2018).Google Scholar | DOI

[31] Thakur, D. S., Function Field Arithmetic (World Scientific Publishing, River Edge NJ, 2004).Google Scholar | DOI

[32] Thakur, D. S., ‘Power sums with applications to multizeta and zeta zero distribution for F[t]’, Finite Fields Appl. 15(4) (2009), 534–552.Google Scholar | DOI

[33] Thakur, D. S., ‘Relations between multizeta values for F[t]’, Int. Math. Res. Not. IMRN (12) (2009), 2318–2346.Google Scholar | DOI

[34] Thakur, D. S., ‘Shuffle relations for function field multizeta values’, Int. Math. Res. Not. IMRN (11) (2010), 1973–1980.Google Scholar

[35] Thakur, D. S., ‘Multizeta values for function fields: a survey’, J. Théor. Nombres Bordeaux 29(3) (2017), 997–1023.Google Scholar | DOI

[36] Zagier, D., ‘Values of zeta functions and their applications’, in ECM Volume (Progress in Math.) vol. 120 (1994), 497–512.Google Scholar

[37] Zhao, J., Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values (Series on Number Theory and Its Applications, 12) (World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2016).Google Scholar | DOI

Cité par Sources :