Mixed Tate motives over $\mathbb{Z}$

Abstract

We prove that the category of mixed Tate motives over $\mathbb{Z}$ is spanned by the motivic fundamental group of $\mathbb{P}^1$ minus three points. We prove a conjecture by M. Hoffman which states that every multiple zeta value is a $\mathbb{Q}$-linear combination of $\zeta(n_1,\ldots, n_r)$, where $n_i\in \{2,3\}$.

  • [Andre] Y. André, "Une introduction aux motifs," Panoramas et Synthèses, vol. 17, 2004.
    @article{Andre,
      author={André, Y.},
      TITLE={Une introduction aux motifs},
      JOURNAL={Panoramas et Synthèses},
      VOLUME={17},
      YEAR={2004},
      }
  • [DataMine] Go to document J. Blümlein, D. J. Broadhurst, and J. A. M. Vermaseren, "The multiple zeta value data mine," Comput. Phys. Comm., vol. 181, iss. 3, pp. 582-625, 2010.
    @article {DataMine, MRKEY = {2578167},
      AUTHOR = {Bl{ü}mlein, J. and Broadhurst, D. J. and Vermaseren, J. A. M.},
      TITLE = {The multiple zeta value data mine},
      JOURNAL = {Comput. Phys. Comm.},
      FJOURNAL = {Computer Physics Communications. An International Journal and Program Library for Computational Physics and Physical Chemistry},
      VOLUME = {181},
      YEAR = {2010},
      NUMBER = {3},
      PAGES = {582--625},
      ISSN = {0010-4655},
      CODEN = {CPHCBZ},
      MRCLASS = {11M32 (11Y70)},
      MRNUMBER = {2578167},
      MRREVIEWER = {Zhonghua Li},
      DOI = {10.1016/j.cpc.2009.11.007},
      }
  • [Br] F. Brown, On the decomposition of motivic multiple zeta values, 2010.
    @misc{Br,
      author={Brown, F.},
      TITLE={On the decomposition of motivic multiple zeta values},
      ARXIV={1102.1310v2},
      YEAR={2010},
      }
  • [D] Go to document P. Deligne, "Le groupe fondamental unipotent motivique de $\Bbb G_m-\mu_N$, pour $N=2,3,4,6$ ou $8$," Publ. Math. Inst. Hautes Études Sci., iss. 112, pp. 101-141, 2010.
    @article {D, MRKEY = {2737978},
      AUTHOR = {Deligne, Pierre},
      TITLE = {Le groupe fondamental unipotent motivique de {$\bold G\sb m-\mu\sb N$},
      pour {$N=2,3,4,6$} ou {$8$}},
      JOURNAL = {Publ. Math. Inst. Hautes Études Sci.},
      FJOURNAL = {Publications Mathématiques. Institut de Hautes Études Scientifiques},
      NUMBER = {112},
      YEAR = {2010},
      PAGES = {101--141},
      ISSN = {0073-8301},
      MRCLASS = {14F42},
      MRNUMBER = {2737978},
      DOI = {10.1007/s10240-010-0027-6},
      ZBLNUMBER = {1218.14016},
      }
  • [DG] Go to document P. Deligne and A. B. Goncharov, "Groupes fondamentaux motiviques de Tate mixte," Ann. Sci. École Norm. Sup., vol. 38, iss. 1, pp. 1-56, 2005.
    @article {DG, MRKEY = {2136480},
      AUTHOR = {Deligne, Pierre and Goncharov, Alexander B.},
      TITLE = {Groupes fondamentaux motiviques de {T}ate mixte},
      JOURNAL = {Ann. Sci. École Norm. Sup.},
      FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
      VOLUME = {38},
      YEAR = {2005},
      NUMBER = {1},
      PAGES = {1--56},
      ISSN = {0012-9593},
      CODEN = {ASENAH},
      MRCLASS = {11G55 (14F42 14G10 19F27)},
      MRNUMBER = {2136480},
      MRREVIEWER = {Tam{á}s Szamuely},
      DOI = {10.1016/j.ansens.2004.11.001},
      ZBLNUMBER = {1084.14024},
      }
  • [GG] Go to document A. B. Goncharov, "Galois symmetries of fundamental groupoids and noncommutative geometry," Duke Math. J., vol. 128, iss. 2, pp. 209-284, 2005.
    @article {GG, MRKEY = {2140264},
      AUTHOR = {Goncharov, A. B.},
      TITLE = {Galois symmetries of fundamental groupoids and noncommutative geometry},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {128},
      YEAR = {2005},
      NUMBER = {2},
      PAGES = {209--284},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {11G55 (11G09 14C30 16W30 19E15 20F34)},
      MRNUMBER = {2140264},
      MRREVIEWER = {Matilde Marcolli},
      DOI = {10.1215/S0012-7094-04-12822-2},
      ZBLNUMBER = {1095.11036},
      }
  • [Hoffman] Go to document M. E. Hoffman, "The algebra of multiple harmonic series," J. Algebra, vol. 194, iss. 2, pp. 477-495, 1997.
    @article {Hoffman, MRKEY = {1467164},
      AUTHOR = {Hoffman, Michael E.},
      TITLE = {The algebra of multiple harmonic series},
      JOURNAL = {J. Algebra},
      FJOURNAL = {Journal of Algebra},
      VOLUME = {194},
      YEAR = {1997},
      NUMBER = {2},
      PAGES = {477--495},
      ISSN = {0021-8693},
      CODEN = {JALGA4},
      MRCLASS = {11M41 (05E05)},
      MRNUMBER = {1467164},
      DOI = {10.1006/jabr.1997.7127},
      ZBLNUMBER = {0881.11067},
      }
  • [Levine] M. Levine, "Tate motives and the vanishing conjectures for algebraic $K$-theory," in Algebraic $K$-Theory and Algebraic Topology, Dordrecht: Kluwer Acad. Publ., 1993, vol. 407, pp. 167-188.
    @incollection {Levine, MRKEY = {1367296},
      AUTHOR = {Levine, Marc},
      TITLE = {Tate motives and the vanishing conjectures for algebraic {$K$}-theory},
      BOOKTITLE = {Algebraic {$K$}-Theory and Algebraic Topology},
      VENUE={{L}ake {L}ouise, {AB},
      1991},
      SERIES = {NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.},
      VOLUME = {407},
      PAGES = {167--188},
      PUBLISHER = {Kluwer Acad. Publ.},
      ADDRESS = {Dordrecht},
      YEAR = {1993},
      MRCLASS = {19E99 (14C35)},
      MRNUMBER = {1367296},
      MRREVIEWER = {Amnon Neeman},
      ZBLNUMBER = {0885.19001},
      }
  • [Rac] Go to document G. Racinet, "Doubles mélanges des polylogarithmes multiples aux racines de l’unité," Publ. Math. Inst. Hautes Études Sci., iss. 95, pp. 185-231, 2002.
    @article {Rac, MRKEY = {1953193},
      AUTHOR = {Racinet, Georges},
      TITLE = {Doubles mélanges des polylogarithmes multiples aux racines de l'unité},
      JOURNAL = {Publ. Math. Inst. Hautes Études Sci.},
      FJOURNAL = {Publications Mathématiques. Institut de Hautes Études Scientifiques},
      NUMBER = {95},
      YEAR = {2002},
      PAGES = {185--231},
      ISSN = {0073-8301},
      MRCLASS = {11G55 (11M41)},
      MRNUMBER = {1953193},
      MRREVIEWER = {Jan Nekov{á}{\v{r}}},
      DOI = {10.1007/s102400200004},
      ZBLNUMBER = {1050.11066},
      }
  • [Soud] Go to document I. Soudères, "Motivic double shuffle," Int. J. Number Theory, vol. 6, iss. 2, pp. 339-370, 2010.
    @article {Soud, MRKEY = {2646761},
      AUTHOR = {Soud{è}res, Ismael},
      TITLE = {Motivic double shuffle},
      JOURNAL = {Int. J. Number Theory},
      FJOURNAL = {International Journal of Number Theory},
      VOLUME = {6},
      YEAR = {2010},
      NUMBER = {2},
      PAGES = {339--370},
      ISSN = {1793-0421},
      MRCLASS = {11G55 (11Mxx 14H10)},
      MRNUMBER = {2646761},
      DOI = {10.1142/S1793042110002995},
      ZBLNUMBER = {05695841},
      }
  • [T] Go to document T. Terasoma, "Mixed Tate motives and multiple zeta values," Invent. Math., vol. 149, iss. 2, pp. 339-369, 2002.
    @article {T, MRKEY = {1918675},
      AUTHOR = {Terasoma, Tomohide},
      TITLE = {Mixed {T}ate motives and multiple zeta values},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {149},
      YEAR = {2002},
      NUMBER = {2},
      PAGES = {339--369},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {11G55 (11M41 19F27)},
      MRNUMBER = {1918675},
      MRREVIEWER = {Jan Nekov{á}{\v{r}}},
      DOI = {10.1007/s002220200218},
      ZBLNUMBER = {1042.11043},
      }
  • [Zagier] Go to document D. B. Zagier, "Evaluation of the multiple zeta values $\zeta(2,\dots,2,3,2,\dots,2)$," Ann. of Math., vol. 175, pp. 977-1000, 2012.
    @article{Zagier,
      author={Zagier, D. B.},
      TITLE={Evaluation of the multiple zeta values $\zeta(2,\dots,2,3,2,\dots,2)$},
      JOURNAL={Ann. of Math.},
      VOLUME={175},
      YEAR={2012},
      PAGES={977-1000},
      DOI = {10.4007/annals.2012.175.2.11},
      }

Authors

Francis Brown

Institut de Mathématiques de Jussieu, 175 Rue du Chevaleret, 75013 Paris, France