##
**Mixed Tate motives over \(\mathbb{Z}\).**
*(English)*
Zbl 1278.19008

In this outstanding paper, the author proves a deep conjecture on multiple zeta values. For an \(r\)-tuple of positive integers \(n_1, \dots, n_r\) with \(n_i\geq 1\) and \(n_r\geq 2\), the multiple zeta value \(\zeta(n_1, \dots, n_r)\) is defined as the iterated multiple sum
\[
\zeta(n_1, \dots, n_r) = \sum_{0<k_1< \dots < k_r} \frac{1}{k_1^{n_1}\cdots k_r^{n_r}}.
\]
Multiple zeta values are real numbers that were originally defined by Euler and are also referred to as Euler/Zagier sums. They have recently drawn increasing attention because they appear in many different problems in mathematics and mathematical physics.

Let \(n=n_1+\dots+n_r\) be the weight of the multiple zeta value \(\zeta(n_1, \dots, n_r)\) and let \(\mathcal{Z}_n\) denote the \(\mathbb{Q}\)-vector space spanned by all multiple zeta values of weight \(n\). It is a deep conjecture of Zagier’s that the dimension of \(\mathcal{Z}_n\) is given by \[ \mathrm{dim}~ \mathcal{Z}_n = d_n \] where \(d_n\) is defined as the coefficient of \(x^n\) in the power series expansion of \(\frac{1}{1-x^2-x^3}\) (see M. E. Hoffman [J. Algebra 194, No. 2, 477–495 (1997; Zbl 0881.11067)] and Y. André [Une introduction aux motifs. Motifs purs, motifs mixtes, périodes. Paris: Société Mathématique de France (2004; Zbl 1060.14001)]). In particular, one would like to find all \(\mathbb{Q}\)-linear relations between multiple zeta values. A proof of the conjecture \(\mathrm{dim}~ \mathcal{Z}_n = d_n\) seems to be out of reach at present.

Nevertheless, it is known that the multiple zeta values in \(\mathbb{R}\) are the images under a \(\mathbb{Q}\)-linear map of certain motivic multiple zeta values \(\zeta^m(n_1, \dots, n_r)\). T. Terasoma [Invent. Math. 149, No. 2, 339–369 (2002; Zbl 1042.11043)] and A. B. Goncharov [Duke Math. J. 128, No. 2, 209–284 (2005; Zbl 1095.11036)] proved independently that the conjectural dimension formula for the space of the motivic multiple zeta values \(\zeta^m(n_1, \dots, n_r)\) implies the upper bound \[ \mathrm{dim}~ \mathcal{Z}_n \leq d_n. \] In the paper under review, the author proves the motivic dimension formula.

In more detail, the author considers the graded \(\mathbb{Q}\)-algebra \(\mathcal{M}\), whose homogeneous elements are the motivic \(\zeta^m(n_1, \dots, n_r)\) in degree \(n=\sum n_i\), and defines a homomorphism \(\mathrm{real}: \mathcal{M} \to \mathbb{R}\) such that \[ \mathrm{real}(\zeta^m(n_1, \dots, n_r))=\zeta(n_1, \dots, n_r). \] The \(\mathbb{Q}\)-linear relations among the \(\zeta^m(n_1, \dots, n_r)\) induce, so called motivic, relations among the \(\zeta(n_1, \dots, n_r)\). The main result of the paper is:

The set of elements \(\zeta^m(n_1, \dots, n_r)\) for which each \(n_i\) is either \(2\) or \(3\) form a basis for the \(\mathbb{Q}\)-vector space of motivic multiple zeta values.

As a first consequence, one obtains a proof of Hoffman’s conjecture that every multiple zeta value is a \(\mathbb{Q}\)-linear combination of values \(\zeta(n_1, \dots, n_r)\) for which each \(n_i\) is either \(2\) or \(3\).

The second important consequence is that the category of mixed Tate motives over \(\mathbb{Z}\) is generated by the motivic fundamental group \(\pi\) of \(\mathbb{P}^1-\{0,1,\infty\}\). More precisely, let \(\mathcal{MT}(\mathbb{Z})\) be the category of mixed Tate motives unramified over \(\mathbb{Z}\). It is a Tannakian category let \(\mathcal{G}_{\mathcal{MT}}\) denote its Galois group. Let \(\mathcal{MT}'(\mathbb{Z})\) be the full Tannakian subcategory generated by \(\pi\) and let \(\mathcal{G}_{\mathcal{MT}'}\) be its Galois group. The author proves the deep conjecture:

The map \(\mathcal{G}_{\mathcal{MT}} \to \mathcal{G}_{\mathcal{MT}'}\) is an isomorphism.

A third important consequence is that all periods of mixed Tate motives over \(\mathbb{Z}\) are \(\mathbb{Q}[\frac{1}{2\pi i}]\)-linear combinations of multiple zeta values.

Finally, we remark that the author’s proof does not seem to yield an effective algorithm that would allow to determine the linear relations between given multiple zeta values. More about possible and necessary calculations and concrete relations can be found in the paper by D. Zagier [Ann. Math. (2) 175, No. 2, 977–1000 (2012; Zbl 1268.11121)]. The reader should also have a look at Deligne’s article in the Séminaire Bourbaki, 64ème année, 2011-2012, no. 1048.

Let \(n=n_1+\dots+n_r\) be the weight of the multiple zeta value \(\zeta(n_1, \dots, n_r)\) and let \(\mathcal{Z}_n\) denote the \(\mathbb{Q}\)-vector space spanned by all multiple zeta values of weight \(n\). It is a deep conjecture of Zagier’s that the dimension of \(\mathcal{Z}_n\) is given by \[ \mathrm{dim}~ \mathcal{Z}_n = d_n \] where \(d_n\) is defined as the coefficient of \(x^n\) in the power series expansion of \(\frac{1}{1-x^2-x^3}\) (see M. E. Hoffman [J. Algebra 194, No. 2, 477–495 (1997; Zbl 0881.11067)] and Y. André [Une introduction aux motifs. Motifs purs, motifs mixtes, périodes. Paris: Société Mathématique de France (2004; Zbl 1060.14001)]). In particular, one would like to find all \(\mathbb{Q}\)-linear relations between multiple zeta values. A proof of the conjecture \(\mathrm{dim}~ \mathcal{Z}_n = d_n\) seems to be out of reach at present.

Nevertheless, it is known that the multiple zeta values in \(\mathbb{R}\) are the images under a \(\mathbb{Q}\)-linear map of certain motivic multiple zeta values \(\zeta^m(n_1, \dots, n_r)\). T. Terasoma [Invent. Math. 149, No. 2, 339–369 (2002; Zbl 1042.11043)] and A. B. Goncharov [Duke Math. J. 128, No. 2, 209–284 (2005; Zbl 1095.11036)] proved independently that the conjectural dimension formula for the space of the motivic multiple zeta values \(\zeta^m(n_1, \dots, n_r)\) implies the upper bound \[ \mathrm{dim}~ \mathcal{Z}_n \leq d_n. \] In the paper under review, the author proves the motivic dimension formula.

In more detail, the author considers the graded \(\mathbb{Q}\)-algebra \(\mathcal{M}\), whose homogeneous elements are the motivic \(\zeta^m(n_1, \dots, n_r)\) in degree \(n=\sum n_i\), and defines a homomorphism \(\mathrm{real}: \mathcal{M} \to \mathbb{R}\) such that \[ \mathrm{real}(\zeta^m(n_1, \dots, n_r))=\zeta(n_1, \dots, n_r). \] The \(\mathbb{Q}\)-linear relations among the \(\zeta^m(n_1, \dots, n_r)\) induce, so called motivic, relations among the \(\zeta(n_1, \dots, n_r)\). The main result of the paper is:

The set of elements \(\zeta^m(n_1, \dots, n_r)\) for which each \(n_i\) is either \(2\) or \(3\) form a basis for the \(\mathbb{Q}\)-vector space of motivic multiple zeta values.

As a first consequence, one obtains a proof of Hoffman’s conjecture that every multiple zeta value is a \(\mathbb{Q}\)-linear combination of values \(\zeta(n_1, \dots, n_r)\) for which each \(n_i\) is either \(2\) or \(3\).

The second important consequence is that the category of mixed Tate motives over \(\mathbb{Z}\) is generated by the motivic fundamental group \(\pi\) of \(\mathbb{P}^1-\{0,1,\infty\}\). More precisely, let \(\mathcal{MT}(\mathbb{Z})\) be the category of mixed Tate motives unramified over \(\mathbb{Z}\). It is a Tannakian category let \(\mathcal{G}_{\mathcal{MT}}\) denote its Galois group. Let \(\mathcal{MT}'(\mathbb{Z})\) be the full Tannakian subcategory generated by \(\pi\) and let \(\mathcal{G}_{\mathcal{MT}'}\) be its Galois group. The author proves the deep conjecture:

The map \(\mathcal{G}_{\mathcal{MT}} \to \mathcal{G}_{\mathcal{MT}'}\) is an isomorphism.

A third important consequence is that all periods of mixed Tate motives over \(\mathbb{Z}\) are \(\mathbb{Q}[\frac{1}{2\pi i}]\)-linear combinations of multiple zeta values.

Finally, we remark that the author’s proof does not seem to yield an effective algorithm that would allow to determine the linear relations between given multiple zeta values. More about possible and necessary calculations and concrete relations can be found in the paper by D. Zagier [Ann. Math. (2) 175, No. 2, 977–1000 (2012; Zbl 1268.11121)]. The reader should also have a look at Deligne’s article in the Séminaire Bourbaki, 64ème année, 2011-2012, no. 1048.

Reviewer: Gereon Quick (Münster)

### MSC:

19E15 | Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects) |

11M32 | Multiple Dirichlet series and zeta functions and multizeta values |

14F42 | Motivic cohomology; motivic homotopy theory |

11G09 | Drinfel’d modules; higher-dimensional motives, etc. |

14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |

### References:

[1] | Y. André, ”Une introduction aux motifs,” Panoramas et Synthèses, vol. 17, 2004. · Zbl 1060.14001 |

[2] | 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. · Zbl 1221.11183 |

[3] | F. Brown, On the decomposition of motivic multiple zeta values, 2010. |

[4] | P. Deligne, ”Le groupe fondamental unipotent motivique de \(\mathbb G_m-\mu_N\), pour \(N=2,3,4,6\) ou \(8\),” Publ. Math. Inst. Hautes Études Sci., iss. 112, pp. 101-141, 2010. · Zbl 1218.14016 |

[5] | 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. · Zbl 1084.14024 |

[6] | A. B. Goncharov, ”Galois symmetries of fundamental groupoids and noncommutative geometry,” Duke Math. J., vol. 128, iss. 2, pp. 209-284, 2005. · Zbl 1095.11036 |

[7] | M. E. Hoffman, ”The algebra of multiple harmonic series,” J. Algebra, vol. 194, iss. 2, pp. 477-495, 1997. · Zbl 0881.11067 |

[8] | 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. · Zbl 0885.19001 |

[9] | G. Racinet, ”Doubles mélanges des polylogarithmes multiples aux racines de l’unité,” Publ. Math. Inst. Hautes Études Sci., iss. 95, pp. 185-231, 2002. · Zbl 1050.11066 |

[10] | I. Soudères, ”Motivic double shuffle,” Int. J. Number Theory, vol. 6, iss. 2, pp. 339-370, 2010. · Zbl 1234.11083 |

[11] | T. Terasoma, ”Mixed Tate motives and multiple zeta values,” Invent. Math., vol. 149, iss. 2, pp. 339-369, 2002. · Zbl 1042.11043 |

[12] | 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. · Zbl 1268.11121 |

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