Mixed Tate motives and multiple zeta values. (English) Zbl 1042.11043

Terasoma, Tomohide

Invent. Math. 149, No. 2, 339-369 (2002).

The author establishes a striking result on Zagier’s conjecture on the mulitple zeta values. Let \(L_n\) be the subspace of \(\mathbb{R}\) generated by the multiple zeta values of weight \(n\) and let \(d_n\) be the positive integer defined by \(\sum_{i=1}^\infty d_it^i= (1-t^2-t^3)^{-1}\). Zagier’s conjecture then asserts the equality \(\dim(L_n)= d_n\). The main result of this paper is to show the inequality \(\dim (L_n)\leq d_n\). The outline of the proof is as follows: Firstly, the multiple zeta values of weight \(n\), which are given by the iterated integrals, can be interpreted as the periods of the relative cohomology \(H^n\) of certain periods of algebraic varieties. Next, regard \(H^n\) as an object in the mixed Tate motives in which the dimensions of the extension groups can be estimated by the known results on algebraic \(K\)-groups. This yields the desired inequality. The same result was independently obtained by A. B. Goncharov in a different way [European Congress of Mathematics, Vol. I, 361–392 (2001; Zbl 1042.11042), see the preceding review].

Reviewer: Masanori Morishita (Kanazawa)

MSC:

11G55	Polylogarithms and relations with \(K\)-theory
19F27	Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)
11M41	Other Dirichlet series and zeta functions

Keywords:

multiple zeta values; iterated integrals; periods of relative cohomology; mixed Tate motives