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Zagier’s conjecture and wedge complexes in algebraic \(K\)-theory. (English) Zbl 0868.19002
In “Polylogarithms, Dedekind zetafunctions, and the algebraic \(K\)-theory of fields” [in: Arithmetic algebraic geometry, Prog. Math. 89, 391-430 (1991; Zbl 0728.11062)], D. Zagier formulated a conjecture about the higher odd \(K\)-theory groups \(K_{2n-1}(F)\) of a number field \(F\) based on Bloch’s earlier work in the case \(n=2\). Beilinson and Deligne reformulated the conjecture as follows:
Let \({\mathcal L}=F^*\otimes \mathbb{Q}\) and let \(\{x\}_1= 1-x\) for \(x\in F\setminus \{0,1\}\). There should exist abelian groups \({\mathcal L}_n\) for \(n\geq 1\) generated by symbols \(\{x\}_n\), \(x\in F\setminus \{0,1\}\) together with a map \(d_n:{\mathcal L}\to\bigwedge^2 (\bigoplus^{n-1}_{l=1}{\mathcal L}_l)\) given by \(\{x\}_n\mapsto x\wedge\{ x\}_{n-1}\). There should exist an isomorphism \(\varphi_n:\ker d_n\to K^{(n)}_{2n-1}(F)\) and the composite of \(\varphi_n\) and the Borel regulator map should be given explicitly in terms of the \(n\)-th polylogarithm.
Using a formalism of multi-relative \(K\)-theory the author constructs for each \(n\geq 1\) complexes \(\widetilde{\mathcal M}_{(n)}\) of \(\mathbb{Q}\)-vector spaces, whose \(p\)-th cohomology \(H^p(\widetilde{\mathcal M}_{(n)})\) maps to \(K^{(n)}_{2n-p}(F)\). This leads to a proof of part of the Zagier conjecture for all \(n\geq 2\), since for \(p=1\) this map is an injection. Moreover, using work of Suslin, Goncharov and Zagier, the author shows that for \(n=2,3\) surjectivity holds as well, i.e., the maps \(H^1(\widetilde{\mathcal M}_{(n)})\to K^{(n)}_{2n-1}(F)\) are isomorphisms for \(n=2,3.\) The same result holds for all \(n\geq 2\) provided \(F=\mathbb{Q}(\zeta_N)\) is a cyclotomic field.

MSC:
19F15 Symbols and arithmetic (\(K\)-theoretic aspects)
11R70 \(K\)-theory of global fields
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
11R42 Zeta functions and \(L\)-functions of number fields
Citations:
Zbl 0728.11062
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References:
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