Abstract
We prove Zagier’s conjecture on the value at $s=4$ of the Dedekind $\zeta$-function of a number field $\mathrm{F}$:
$$\zeta_{\mathrm{F}}(4) = \pi^{4(r_1+r_2)} |d_{\mathrm{F}}|^{-1/2} \cdot \mathrm{det}\Bigl(\mathcal{L}_4(\sigma_i(y_j))\Bigr), \quad 1\le i,j \le r_2.$$
Here $\{\sigma_2\}$ is the set of all complex embeddings of $\mathrm{F}$, up to the conjugation. Namely, up to a standard factor $\zeta_{\mathrm{F}}(4)$ is equal to a $r_2\times r_2$ determinant, whose entries are $\mathbb{Q}$-linear combinations of the values of a single-valued version of $\mathcal{L}_4(z)$ of the classical $4$-logarithm at some numbers in $\sigma_j(\mathrm{F})$. These linear combinations are subject to a specific condition discovered by Zagier.
For an infinite field $\mathrm{F}$, we define a map of $K$-groups $K_{8-i}(\mathrm{F})$, $i=1,\ldots,4$, to the $i$-th cohomology of the weight $4$ polylogarithmic motivic complex $\mathcal{B}^\bullet(\mathrm{F},4)$. When $\mathrm{F}$ is the function field of a complex variety, composing the map with the regulator map on the polylogarithmic complex to the Deligne cohomology, we get a $\mathbb{Q}^\times$-multiple of Beilinson’s regulator. This implies that the comdposition $K_7(\mathbb{C}) \to H^1\mathcal{B}^\bullet(\mathbb{C},4) \to \mathbb{R}$, where the second map is given by $\mathcal{L}_4(z)$, is a $\mathbb{Q}^\times$-multiple of Borel’s regulator. This plus Borel’s theorem on the ranks of algebraic $K$-groups of number fields implies Zagier’s conjecture.
We get a strong evidence for a part of Freenss Conjecture describing the weight four part $\mathcal{L}_4(\mathrm{F})$ of the motivic Lie coalgebra of $\mathrm{F}$ via higher Bloch groups $\mathcal{B}_j(\mathrm{F})$ as an extension:
$$0\to \mathcal{B}_4(\mathrm{F}) \to \mathcal{L}_4(\mathrm{F} \to \Lambda^2\mathcal{B}_2(\mathrm{F})\to 0.$$
The main tools are motivic correlators and a new link of cluster varieties to polylogarithms.
