Motivic correlators, cluster varieties and Zagier's conjecture on zeta(F,4)

Abstract

We prove Zagier's conjecture on the value at s=4 of the Dedekind zeta-function of a number field F. For any field F, we define a map from the appropriate pieces of algebraic K-theory of F to the cohomology of the weight 4 polylogarithmic motivic complex. When F is the function field of a complex variety, composing this map with the regulator map on the polylogarithmic complex to the Deligne cohomology, we get a rational multiple of Beilinson's regulator. This plus Borel's theorem implies Zagier's conjecture. Another application is a formula expressing the value at s=4 of the L-function of an elliptic curve E over Q via generalized Eisenstein-Kronecker series. We get a strong evidence for the part of Freeness Conjecture describing the weight four part of the motivic Lie coalgebra of F via higher Bloch groups. Our main tools are motivic correlators and a new link of cluster varieties to polylogarithms.

arXiv commentVersion 5. Final version, accepted for publication in Annals of Mathematics. 111 pages

References in corpus (9)

• Discrete Heisenberg-Weyl Group and Modular Group
• Quantum Dilogarithm
• FI-modules and stability for representations of symmetric groups
• Motivic Amplitudes and Cluster Coordinates
• The quantum dilogarithm and representations quantum cluster varieties
• The pentagon relation for the quantum dilogarithm and quantized M_{0,5}
• Multiple polylogarithms in weight 4
• Sur la conjecture de Zagier pour n=4
• On the functional equations for polylogarithms in one variable

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• Explicit formulas for Grassmannian polylogarithms
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• Motivic fundamental groups of CM elliptic curves and geometry of Bianchi hyperbolic threefolds