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

Alexander B. Goncharov, Daniil Rudenko

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.

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.

Paper Structure

This paper contains 139 sections, 71 theorems, 557 equations, 13 figures.

Table of Contents

  1. Introduction and the architecture of the proof
  2. The classical polylogarithms and algebraic $K$-theory
  3. 1. The classical $n$-logarithm.
  4. 2. Zagier's conjecture.
  5. 3. Functional equations for polylogarithms.
  6. Conventions.
  7. 4. Polylogarithmic motivic complexes Gon95.
  8. 6. Main results.
  9. 7. The motivic Tate Lie algebra of a field ${\rm F}$.
  10. Functional equations for polylogarithms and motivic Tate Lie coalgebras
  11. 1. Constructing the Lie coalgebra $\mathbb{L}_{\leq 4}({\rm F})$.
  12. 2. An alternative form of relation ${\bf Q}_4$.
  13. 3. The cobracket maps.
  14. 4. Elements $\{x,y\}_{m-1,1}$ and multiple polylogarithms.
  15. 5. Motivic correlators and the Lie coalgebra ${\Bbb L}_{\leq 4}({\rm F})$.
  16. ...and 124 more sections

Key Result

Theorem 1.1

Let ${\rm F}$ be a number field, $[{\rm F}:{\mathbb Q}]=r_1+2r_2$, and the set $\{\sigma_j\}$ of all embeddings ${\rm F} \to {\mathbb C}$ is numbered so that $\overline \sigma_{r_1+i} = \sigma_{r_1+r_2+i}$. Let $d_{\rm F}$ be the discriminant of ${\rm F}$. Then there exist elements $y_1, \dots , y_{

Figures (13)

  • Figure 1: The specialization of relation (\ref{['TRIF']}) to this configuration is the 22-term relation for ${\cal L}_3$.
  • Figure 2: Geometry of relation ${\bf Q}_3$. The diagonal $x_1x_4$ divides the hexagon into quadrangles. The arguments of $\left \{[x_1, x_2,x_3,x_4],[x_4, x_5, x_6, x_1] \right\}_{2,1}$ are the cross-ratios assigned to them, and $[x_1, x_2,x_3,x_4, x_5, x_6]$ is their ratio. Elements $\{[x_1,x_2,x_4,x_5]\}_3$ and $\{[x_1,x_3,x_4,x_5]\}_3$ correspond to the quadrangles in the two hexagons on the right.
  • Figure 3: Geometry of relation ${\bf Q}_4$. The diagonal $x_1x_4$ divides the heptagon into a quadrangle and a pentagon. Deleting one of three vertices $\{x_5, x_6, x_7\}$ of the pentagon we get another quadrangle: on the picture we deleted $x_6$. The arguments of $\left \{[x_1, x_2,x_3,x_4],[x_4, x_5, \widehat{x}_6, x_7, x_1] \right\}_{3,1}$ are cross-ratios assigned to these quadrangles.
  • Figure 4: A data defining a weight 5 motivic correlator.
  • Figure 5: The element $W_t$ is assigned to a triangle $t$ decorated by a configuration of vectors $(l_1, l_2, l_3)$. The elements $\omega(l_i, l_j)$ are assigned to the sides of the triangle.
  • ...and 8 more figures

Theorems & Definitions (144)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Conjecture 1.4
  • Conjecture 1.5
  • Conjecture 1.6
  • Conjecture 1.7
  • Theorem 1.8
  • Definition 1.9
  • Definition 1.10
  • ...and 134 more