Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2018
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/1803.08585 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913635182313472 |
|---|---|
| author | Goncharov, Alexander B. Rudenko, Daniil |
| author_facet | Goncharov, Alexander B. Rudenko, Daniil |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1803_08585 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Motivic correlators, cluster varieties and Zagier's conjecture on zeta(F,4) Goncharov, Alexander B. Rudenko, Daniil Number Theory Algebraic Geometry 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. |
| title | Motivic correlators, cluster varieties and Zagier's conjecture on zeta(F,4) |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/1803.08585 |