On the $K_4$ group of modular curves
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2020
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| _version_ | 1866916925513138176 |
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| author | Brunault, François |
| author_facet | Brunault, François |
| contents | We construct elements in the $K_4$ group of modular curves using the polylogarithmic complexes of weight 3 defined by Goncharov and De Jeu. The construction is uniform in the level and relies on new modular units arising as cross-ratios of division values of the Weierstrass $\wp$ function. These units provide explicit triangulations of the $3$-term relations in $K_2$, which in turn give rise to elements in $K_4$. Based on numerical computations and on results of Wang, we conjecture that these elements are proportional to the Beilinson elements defined via Eisenstein symbols. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2009_07614 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On the $K_4$ group of modular curves Brunault, François Number Theory K-Theory and Homology 11G55 (Primary) 11F67, 11G16, 19F27 (Secondary) We construct elements in the $K_4$ group of modular curves using the polylogarithmic complexes of weight 3 defined by Goncharov and De Jeu. The construction is uniform in the level and relies on new modular units arising as cross-ratios of division values of the Weierstrass $\wp$ function. These units provide explicit triangulations of the $3$-term relations in $K_2$, which in turn give rise to elements in $K_4$. Based on numerical computations and on results of Wang, we conjecture that these elements are proportional to the Beilinson elements defined via Eisenstein symbols. |
| title | On the $K_4$ group of modular curves |
| topic | Number Theory K-Theory and Homology 11G55 (Primary) 11F67, 11G16, 19F27 (Secondary) |
| url | https://arxiv.org/abs/2009.07614 |