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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2405.13853 |
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| _version_ | 1866929354216308736 |
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| author | Charlton, Steven |
| author_facet | Charlton, Steven |
| contents | We prove that the weight 6, depth 3, multiple polylogarithm $ \mathrm{Li}_{4,1,1}((xyz)^{-1}, x, y) $, or rather its more natural `divergent' incarnation $ \mathrm{Li}_{3;1,1,1}(x,y,z) $, satisfies the 6-fold anharmonic symmetries of the dilogarithm $ \mathrm{Li}_2 $, $ λ\mapsto 1-λ$ and $ λ\mapsto λ^{-1} $, in each of $x$, $y$ and $z$ independently, modulo terms of depth $ \leq2 $. This establishes the `higher Zagier' part of the weight 6, depth 3, reduction conjectured by Matveiakin and Rudenko. Together with their proof of the `higher Gangl' part of the weight 6, depth 3, reduction (which is formulated modulo the `higher Zagier' part), we establish Goncharov's Depth Conjecture in the case of weight 6, depth 3. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_13853 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Symmetries of weight 6 multiple polylogarithms and Goncharov's Depth Conjecture Charlton, Steven Number Theory 11G55 We prove that the weight 6, depth 3, multiple polylogarithm $ \mathrm{Li}_{4,1,1}((xyz)^{-1}, x, y) $, or rather its more natural `divergent' incarnation $ \mathrm{Li}_{3;1,1,1}(x,y,z) $, satisfies the 6-fold anharmonic symmetries of the dilogarithm $ \mathrm{Li}_2 $, $ λ\mapsto 1-λ$ and $ λ\mapsto λ^{-1} $, in each of $x$, $y$ and $z$ independently, modulo terms of depth $ \leq2 $. This establishes the `higher Zagier' part of the weight 6, depth 3, reduction conjectured by Matveiakin and Rudenko. Together with their proof of the `higher Gangl' part of the weight 6, depth 3, reduction (which is formulated modulo the `higher Zagier' part), we establish Goncharov's Depth Conjecture in the case of weight 6, depth 3. |
| title | Symmetries of weight 6 multiple polylogarithms and Goncharov's Depth Conjecture |
| topic | Number Theory 11G55 |
| url | https://arxiv.org/abs/2405.13853 |