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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2402.13406 |
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| _version_ | 1866908330237100032 |
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| author | Sakugawa, Kenji |
| author_facet | Sakugawa, Kenji |
| contents | Let $p$ be a prime number and let $V$ be a continuous representation of $\mathrm{Gal}(\overline {\mathbf Q}/\mathbf Q)$ on a finite dimensional $\mathbf Q_p$-vector space, which is geometric. One of the Bloch-Kato conjectures for $V$ predicts that the rank of the Hasse-Weil $L$-function of $V$ at $s=0$ coincides with the rank of Blcoh-Kato Selmer group of $V^\vee(1)$. In this paper, we prove that the depth-weight compatibility on the fundamental Lie algebra of the mixed Tate motives over $\mathbf Z$ implies the Bloch-Kato conjecture for the $p$-adic Galois representations associated with full-level Hecke eigen cuspforms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_13406 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The depth-weight compatibility on the motivic fundamental Lie algebra and the Bloch-Kato conjecture for modular forms Sakugawa, Kenji Number Theory Let $p$ be a prime number and let $V$ be a continuous representation of $\mathrm{Gal}(\overline {\mathbf Q}/\mathbf Q)$ on a finite dimensional $\mathbf Q_p$-vector space, which is geometric. One of the Bloch-Kato conjectures for $V$ predicts that the rank of the Hasse-Weil $L$-function of $V$ at $s=0$ coincides with the rank of Blcoh-Kato Selmer group of $V^\vee(1)$. In this paper, we prove that the depth-weight compatibility on the fundamental Lie algebra of the mixed Tate motives over $\mathbf Z$ implies the Bloch-Kato conjecture for the $p$-adic Galois representations associated with full-level Hecke eigen cuspforms. |
| title | The depth-weight compatibility on the motivic fundamental Lie algebra and the Bloch-Kato conjecture for modular forms |
| topic | Number Theory |
| url | https://arxiv.org/abs/2402.13406 |