On the Tamagawa number conjecture for newforms at Eisenstein primes
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866912580863262720 |
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| author | Yin, Mulun |
| author_facet | Yin, Mulun |
| contents | We extend the results of [CGLS22] to higher weight modular forms and prove a rank $0$ Tamagawa number formula (also known as the Bloch-Kato conjecture) for modular forms at good Eisenstein primes, under some technical assumption on periods. Under standard hypotheses (i.e. the injectivity of the $p$-adic Abel-Jabobi map and the non-degeneracy of the Gillet-Soulé height pairing), we also discuss some partial results towards a rank $1$ result. A conditional higher weight $p$-converse theorem to Gross-Zagier-Zhang-Kolyvagin-Nekovář is also obtained as a consequence of the anticyclotomic Iwasawa Main Conjectures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_24193 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Tamagawa number conjecture for newforms at Eisenstein primes Yin, Mulun Number Theory 11G40 We extend the results of [CGLS22] to higher weight modular forms and prove a rank $0$ Tamagawa number formula (also known as the Bloch-Kato conjecture) for modular forms at good Eisenstein primes, under some technical assumption on periods. Under standard hypotheses (i.e. the injectivity of the $p$-adic Abel-Jabobi map and the non-degeneracy of the Gillet-Soulé height pairing), we also discuss some partial results towards a rank $1$ result. A conditional higher weight $p$-converse theorem to Gross-Zagier-Zhang-Kolyvagin-Nekovář is also obtained as a consequence of the anticyclotomic Iwasawa Main Conjectures. |
| title | On the Tamagawa number conjecture for newforms at Eisenstein primes |
| topic | Number Theory 11G40 |
| url | https://arxiv.org/abs/2410.24193 |