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arXiv · 2410.24193

On the Tamagawa number conjecture for newforms at Eisenstein primes

Mulun Yin
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Abstract

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-Soul\'e 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-Nekov\'a\v{r} is also obtained as a consequence of the anticyclotomic Iwasawa Main Conjectures.

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Mulun Yin. 2024-10-31. On the Tamagawa number conjecture for newforms at Eisenstein primes. https://arxiv.org/abs/2410.24193

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