Adjoint motives of modular forms and the Tamagawa number conjecture
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917140562444288 |
|---|---|
| author | Diamond, Fred Flach, Matthias Guo, Li |
| author_facet | Diamond, Fred Flach, Matthias Guo, Li |
| contents | Let $f$ be a newform of weight $k\geq 2$, level $N$ with coefficients in a number field $K$, and $A$ the adjoint motive of the motive $M$ associated to $f$. We carefully discuss the construction of the realisations of $M$ and $A$, as well as natural integral structures in these realisations. We then use the method of Taylor and Wiles to verify the $λ$-part of the Tamagawa number conjecture of Bloch and Kato for $L(A,0)$ and $L(A,1)$. Here $λ$ is any prime of $K$ not dividing $Nk!$, and so that the mod $λ$ representation associated to $f$ is absolutely irreducible when restricted to the Galois group over $\mathbb{Q}(\sqrt{(-1)^{(\ell-1)/2}\ell})$ where $λ\mid \ell$. The method also establishes modularity of all lifts of the mod $λ$ representation which are crystalline of Hodge-Tate type $(0,k-1)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_02348 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Adjoint motives of modular forms and the Tamagawa number conjecture Diamond, Fred Flach, Matthias Guo, Li Number Theory 11F80, 11G40, 14G10, 14F, 19F27 Let $f$ be a newform of weight $k\geq 2$, level $N$ with coefficients in a number field $K$, and $A$ the adjoint motive of the motive $M$ associated to $f$. We carefully discuss the construction of the realisations of $M$ and $A$, as well as natural integral structures in these realisations. We then use the method of Taylor and Wiles to verify the $λ$-part of the Tamagawa number conjecture of Bloch and Kato for $L(A,0)$ and $L(A,1)$. Here $λ$ is any prime of $K$ not dividing $Nk!$, and so that the mod $λ$ representation associated to $f$ is absolutely irreducible when restricted to the Galois group over $\mathbb{Q}(\sqrt{(-1)^{(\ell-1)/2}\ell})$ where $λ\mid \ell$. The method also establishes modularity of all lifts of the mod $λ$ representation which are crystalline of Hodge-Tate type $(0,k-1)$. |
| title | Adjoint motives of modular forms and the Tamagawa number conjecture |
| topic | Number Theory 11F80, 11G40, 14G10, 14F, 19F27 |
| url | https://arxiv.org/abs/2512.02348 |