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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2501.13585 |
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| _version_ | 1866917947298021376 |
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| author | Yang, Siqi |
| author_facet | Yang, Siqi |
| contents | Let $p$ be a prime, $F$ be a totally real field in which $p$ is unramified and $ρ: \mathrm{Gal}(\overline{F}/F)\rightarrow \mathrm{GL}_2(\overline{\mathbb{F}}_p)$ be a totally odd, irreducible, continuous representation. The geometric Serre weight conjecture formulated by Diamond and Sasaki can be viewed as a geometric variant of the Buzzard-Diamond-Jarvis conjecture, where they have the notion of geometric modularity in the sense that $ρ$ arises from a mod $p$ Hilbert modular form and algebraic modularity in the sense of Buzzard-Diamond-Jarvis. Diamond and Sasaki conjecture that if $ρ$ is geometrically modular of weight $(k,l)\in \mathbb{Z}^Σ_{\geq 2}\times\mathbb{Z}^Σ$ and $k$ lies in the minimal cone, then $ρ$ is algebraically modular of the same weight, where $Σ$ is the set of embeddings from $F$ into $\overline{\mathbb{Q}}$. We prove the conjecture without parity hypotheses for real quadratic fields $F$ in which $p \geq 5$ is inert, and for totally real fields $F$ in which $p \geq \min\{5, [F:\mathbb{Q}]\}$ totally splits. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_13585 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the geometric Serre weight conjecture for Hilbert modular forms Yang, Siqi Number Theory Let $p$ be a prime, $F$ be a totally real field in which $p$ is unramified and $ρ: \mathrm{Gal}(\overline{F}/F)\rightarrow \mathrm{GL}_2(\overline{\mathbb{F}}_p)$ be a totally odd, irreducible, continuous representation. The geometric Serre weight conjecture formulated by Diamond and Sasaki can be viewed as a geometric variant of the Buzzard-Diamond-Jarvis conjecture, where they have the notion of geometric modularity in the sense that $ρ$ arises from a mod $p$ Hilbert modular form and algebraic modularity in the sense of Buzzard-Diamond-Jarvis. Diamond and Sasaki conjecture that if $ρ$ is geometrically modular of weight $(k,l)\in \mathbb{Z}^Σ_{\geq 2}\times\mathbb{Z}^Σ$ and $k$ lies in the minimal cone, then $ρ$ is algebraically modular of the same weight, where $Σ$ is the set of embeddings from $F$ into $\overline{\mathbb{Q}}$. We prove the conjecture without parity hypotheses for real quadratic fields $F$ in which $p \geq 5$ is inert, and for totally real fields $F$ in which $p \geq \min\{5, [F:\mathbb{Q}]\}$ totally splits. |
| title | On the geometric Serre weight conjecture for Hilbert modular forms |
| topic | Number Theory |
| url | https://arxiv.org/abs/2501.13585 |