Determination of certain mod $p$ Galois representations using local constancy
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866916297486368768 |
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| author | Ganguli, Abhik Kumar, Suneel |
| author_facet | Ganguli, Abhik Kumar, Suneel |
| contents | Let $p \geq 5$ be a prime. Let $k = b + c(p-1)$ be an integer in $[2p+2, p^2 - p +3]$, where $b \in [2,p]$ and $c \in [2, p-1]$. We prove local constancy in the weight space of the mod $p$ reduction of certain two-dimensional crystalline representations of $\mathrm{Gal}(\bar{\mathbb{Q}}_p/\mathbb{Q}_p)$, where the slope $ν(a_p)$ is constrained to be in $(1, c)$ and non-integral. We use the mod $p$ local Langlands correspondence for $\text{GL}_{2} (\mathbb{Q}_{p})$ to compute the mod $p$ reductions explicitly, thereby also giving a lower bound on the radius of constancy around the weights $k$ in the above range and under additional conditions on the slope. As an application of local constancy, we obtain explicit mod $p$ reductions at many new values of $k$ and $a_p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_15600 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Determination of certain mod $p$ Galois representations using local constancy Ganguli, Abhik Kumar, Suneel Number Theory 11F80, 11F70, 11F33 Let $p \geq 5$ be a prime. Let $k = b + c(p-1)$ be an integer in $[2p+2, p^2 - p +3]$, where $b \in [2,p]$ and $c \in [2, p-1]$. We prove local constancy in the weight space of the mod $p$ reduction of certain two-dimensional crystalline representations of $\mathrm{Gal}(\bar{\mathbb{Q}}_p/\mathbb{Q}_p)$, where the slope $ν(a_p)$ is constrained to be in $(1, c)$ and non-integral. We use the mod $p$ local Langlands correspondence for $\text{GL}_{2} (\mathbb{Q}_{p})$ to compute the mod $p$ reductions explicitly, thereby also giving a lower bound on the radius of constancy around the weights $k$ in the above range and under additional conditions on the slope. As an application of local constancy, we obtain explicit mod $p$ reductions at many new values of $k$ and $a_p$. |
| title | Determination of certain mod $p$ Galois representations using local constancy |
| topic | Number Theory 11F80, 11F70, 11F33 |
| url | https://arxiv.org/abs/2406.15600 |