Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2310.11522 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909232662577152 |
|---|---|
| author | Blanco-Chacón, Iván Dieulefait, Luis |
| author_facet | Blanco-Chacón, Iván Dieulefait, Luis |
| contents | Let $F/\mathbb{Q}$ be any totally real number field and $\frak{N}$ an ideal of its ring of integers of norm $N$ and define, for every even $n$, the $[F:\mathbb{Q}]$-dimensional multiweight $\textbf{n}=(n,...,n)$. We prove that for a non CM Hilbert cuspidal Hecke eigenform for $F$, say $f\in S_{\textbf{k}}(Γ_0(\frak{N}))$ with $k>2$ even, and a prime $p>\max\{k+1,6\}$ totally split in $F$ such that $p\nmid N$ and such that the residual mod $p$ representation $\overlineρ_f$ satisfies that $\mathrm{SL}_2(\mathbb{F}_p)\subseteq \mathrm{Im}(\overlineρ_f)$, there exists a lift $ρ_g$ associated to a Hilbert modular cuspform for $F$, say $g\in S_{\textbf{2}}(\frak{N}p^2,ε)$ for some Nebentypus character $ε$ which is supercuspidal at each prime of $F$ over $p$. We also observe that our techniques provide an alternative proof to the corresponding statement for classical Hecke cuspforms already proved by Khare \cite{khare} with classical techniques. Finally, we take the opportunity to include a corrigenda for \cite{dieulefait} which follows from our main result, which provides a congruence that puts the micro good dihedral prime in the level. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_11522 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Modular supercuspidal lifts of weight $2$ Blanco-Chacón, Iván Dieulefait, Luis Number Theory Let $F/\mathbb{Q}$ be any totally real number field and $\frak{N}$ an ideal of its ring of integers of norm $N$ and define, for every even $n$, the $[F:\mathbb{Q}]$-dimensional multiweight $\textbf{n}=(n,...,n)$. We prove that for a non CM Hilbert cuspidal Hecke eigenform for $F$, say $f\in S_{\textbf{k}}(Γ_0(\frak{N}))$ with $k>2$ even, and a prime $p>\max\{k+1,6\}$ totally split in $F$ such that $p\nmid N$ and such that the residual mod $p$ representation $\overlineρ_f$ satisfies that $\mathrm{SL}_2(\mathbb{F}_p)\subseteq \mathrm{Im}(\overlineρ_f)$, there exists a lift $ρ_g$ associated to a Hilbert modular cuspform for $F$, say $g\in S_{\textbf{2}}(\frak{N}p^2,ε)$ for some Nebentypus character $ε$ which is supercuspidal at each prime of $F$ over $p$. We also observe that our techniques provide an alternative proof to the corresponding statement for classical Hecke cuspforms already proved by Khare \cite{khare} with classical techniques. Finally, we take the opportunity to include a corrigenda for \cite{dieulefait} which follows from our main result, which provides a congruence that puts the micro good dihedral prime in the level. |
| title | Modular supercuspidal lifts of weight $2$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2310.11522 |