Weak abelian direct summands and irreducibility of Galois representations
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866917676917456896 |
|---|---|
| author | Böckle, Gebhard Hui, Chun-Yin |
| author_facet | Böckle, Gebhard Hui, Chun-Yin |
| contents | Let $ρ_\ell$ be a semisimple $\ell$-adic representation of a number field $K$ that is unramified almost everywhere. We introduce a new notion called weak abelian direct summands of $ρ_\ell$ and completely characterize them, for example, if the algebraic monodromy of $ρ_\ell$ is connected. If $ρ_\ell$ is in addition $E$-rational for some number field $E$, we prove that the weak abelian direct summands are locally algebraic (and thus de Rham). We also show that the weak abelian parts of a connected semisimple Serre compatible system form again such a system.
Using our results on weak abelian direct summands, when $K$ is totally real and $ρ_\ell$ is the three-dimensional $\ell$-adic representation attached to a regular algebraic cuspidal automorphic, not necessarily polarizable representation $π$ of $\mathrm{GL}_3(\mathbb{A}_K)$ together with an isomorphism $\mathbb{C}\simeq \overline{\mathbb{Q}}_\ell$, we prove that $ρ_\ell$ is irreducible. We deduce in this case also some $\ell$-adic Hodge theoretic properties of $ρ_\ell$ if $\ell$ belongs to a Dirichlet density one set of primes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_08954 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weak abelian direct summands and irreducibility of Galois representations Böckle, Gebhard Hui, Chun-Yin Number Theory Algebraic Geometry Representation Theory Let $ρ_\ell$ be a semisimple $\ell$-adic representation of a number field $K$ that is unramified almost everywhere. We introduce a new notion called weak abelian direct summands of $ρ_\ell$ and completely characterize them, for example, if the algebraic monodromy of $ρ_\ell$ is connected. If $ρ_\ell$ is in addition $E$-rational for some number field $E$, we prove that the weak abelian direct summands are locally algebraic (and thus de Rham). We also show that the weak abelian parts of a connected semisimple Serre compatible system form again such a system. Using our results on weak abelian direct summands, when $K$ is totally real and $ρ_\ell$ is the three-dimensional $\ell$-adic representation attached to a regular algebraic cuspidal automorphic, not necessarily polarizable representation $π$ of $\mathrm{GL}_3(\mathbb{A}_K)$ together with an isomorphism $\mathbb{C}\simeq \overline{\mathbb{Q}}_\ell$, we prove that $ρ_\ell$ is irreducible. We deduce in this case also some $\ell$-adic Hodge theoretic properties of $ρ_\ell$ if $\ell$ belongs to a Dirichlet density one set of primes. |
| title | Weak abelian direct summands and irreducibility of Galois representations |
| topic | Number Theory Algebraic Geometry Representation Theory |
| url | https://arxiv.org/abs/2404.08954 |