Galois representations with large image in the global Langlands correspondence
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912293626839040 |
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| author | Zenteno, Adrian |
| author_facet | Zenteno, Adrian |
| contents | The global Langlands conjecture for $\text{GL}_n$ over a number field $F$ predicts a correspondence between certain algebraic automorphic representations $π$ of $\text{GL}_n(\mathbb{A}_F)$ and certain families $\{ ρ_{π,\ell} \}_\ell$ of $n$-dimensional $\ell$-adic Galois representations of $\text{Gal}(\overline{F}/F)$. In general, it is expected that the image of the residual Galois representation $\overlineρ_{π,\ell}$ of $ρ_{π,\ell}$ should be as large as possible for almost all primes $\ell$, unless there is an automorphic reason for the image to be small.
In this paper, we study the images of certain compatible systems of Galois representations $\{ρ_{π,\ell} \}_\ell$ associated to regular algebraic, polarizable, cuspidal automorphic representations $π$ of $\text{GL}_n(\mathbb{A}_F)$ by using only standard techniques and currently available tools (e.g., Fontaine-Laffaille theory, Serre's modularity conjecture, classification of the maximal subgroups of Lie type groups, and known results about irreducibility of automorphic Galois representations and Langlands functoriality). In particular, when $F$ is a totally real field and $n$ is an odd prime number $\leq 293$, we prove that (under certain automorphic conditions) the images of the residual representations $\overlineρ_{π,\ell}$ are as large as possible for infinitely many primes $\ell$. In fact, we prove the large image conjecture (i.e., large image for almost all primes $\ell$) when $F=\mathbb{Q}$ and $n=5$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_02493 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Galois representations with large image in the global Langlands correspondence Zenteno, Adrian Number Theory Group Theory 11F80, 11F70, 20G40, 20G41 The global Langlands conjecture for $\text{GL}_n$ over a number field $F$ predicts a correspondence between certain algebraic automorphic representations $π$ of $\text{GL}_n(\mathbb{A}_F)$ and certain families $\{ ρ_{π,\ell} \}_\ell$ of $n$-dimensional $\ell$-adic Galois representations of $\text{Gal}(\overline{F}/F)$. In general, it is expected that the image of the residual Galois representation $\overlineρ_{π,\ell}$ of $ρ_{π,\ell}$ should be as large as possible for almost all primes $\ell$, unless there is an automorphic reason for the image to be small. In this paper, we study the images of certain compatible systems of Galois representations $\{ρ_{π,\ell} \}_\ell$ associated to regular algebraic, polarizable, cuspidal automorphic representations $π$ of $\text{GL}_n(\mathbb{A}_F)$ by using only standard techniques and currently available tools (e.g., Fontaine-Laffaille theory, Serre's modularity conjecture, classification of the maximal subgroups of Lie type groups, and known results about irreducibility of automorphic Galois representations and Langlands functoriality). In particular, when $F$ is a totally real field and $n$ is an odd prime number $\leq 293$, we prove that (under certain automorphic conditions) the images of the residual representations $\overlineρ_{π,\ell}$ are as large as possible for infinitely many primes $\ell$. In fact, we prove the large image conjecture (i.e., large image for almost all primes $\ell$) when $F=\mathbb{Q}$ and $n=5$. |
| title | Galois representations with large image in the global Langlands correspondence |
| topic | Number Theory Group Theory 11F80, 11F70, 20G40, 20G41 |
| url | https://arxiv.org/abs/2306.02493 |