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arXiv 2306.02493math.NTmath.GR
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全局Langlands对应中具有大像的Galois表示

Galois representations with large image in the global Langlands correspondence

Adrian Zenteno

备注：
This article supersedes arXiv:2008.00556. Accepted in the Fields Institute Monographs
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AI总结：

研究GL_n上代数自守表示关联的Galois表示残差表示像的大小，利用标准技术证明对全实域及奇素数n≤293时无穷多素数ℓ像最大，并在F=ℚ, n=5时证明大像猜想。

摘要：

数域$F$上$\text{GL}_n$的全局Langlands猜想预测了$\text{GL}_n(\mathbb{A}_F)$的某些代数自守表示$\pi$与$\text{Gal}(\overline{F}/F)$的$n$维$\ell$-adic Galois表示族$\{ \rho_{\pi,\ell} \}_\ell$之间的对应。一般而言，预期残差Galois表示$\overline{\rho}_{\pi,\ell}$的像在几乎所有素数$\ell$处应尽可能大，除非存在自守原因导致像较小。本文中，我们仅使用标准技术和当前可用工具（例如Fontaine-Laffaille理论、Serre模性猜想、Lie型群极大子群分类、以及自守Galois表示不可约性和Langlands函子性的已知结果），研究与$\text{GL}_n(\mathbb{A}_F)$的正则代数、可极化、尖点自守表示$\pi$相关联的Galois表示相容系统$\{\rho_{\pi,\ell} \}_\ell$的像。特别地，当$F$是全实域且$n$是$\leq 293$的奇素数时，我们证明（在某种自守条件下）残差表示$\overline{\rho}_{\pi,\ell}$的像在无穷多个素数$\ell$处尽可能大。事实上，当$F=\mathbb{Q}$且$n=5$时，我们证明了大像猜想（即几乎所有素数$\ell$处像最大）。

英文摘要：

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 $n=5$.

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