论文标题
在功能字段上尖锐自动形态表示的循环基础变化
Cyclic base change of cuspidal automorphic representations over function fields
论文作者
论文摘要
让$ g $是全球功能字段$ k $的分裂半简单组。考虑到满足技术假设的$ g $的cuspidal自动形态表示$π$π$，我们证明，对于几乎所有的质量$ \ ell $，沿任何$ \ mathbb {z}/\ ell \ ell \ ell \ mathbb {z} $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ \ mathbb {z}/\ mathbb {z}/\ mathbb {z}/\ mathbb {z}/\ mathbb {z}/\ mathbb {z} $ k $ k $ k $ k $ k $ k $的基本更改的提升。我们的证明不依赖任何痕量公式。取而代之的是，它基于模块化提升定理，以及史密斯理论的论点，以获得残留表示的基础变化。作为一个应用程序，我们还证明，对于本地函数字段$ f $上的任何分裂的半胶合组$ g $，几乎所有Primes $ \ ell $，$ G（f）$的任何不可允许的可允许的代表都允许沿任何$ \ MATHBB {z}/\ ell \ ell \ ell \ ell \ ell \ althbb {z} $ f $ f $ f $ f $。最后，我们针对称为Toral SuperCuspidal表示形式的一类表示，更明确地表征了本地基础更改。
Let $G$ be a split semi-simple group over a global function field $K$. Given a cuspidal automorphic representation $Π$ of $G$ satisfying a technical hypothesis, we prove that for almost all primes $\ell$, there is a cyclic base change lifting of $Π$ along any $\mathbb{Z}/\ell\mathbb{Z}$-extension of $K$. Our proof does not rely on any trace formulas; instead it is based on modularity lifting theorems, together with a Smith theory argument to obtain base change for residual representations. As an application, we also prove that for any split semisimple group $G$ over a local function field $F$, and almost all primes $\ell$, any irreducible admissible representation of $G(F)$ admits a base change along any $\mathbb{Z}/\ell\mathbb{Z}$-extension of $F$. Finally, we characterize local base change more explicitly for a class of representations called toral supercuspidal representations.
