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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2205.04499 |
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| _version_ | 1866929596722577408 |
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| author | Böckle, Gebhard Feng, Tony Harris, Michael Khare, Chandrashekhar Thorne, Jack A. |
| author_facet | Böckle, Gebhard Feng, Tony Harris, Michael Khare, Chandrashekhar Thorne, Jack A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_04499 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Cyclic base change of cuspidal automorphic representations over function fields Böckle, Gebhard Feng, Tony Harris, Michael Khare, Chandrashekhar Thorne, Jack A. Number Theory 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. |
| title | Cyclic base change of cuspidal automorphic representations over function fields |
| topic | Number Theory |
| url | https://arxiv.org/abs/2205.04499 |