论文标题
史密斯理论和循环基础变化功能
Smith theory and cyclic base change functoriality
论文作者
论文摘要
Lafforgue和Genestier-Lafforgue已为任意还原组的全局和（半密码）局部Langlands对应于功能领域。我们建立了有关循环基础变化功能的这些对应关系的各种属性：对于$ \ Mathbb {z}/p \ Mathbb {z} $ - 全局函数字段的扩展，我们证明了Mod $ p $ pu $ automormorphical for to nutauctiational farmy of to nutuctiational farmy fortions。对于$ \ mathbb {z}/p \ mathbb {z} $ - 本地函数字段的扩展，我们为任何还原组的mod $ p $ bernstein中心构建了基本变化同构同构。然后，我们用它来证明沿$ \ mathbb {z}/p \ mathbb {z} $ - 所有足够大$ p $的扩展，并且Tate共同体沿基本变化下降，验证了tre tremumann-verkateShatesh的A的功能版本。 这些证明基于shtukas模量空间的均等本地化参数。他们还借鉴了代表理论的新工具，包括奇偶校验和史密斯 - 特雷曼理论。特别是，我们使用它们来建立与Gus Lonergan的联合附录中的Mod $ p $球形Hecke代数的基本变化同态同态的分类。
Lafforgue and Genestier-Lafforgue have constructed the global and (semisimplified) local Langlands correspondences for arbitrary reductive groups over function fields. We establish various properties of these correspondences regarding functoriality for cyclic base change: For $\mathbb{Z}/p\mathbb{Z}$-extensions of global function fields, we prove the existence of base change for mod $p$ automorphic forms on arbitrary reductive groups. For $\mathbb{Z}/p\mathbb{Z}$-extensions of local function fields, we construct a base change homomorphism for the mod $p$ Bernstein center of any reductive group. We then use this to prove existence of local base change for mod $p$ irreducible representation along $\mathbb{Z}/p\mathbb{Z}$-extensions for all large enough $p$, and that Tate cohomology realizes descent along base change, verifying a function field version of a conjecture of Treumann-Venkatesh. The proofs are based on equivariant localization arguments for the moduli spaces of shtukas. They also draw upon new tools from representation theory, including parity sheaves and Smith-Treumann theory. In particular, we use these to establish a categorification of the base change homomorphism for mod $p$ spherical Hecke algebras, in a joint appendix with Gus Lonergan.
