论文标题
Modulo $ \ ell $区别问题
Modulo $\ell$ distinction problems
论文作者
论文摘要
令$ f $是一个非架构的本地特征领域，其特征与2 $ p $不同。本文涉及与Galois互动区分的连接还原组$ G $的$ \ ell $模块化表示，其$ \ ell $与$ p $不同。我们首先证明了一般定理，允许提起超舒张$ \ overline {\ mathbb {f}} _ {\ ell} $ - $ \ mathrm {gl} _n（f）的表示，由杰出的超级cus $ h $与杰出的supercuspidal juble划分。 $ \ OVERLINE {\ MATHBB {q}} _ {\ ell} $ - 表示。给定二次字段扩展$ e/f $和一个不可约的$ \ operline {\ mathbb {f}} _ {\ ell} $ - 表示$ \ mathrm {gl} _n（e）$的$ \ mathrm {gl} _n（e）$，我们在jacquet swiment $ in Is in Irreld commoter中commote $ \ mathrm {gl} _n（e）结合自我双对偶，然后$π$是$ \ mathrm {gl} _n（f）$ - 区分或$（\ mathrm {gl} _n（f），ω__{e/f}）$ - dickite本地阶级字段理论），但不是两者，它将Sécherre的结果扩展到了$ p = 2 $的情况。我们将提升定理的另一项应用程序用于以单一互动为特征的超舒张表示形式，将Zou的结果扩展到$ p = 2 $。 之后，我们对$ \ mathrm {gl} _2（f）$ - $ \ mathrm {gl} _2（e）$的杰出表示。使用此分类，我们讨论了$ \ mathrm {pgl} _2 $的Prasad猜想的模块化版本。我们表明，“古典” Prasad猜想在模块化环境中失败。我们建议使用非静态Weil-Deligne表示的解决方案。最后，我们应用Anandavardhanan和Prasad的限制方法来对$ \ mathrm {sl} _2（f）$ - $ \ mathrm {sl} _2（e）$的区分模块化表示。
Let $F$ be a non-archimedean local field of characteristic different from 2 and residual characteristic $p$. This paper concerns the $\ell$-modular representations of a connected reductive group $G$ distinguished by a Galois involution, with $\ell$ an odd prime different from $p$. We start by proving a general theorem allowing to lift supercuspidal $\overline{\mathbb{F}}_{\ell}$-representations of $\mathrm{GL}_n(F)$ distinguished by an arbitrary closed subgroup $H$ to a distinguished supercuspidal $\overline{\mathbb{Q}}_{\ell}$-representation. Given a quadratic field extension $E/F$ and an irreducible $\overline{\mathbb{F}}_{\ell}$-representation $π$ of $\mathrm{GL}_n(E)$, we verify the Jacquet conjecture in the modular setting that if the Langlands parameter $ϕ_π$ is irreducible and conjugate-self-dual, then $π$ is either $\mathrm{GL}_n(F)$-distinguished or $(\mathrm{GL}_n(F),ω_{E/F})$-distinguished (where $ω_{E/F}$ is the quadratic character of $F^\times$ associated to the quadratic field extension $E/F$ by the local class field theory), but not both, which extends one result of Sécherre to the case $p=2$. We give another application of our lifting theorem for supercuspidal representations distinguished by a unitary involution, extending one result of Zou to $p=2$. After that, we give a complete classification of the $\mathrm{GL}_2(F)$-distinguished representations of $\mathrm{GL}_2(E)$. Using this classification we discuss a modular version of the Prasad conjecture for $\mathrm{PGL}_2$. We show that the "classical" Prasad conjecture fails in the modular setting. We propose a solution using non-nilpotent Weil-Deligne representations. Finally, we apply the restriction method of Anandavardhanan and Prasad to classify the $\mathrm{SL}_2(F)$-distinguished modular representations of $\mathrm{SL}_2(E)$.
