arXiv 2109.012102021-09math.NTmath.RT

Fargues-Scholze与Gan-Takeda局部Langlands的兼容性

Compatibility of the Fargues-Scholze and Gan-Takeda Local Langlands

Linus Hamann

备注：: v4: Clarifications and corrections added to published version after extensive A.I. review

AI总结：

本文证明了在非分歧且p>2的条件下，Fargues-Scholze构造的半单Weil参数与Gan-Takeda关于GSp4及其非分裂内形式的局部Langlands对应相兼容，并利用局部Shimura簇上同调验证了Kottwitz猜想的强形式。

摘要：

给定一个素数 $p$、有限扩张 $L/\mathbb{Q}_{p}$、连通 $p$-进约化群 $G/L$ 以及 $G(L)$ 的光滑不可约表示 $π$，Fargues-Scholze 最近为这样的 $π$ 附加了一个半单 Weil 参数，为局部 Langlands 对应提供了一个一般性候选。自然地，人们会问这种构造在半单化之后是否与已知的对应实例兼容。对于 $G = \mathrm{GL}_{n}$ 及其内形式，Fargues-Scholze 和 Hansen-Kaletha-Weinstein 证明了该对应与 Harris-Taylor/Henniart 的对应是兼容的。我们验证了对于 $G = \mathrm{GSp}_{4}$ 及其唯一的非分裂内形式 $G = \mathrm{GU}_{2}(D)$（其中 $D$ 是 $L$ 上的四元数除法代数）具有类似的兼容性，假设 $L/\mathbb{Q}_{p}$ 是非分歧的且 $p > 2$。在这种情况下，局部 Langlands 对应已由 Gan-Takeda 和 Gan-Tantona 构造。类似于 $\mathrm{GL}_{n}$ 及其内形式的情况，这种兼容性的证明是通过描述 Weil 群在与 $\mathrm{GSp}_{4}$ 相关的局部 Shimura 簇上同调上的作用来实现的，使用了 Shen 关于阿贝尔型 Shimura 簇的基本一致化，结合了 Kret-Shin 和 Sorensen 关于全实域上与 $\mathrm{GSp}_{4}$ 内形式相关的全局 Shimura 簇上同调中 Galois 表示的各种全局结果。在证明参数相同之后，我们应用了 Hansen 最近探索的 Fargues-Scholze 构造几何中的一些思想，对该局部 Shimura 簇的上同调给出了更精确的描述，并在过程中验证了 Kottwitz 猜想的强形式。

英文摘要：

Given a prime $p$, a finite extension $L/\mathbb{Q}_{p}$, a connected $p$-adic reductive group $G/L$, and a smooth irreducible representation $π$ of $G(L)$, Fargues-Scholze recently attached a semisimple Weil parameter to such $π$, giving a general candidate for the local Langlands correspondence. It is natural to ask whether this construction is compatible with known instances of the correspondence after semisimplification. For $G = \mathrm{GL}_{n}$ and its inner forms, Fargues-Scholze and Hansen-Kaletha-Weinstein showed that the correspondence is compatible with the correspondence of Harris-Taylor/Henniart. We verify a similar compatibility for $G = \mathrm{GSp}_{4}$ and its unique non-split inner form $G = \mathrm{GU}_{2}(D)$, where $D$ is the quaternion division algebra over $L$, assuming that $L/\mathbb{Q}_{p}$ is unramified and $p > 2$. In this case, the local Langlands correspondence has been constructed by Gan-Takeda and Gan-Tantono. Analogous to the case of $\mathrm{GL}_{n}$ and its inner forms, this compatibility is proven by describing the Weil group action on the cohomology of a local Shimura variety associated to $\mathrm{GSp}_{4}$, using basic uniformization of abelian type Shimura varieties due to Shen, combined with various global results of Kret-Shin and Sorensen on Galois representations in the cohomology of global Shimura varieties associated to inner forms of $\mathrm{GSp}_{4}$ over a totally real field. After showing the parameters are the same, we apply some ideas from the geometry of the Fargues-Scholze construction explored recently by Hansen, to give a more precise description of the cohomology of this local Shimura variety, verifying a strong form of the Kottwitz conjecture in the process.

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