##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory Seminar

## Linus Hamann

#### Princeton

## Compatibility of the Fargues-Scholze and Gan-Takeda local Langlands

##### Abstract:

Given a prime $p$, a finite extension $L/\mathbb{Q}_{p}$, a connected $p$-adic reductive group $G/L$, and a smooth irreducible representation $\pi$ of $G(L)$, Fargues-Scholze recently attached a semisimple Weil parameter to such $\pi$, 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 $G = GL_{n}$ and its inner forms, Fargues-Scholze and Hansen-Kaletha-Weinstein show that the correspondence is compatible with the correspondence of Harris-Taylor/Henniart. We verify a similar compatibility for $G = GSp_{4}$ and its unique non-split inner form $G = GU_{2}(D)$, 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 $GL_{n}$ and its inner forms, this compatibility is proven by describing the Weil group action on the cohomology of a local Shimura variety associated to $GSp_{4}$, 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 $GSp_{4}$ 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.

### November 4, 2021

### 2:00 PM

### APM 6402 and Zoom; see https://www.math.ucsd.edu/$\sim$nts/

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