Lefschetz trace formula and $$\ell $$-adic cohomology of Rapoport–Zink tower for GSp(4)

Mieda, Yoichi

doi:10.1007/s00208-021-02342-z

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2022

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“…), and one can compute the supercuspidal part of i * 1 T V i b! (−) applied to the outer terms of this sequence using a result of Ito-Mieda (Theorem 3.1 of [Mie21]; note that St(ρ, 1) = ρ disc and Sp(ρ, 1) = ρ nt in Mieda's notation). This yields a two-term filtration on the supercuspidal part of A with comprehensible graded pieces.…”

Section: More Examplesmentioning

confidence: 99%

On the Kottwitz conjecture for local shtuka spaces

Kaletha

Hansen

Weinstein³

2022

Forum of Mathematics, Pi

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Kottwitz’s conjecture describes the contribution of a supercuspidal representation to the cohomology of a local Shimura variety in terms of the local Langlands correspondence. A natural extension of this conjecture concerns Scholze’s more general spaces of local shtukas. Using a new Lefschetz–Verdier trace formula for v-stacks, we prove the extended conjecture, disregarding the action of the Weil group, and modulo a virtual representation whose character vanishes on the locus of elliptic elements. As an application, we show that, for an irreducible smooth representation of an inner form of $\operatorname {\mathrm {GL}}_n$ , the L-parameter constructed by Fargues–Scholze agrees with the usual semisimplified parameter arising from local Langlands.

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Section: More Examplesmentioning

confidence: 99%