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arXiv · 1810.06739

Geometric stabilisation via p-adic integration

Michael Groechenig·Dimitri Wyss·Paul Ziegler
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Abstract

In this article we give a new proof of Ngô's Geometric Stabilisation Theorem, which implies the Fundamental Lemma. This is a statement which relates the cohomology of Hitchin fibres for a quasi-split reductive group scheme $G$ to the cohomology of Hitchin fibres for the endoscopy groups $H_κ$. Our proof avoids the Decomposition and Support Theorem, instead the argument is based on results for $p$-adic integration on coarse moduli spaces of Deligne-Mumford stacks. Along the way we establish a description of the inertia stack of the (anisotropic) moduli stack of $G$-Higgs bundles in terms of endoscopic data, and extend duality for generic Hitchin fibres of Langlands dual group schemes to the quasi-split case.

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Michael Groechenig, Dimitri Wyss, Paul Ziegler. 2019-10-28. Geometric stabilisation via p-adic integration. https://arxiv.org/abs/1810.06739

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