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arXiv · 2205.03103

Endoscopy on $\mathrm{SL}_2$-eigenvarieties

Christian Johansson·Judith Ludwig
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Abstract

In this paper, we study p-adic endoscopy on eigenvarieties for $\mathrm{SL}_2$ over totally real fields, taking a geometric perspective. We show that non-automorphic members of endoscopic L-packets of regular weight contribute eigenvectors to overconvergent cohomology at critically refined endoscopic points on the eigenvariety, and we precisely quantify this contribution. This gives a new perspective on and generalizes previous work of the second author. Our methods are geometric, and are based on showing that the $\mathrm{SL}_2$-eigenvariety is locally a quotient of an eigenvariety for $\mathrm{GL}_2$, which allows us to explicitly describe the local geometry of the $\mathrm{SL}_2$-eigenvariety. In particular, we show that it often fails to be Gorenstein at such points.

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Christian Johansson, Judith Ludwig. 2024-03-20. Endoscopy on $\mathrm{SL}_2$-eigenvarieties. https://arxiv.org/abs/2205.03103

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