Endoscopy on SL2-eigenvarieties

Journal article, 2024

In this paper, we study p -adic endoscopy on eigenvarieties for <> SL 2 \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 <> SL 2 \mathrm{SL}_{2} -eigenvariety is locally a quotient of an eigenvariety for <> GL 2 \mathrm{GL}_{2}, which allows us to explicitly describe the local geometry of the <> SL 2 \mathrm{SL}_{2} -eigenvariety. In particular, we show that it often fails to be Gorenstein.