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

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.

60 pages. Version 4: Final version, to appear in Crelle. Version 3: Minor changes. Version 2: Minor updates and corrections