Endoscopy on SL₂-eigenvarieties

Journal article, 2024

In this paper, we study p-adic endoscopy on eigenvarieties for SL2 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 SL2-eigenvariety is locally a quotient of an eigenvariety for GL(2), which allows us to explicitly describe the local geometry of the SL2-eigenvariety. In particular, we show that it often fails to be Gorenstein.