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

TL;DR

This paper investigates p-adic endoscopy on $ ext{SL}_2$ eigenvarieties over totally real fields, revealing how non-automorphic members of endoscopic L-packets contribute to overconvergent cohomology and analyzing the local geometric structure.

Contribution

It provides a geometric analysis of $ ext{SL}_2$-eigenvarieties, showing their local structure as quotients of $ ext{GL}_2$ eigenvarieties and quantifying contributions of non-automorphic L-packet members.

Findings

Non-automorphic members contribute eigenvectors at endoscopic points.

The $ ext{SL}_2$-eigenvariety often fails to be Gorenstein at these points.

The eigenvariety is locally a quotient of a $ ext{GL}_2$ eigenvariety.

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.