Endoscopy on \(\mathrm{SL}_2\)-eigenvarieties. (English) Zbl 07891383
Summary: In this paper, we study \(\mathrm{p} \)-adic endoscopy on eigenvarieties for \(\mathrm{SL}_2\) over totally real fields, taking a geometric perspective. We show that non-automorphic members of endoscopic \(\mathrm{L} \)-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.
MSC:
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
| 11F80 | Galois representations |
| 11F33 | Congruences for modular and \(p\)-adic modular forms |
| 11F85 | \(p\)-adic theory, local fields |
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
Software:
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