Extended eigenvarieties for overconvergent cohomology
Christian Johansson, James Newton

TL;DR
This paper extends the construction of eigenvarieties for reductive groups over number fields, generalizing previous work on eigencurves and providing new perspectives on their boundary and associated Galois representations.
Contribution
It develops a general framework for extended eigenvarieties for arbitrary reductive groups, including new constructions for GL(2) and inner forms, and introduces locally analytic distribution modules with characteristic p coefficients.
Findings
Constructed extended eigenvarieties for reductive groups over number fields.
Provided a new construction of the extended eigencurve for GL(2)/Q.
Built Galois representations over the extended eigenvariety for GL(n) over certain fields.
Abstract
Recently, Andreatta, Iovita and Pilloni have constructed spaces of overconvergent modular forms in characteristic p, together with a natural extension of the Coleman-Mazur eigencurve over a compactified (adic) weight space. Similar ideas have also been used by Liu, Wan and Xiao to study the boundary of the eigencurve. This all goes back to an idea of Coleman. In this article, we construct natural extensions of eigenvarieties for arbitrary reductive groups G over a number field which are split at all places above p. If G is GL(2)/Q, then we obtain a new construction of the extended eigencurve of Andreatta-Iovita-Pilloni. If G is an inner form of GL(2) associated to a definite quaternion algebra, our work gives a new perspective on some of the results of Liu-Wan-Xiao. We build our extended eigenvarieties following Hansen's construction using overconvergent cohomology. One key…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
