Une interpr\'etation modulaire de la vari\'et\'e trianguline

TL;DR

This paper constructs a new eigenvariety analogue using patching modules, linking it to trianguline Galois representations and exploring its implications for modularity, local rings, and conjectures in p-adic representation theory.

Contribution

It introduces a novel construction of an eigenvariety analogue via patching modules and establishes its connection with trianguline Galois representations and various conjectures.

Findings

The patched eigenvariety matches a union of irreducible components of trianguline Galois representations.

The work relates the eigenvariety to modularity conjectures in the crystalline case.

It discusses implications for conjectures by Breuil, Bella"iche, and Chenevier.

Abstract

Using a patching module constructed in recent work of Caraiani, Emerton, Gee, Geraghty, Pa{\v{s}}k{\=u}nas and Shin we construct some kind of analogue of an eigenvariety. We can show that this patched eigenvariety agrees with a union of irreducible components of a space of trianguline Galois representations. Building on this we discuss the relation with the modularity conjectures for the crystalline case, a conjecture of Breuil on the locally analytic socle of representations occurring in completed cohomology and with a conjecture of Bella\"iche and Chenevier on the complete local ring at certain points of eigenvarieties.