On the infinite fern of Galois representations of unitary type

TL;DR

This paper investigates the structure and dimensions of Galois deformation spaces of unitary type, introduces an infinite fern analogue, and proves key properties of crystalline deformations and eigenvarieties, with applications to automorphic representations.

Contribution

It establishes lower bounds on the dimensions of Zariski-closures of modular points, introduces an infinite fern analogue for unitary groups, and proves new properties of crystalline deformations and eigenvarieties.

Findings

Each irreducible component of the Zariski-closure has dimension at least 6[F:Q].

Any first order deformation of a generic crystalline representation is a linear combination of trianguline deformations.

Unitary eigenvarieties are etale over the weight space at non-critical classical points.

Abstract

Let E be a CM number field, F its maximal totally real subfield, c the generator of Gal(E/F), p an odd prime totally split in E, and S a finite set of places of E containing the places above p. Let r : G_{E,S} --> GL_3(F_p^bar) be a modular, absolutely irreducible, Galois representation of type U(3), i.e. such that r^* = r^c, and let X(r) be the rigid analytic generic fiber of its universal G_{E,S}-deformation of type U(3). We show that each irreducible component of the Zariski-closure of the modular points in X(r) has dimension at least 6[F:Q]. We study an analogue of the infinite fern of Gouvea-Mazur in this context and deal with the Hilbert modular case as well. As important steps, we prove that any first order deformation of a generic enough crystalline representation of Gal(Q_p^bar/Q_p) (of any dimension) is a linear combination of trianguline deformations, and that unitary eigenvarieties (of any rank) are etale over the weight space at the non-critical classical points. As another application, we obtain a general theorem about the image of the localization at p of the p-adic Adjoint' Selmer group of the p-adic Galois representations attached to any cuspidal, cohomological, automorphic representation Pi of GL_n(A_E) such that Pi^* = Pi^c (for any n).