Abstract
AbstractLet K be an arbitrary number field, and let ρ : Gal($\math{\bar{K}}$/K) → GL2(E) be a nearly ordinary irreducible geometric Galois representation. In this paper, we study the nearly ordinary deformations of ρ. When K is totally real and ρ is modular, results of Hida imply that the nearly ordinary deformation space associated to ρ contains a Zariski dense set of points corresponding to ‘automorphic’ Galois representations. We conjecture that if K is not totally real, then this is never the case, except in three exceptional cases, corresponding to: (1) ‘base change’, (2) ‘CM’ forms, and (3) ‘even’ representations. The latter case conjecturally can only occur if the image of ρ is finite. Our results come in two flavours. First, we prove a general result for Artin representations, conditional on a strengthening of the Leopoldt Conjecture. Second, when K is an imaginary quadratic field, we prove an unconditional result that implies the existence of ‘many’ positive-dimensional components (of certain deformation spaces) that do not contain infinitely many classical points. Also included are some speculative remarks about ‘p-adic functoriality’, as well as some remarks on how our methods should apply to n-dimensional representations of Gal($\math{\bar{\QQ}}$/ℚ) when n > 2.
Highlights
Let K be a number field, K/K be an algebraic closure of K, and p a prime that splits completely in K
Fix ρ : Gal(K/K) → GL2(E), a continuous “nearly ordinary” absolutely irreducible p-adic Galois representation unramified outside a finite set of places of K, where E is a field extension of Qp
We may form R, the universal deformation ring of continuous nearly ordinary deformations of the Galois representation ρ; this is a complete noetherian local ring that comes along with a universal nearly ordinary deformation of our representation ρ, ρuniv : GK,S → GL2(R), from which we may recover ρ as a specialization at some E-valued point of Spec(R)
Summary
Let K be a number field, K/K be an algebraic closure of K, and p a prime that splits completely in K. Suppose that ρ : Gal(K/K) → GL2(E) is continuous, irreducible, nearly ordinary, has finite image, and admits infinitesimally classical deformations. Suppose that ρ : Gal(K/K) → GL2(E) is continuous, irreducible, nearly ordinary, is unramified except at finitely many places, and admits infinitesimally classical deformations. Our result (Theorem 1.2), comprehensive, is contingent upon a conjecture, and is only formulated for representations with finite image To offer another angle on the study of deformation spaces (of nearly ordinary Galois representations) containing few classical automorphic forms, we start with a residual representation ρ : Gal(K/K) → GL2(F) where F is a finite field of characteristic p, and ρ is irreducible and nearly ordinary at v|p (and satisfies some supplementary technical hypotheses).
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have