Nearly ordinary Galois deformations over arbitrary number fields
By Frank Calegari, Barry Mazur. Journal of the Institute of Mathematics of Jussieu, 2007
63 citations recorded by the source corpus.
Abstract Let 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.
Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/90cf5546d1cb6c76968f077276eb56febd49e0cd
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In Algebra and Number Theory
- Akshay Venkatesh
- Nicolas Bergeron
- Frank Calegari
- P. Michel
- R. Munshi
- V. Blomer
- Nicolas Templier
- Gergely Harcos
In Geometry and Topology
- Akshay Venkatesh
- Nicolas Bergeron
- Frank Calegari
- P. Michel
- R. Munshi
- Nicolas Templier
- V. Blomer
- Gergely Harcos
In Mathematical Physics
- Akshay Venkatesh
- Nicolas Bergeron
- Frank Calegari
- P. Michel
- R. Munshi
- V. Blomer
- Nicolas Templier
- Gergely Harcos
Citations in the corpus
- A torsion Jacquet--Langlands correspondence (2012, background): “10 of [21], with the geometry of the Bianchi manifolds for these fields given by [26].”
- The asymptotic growth of torsion homology for arithmetic groups (2010, methodology): “Numerical experiments ([23], and unpublished data computed by Calegari-Dunfield in connection with [11]) as well as the work of Calegari-Mazur [15] suggest that arithmetic hyperbolic 3-manifolds should have a lot of torsion in their homology.”