Even Galois representations and the Fontaine–Mazur conjecture

2010/11/17 by Frank Calegari · 29 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Combinatorics #Conjecture #Galois group #Galois module #Geometry #Mathematics #Pure mathematics #Quadratic equation #math.NT

paper · pdf · doi:10.1007/s00222-010-0297-0

published in Inventiones mathematicae 185(1), 1-16 (Springer Science+Business Media) · Updated to take into account suggestions of the referee; the main theorems remain unchanged

openalex publication_date 2010/11/17 · arxiv created 2011/09/29 · arxiv updated 2015/05/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We prove, under mild hypotheses, that there are no irreducible two-dimensional_even_ Galois representations of \Gal(\Qbar/\Q) which are de Rham with distinct Hodge--Tate weights. This removes the "ordinary" hypothesis required in previous work of the author. We construct examples of irreducible two-dimensional residual representations that have no characteristic zero geometric (= de Rham) deformations.

Citations

• A Cuspidality Criterion for the Exterior Square Transfer of Cusp Forms on GL(4)
• On Euler Products and the Classification of Automorphic Forms II
• A Family of Calabi–Yau Varieties and Potential Automorphy II
• Galois representations arising from some compact Shimura varieties
• Smoothness, semi-stability and alterations
• Automorphy for some l -adic lifts of automorphic mod l Galois representations. II
• The Sato-Tate conjecture for Hilbert modular forms
• A family of Calabi-Yau varieties and potential automorphy
• Serre’s modularity conjecture (I)
• Automorphy for some l -adic lifts of automorphic mod l Galois representations