Even Galois Representations and the Fontaine-Mazur Conjecture

2009/07/20 by Frank Calegari, Calegari, Frank
Mathematics · #11F80 #11R39 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11F80 #msc:11R39

paper · pdf · doi:10.48550/arxiv.0907.3427

Revised Version

openalex publication_date 2009/07/20 · arxiv created 2010/11/10 · arxiv updated 2010/11/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove some cases of the Fontaine-Mazur conjecture for even Galois representations. In particular, we prove, under mild hypotheses, that there are no irreducible two-dimensional ordinary even Galois representations of \Gal(\Qbar/\Q) with distinct Hodge-Tate weights. If K/\Q is an imaginary quadratic field, we also prove (again, under certain hypotheses) that \Gal(\Qbar/K) does not admit irreducible two-dimensional ordinary Galois representations of non-parallel weight. Finally, we prove that any weakly compatible family of two dimensional irreducible Galois representations of \Gal(\Qbar/\Q) is, up to twist, either modular or finite.

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