Abstract
We prove, under mild hypotheses, that there are no irreducible two-dimensional ordinary even Galois representations of \(\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\) with distinct Hodge–Tate weights. This is in accordance with the Fontaine–Mazur conjecture. If K/Q is an imaginary quadratic field, we also prove (again, under certain hypotheses) that \(\mathrm{Gal}(\overline{\mathbf{Q}}/K)\) does not admit irreducible two-dimensional ordinary Galois representations of non-parallel weight.
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Calegari, F. Even Galois representations and the Fontaine–Mazur conjecture. Invent. math. 185, 1–16 (2011). https://doi.org/10.1007/s00222-010-0297-0
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DOI: https://doi.org/10.1007/s00222-010-0297-0