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Relative deformation theory, relative Selmer groups, and lifting irreducible Galois representations. (English) Zbl 1522.11046

Summary: We study irreducible odd mod \(p\) Galois representations \(\bar{\rho}:\mathrm{Gal}(\overline{F}/ F)\to G(\overline{\mathbb{F}}_p)\), for \(F\) a totally real number field and \(G\) a general reductive group. For \(p{\gg_{G,F}}0\), we show that any \(\bar{\rho}\) that lifts locally, and at places above \(p\) to de Rham and Hodge-Tate regular representations, has a geometric \(p\)-adic lift. We also prove non-geometric lifting results without any oddness assumption.

MSC:

11F80 Galois representations
11R34 Galois cohomology
11R39 Langlands-Weil conjectures, nonabelian class field theory
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