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Relative deformation theory, relative Selmer groups, and lifting irreducible Galois representations
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abstract
We study irreducible odd mod $p$ Galois representations $\bar{\rho} \colon \mathrm{Gal}(\overline{F}/F) \to G(\overline{\mathbb{F}}_p)$, 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.
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Lifting $G$-irreducible but $\mathrm{GL}_n$-reducible Galois representations
For every even N at least 6, the paper produces infinitely many SO_{N+1}-valued residual Galois representations that are SO_{N+1}-irreducible but GL_{N+1}-reducible, each with a geometric lift of Zariski-dense image.
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