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Ramified lifts of Galois representations and dimension of ordinary deformation rings
Abstract
Let $F$ be a totally real number field, $k$ a finite field of characteristic $p$ and $\ol{\rho}: \tr{Gal}(\ol F/F) \ra GL_n(k)$ a continuous Galois representation. Under some technical hypotheses on $\ol{\rho}$ we extend the method of Khare and Ramakrishna of constructing ramified characteristic zero lifts of $\rh$ from the setting of $GL_2$ to $GL_n$. As an application of this method we prove the existence of closed points $x \in \tr{Spec}(R^{\tr{ord}}[1/p])$, on a certain nearly ordinary deformation ring $R^{\tr{ord}}$, such that Mazur's dimension conjecture is true locally at $x$. In the process we obtain examples of ordinary Galois representations $\rho: G_{F, S\cup Q} \ra GL_n(\mc O)$, where $\mc O$ is a finite extension of $\Z_p$, such that its adjoint Selmer group $H^1_{\mc L} (G_{F, S\cup Q}, ad^0\, \rho \otimes \Q_p/\Z_p)$ is finite. We then determine the exact size of these Selmer groups in terms of the ramification index of the primes in the auxiliary set.
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