
arXiv: 1409.1834
Let$p\geqslant 5$be a prime, and let${\mathcal{O}}$be the ring of integers of a finite extension$K$of$\mathbb{Q}_{p}$with uniformizer${\it\pi}$. Let${\it\rho}_{n}:G_{\mathbb{Q}}\rightarrow \mathit{GL}_{2}\left({\mathcal{O}}/({\it\pi}^{n})\right)$have modular mod-${\it\pi}$reduction$\bar{{\it\rho}}$, be ordinary at$p$, and satisfy some mild technical conditions. We show that${\it\rho}_{n}$can be lifted to an${\mathcal{O}}$-valued characteristic-zero geometric representation which arises from a newform. This is new in the case when$K$is a ramified extension of$\mathbb{Q}_{p}$. We also show that a prescribed ramified complete discrete valuation ring${\mathcal{O}}$is the weight-$2$deformation ring for$\bar{{\it\rho}}$for a suitable choice of auxiliary level. This implies that the field of Fourier coefficients of newforms of weight 2, square-free level, and trivial nebentype that give rise to semistable$\bar{{\it\rho}}$of weight 2 can have arbitrarily large ramification index at$p$.
11F80, Mathematics - Number Theory, Galois representations, 11R, Representation-theoretic methods; automorphic representations over local and global fields, QA1-939, FOS: Mathematics, 11F, Number Theory (math.NT), Mathematics
11F80, Mathematics - Number Theory, Galois representations, 11R, Representation-theoretic methods; automorphic representations over local and global fields, QA1-939, FOS: Mathematics, 11F, Number Theory (math.NT), Mathematics
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