LIFTING TORSION GALOIS REPRESENTATIONS
LIFTING TORSION GALOIS REPRESENTATIONS
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 ${\it\rho}_{n}$ …