Infinitely ramified Galois representations

Abstract

In this paper we show how to construct, for most p >= 5, two types of surjective representations \rho:G_Q=Gal(\bar{Q}/Q) -> GL_2(Z_p) that are ramified at an infinite number of primes. The image of inertia at almost all of these primes will be torsion-free. The first construction is unconditional. The catch is that we cannot say whether \rho|_{G_p=Gal(\bar{Q_p}/Q_p) is crystalline or even potentially semistable. The second construction assumes the Generalized Riemann Hypothesis (GRH). With this assumption we can further arrange that \rho|_{G_p} is crystalline at p. We remark that infinitely ramified *reducible* representations have been previously constructed by more elementary means.