  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.
