Infinitely ramified Galois representations
Ravi Ramakrishna
Abstract
In this paper we show how to construct, for most p >= 5, two types of surjective representations ρ:GQ=Gal(Q/Q) -> GL2(Zp) 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 ρ|Gp=Gal(Qp/Qp) is crystalline or even potentially semistable. The second construction assumes the Generalized Riemann Hypothesis (GRH). With this assumption we can further arrange that ρ|Gp is crystalline at p. We remark that infinitely ramified *reducible* representations have been previously constructed by more elementary means.
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