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Infinitely ramified Galois representations. (English) Zbl 1078.11510
Summary: In this paper we show how to construct, for most $$p \geq 5$$, two types of surjective representations $$\rho: G_{\mathbb Q}=\text{Gal}(\overline{\mathbb Q}/\mathbb Q)\to\text{GL}_2(\mathbb Z_p)$$ 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 $$\rho|_{G_p=\text{Gal}}(\overline{\mathbb Q_p}/\mathbb Q_p)$$ 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.

##### MSC:
 11F80 Galois representations 11R32 Galois theory 11R34 Galois cohomology
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