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.


11F80 Galois representations
11R32 Galois theory
11R34 Galois cohomology
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