Galois representations of dihedral type over Q_p

Monodromy of subrepresentations and irreducibility of low degree automorphic Galois representations

“…For case (I), we can lift $$G$$ to $$\mathrm{GL}_2(\mathbb {C})$$ so that $$$$\begin{equation*} \ker \bar{\rho }_\lambda =\ker (\widetilde{\phi _\lambda }|_{D_p}). \end{equation*}$$$$For case (II), we can find a good lift of $$\widetilde{\phi }_\lambda |_{D_p}$$ to $$\mathrm{GL}_2(\mathbb {C})$$ such that the Artin conductor is bounded in terms of $$|\widetilde{\phi }_\lambda (I_p^w)|$$ (the order of the image of the wild inertia) by [31, Theorem 1, Proposition 1]. Since () is a subrepresentation of $$\operatorname{End}(\overline{V}_\lambda ^{\operatorname{ss}}\otimes \overline{\mathbb {F}}_\ell )$$, the order $$|\widetilde{\phi }_\lambda (I_p^w)|$$ is bounded by a constant independent of $$\lambda$$ by §4.1(ii).…”

“…The degree 𝑘 ∶= 𝑑𝑒g(𝑋 conn ∕𝑋) = |𝐆 𝜆 ∕𝐆 • 𝜆 | is independent of 𝜆 by assertions (i) and (ii) above. In this proof, we call 𝑘 the degree of (31). Consider 𝓁 > max{𝑘, 𝑛} from now on and we prove Theorem 1.2 by induction on the degree 𝑘.…”