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Let ${\Bbb Q}_p$ be a local field. For a prime $q \not= p$, we consider an irreducible residual Galois representation $\overline{\rho} \colon {\mathrm{Gal}}(\overline{{\Bbb Q}}_p/{{\Bbb Q}_p}) \to \...

Let $\bar{\rho}: G_K\to PGL_n(\mathbb{C})$ be projective representation of the absolute Galois group of a number field $K$ and $\varphi\in Aut(G_K)$. A theorem of Tate tells us that we can always ...