Let $\mathbb{F}_q$ be a finite field, $\chi$ a complex (or $\overline{\mathbb{Q}_\ell}$) character of $\mathbb{F}_q^\times$, and $\mathcal{L}_\chi$ the Kummer sheaf on the multiplicative group $\mathbb{G}_m/\mathbb{F}_q$. The literature I could find and actually read say that the local monodromy of $\mathcal{L}_\chi$ at $0$ and $\infty$ are both tame, but they don't say any further about what these local monodromy are, as representations of the pro-cyclic groups $I(0)/P(0)$ (the inertia group at $0$ mod the wild inertia group at $0$) and $I(\infty)/P(\infty)$. Is there a simple description of them (like "a character of the group $I/P$ order $d$, where $d$ is..."), or at least a good introductory reference about these sheaves and their monodromy in English? (I can't read French at all, and I am not convinced that the French needed to understand SGA 4$\frac{1}{2}$ is easy to learn, as some people claim.)



You must log in to answer this question.