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BIRS Workshop Lecture Videos

Compatible systems of Galois representations of global function fields Boeckle, Gebhard


Let $K$ be a finitely generated infinite field over the finite prime field $\mathbb{F}_p$ with separable closure $K^s$ and let $X$ be a smooth projective variety over $K$. By Deligne the cohomology groups $H^i_{\mathrm{\acute{e}t}}(X_{K^s},\mathbb{Q}_\ell)$ for varying primes $\ell\neq p$ form a ($\mathbb{Q}$-rational) compatible system of Galois representations of $\mathrm{Gal}(K^s/K)$ and its restriction to the geometric Galois group $G^{\mathrm{geo}}_K=\mathrm{Gal}(K^s/K\mathbb{F}_p^s)$ is semisimple. Using mainly algebraic geometry, representation theory and Bruhat-Tits theory, Cadoret, Hui and Tamagawa showed recently that also the family of reductions $H^i_{\mathrm{\acute{e}t}}(X_{K^s},\mathbb{F}_\ell)$ is semisimple as a representation of $G^{\mathrm{geo}}_K$ for almost all $\ell$, the key case being that of a global function field $K$. This has important consequence for the image of $G^{\mathrm{geo}}_K$ for its action on the adelic module~$H^i_{\mathrm{\acute{e}t}}(X_{K^s},\mathbb{A}_{\mathbb{Q}})$. In joint work with W. Gajda and S. Petersen, using automorphic methods as a main tool, we prove the analog of the above result for any $E$-rational compatible system of Galois representations of a global function field. In the talk I shall explain the context, indicate the applications and sketch how automorphic methods come to bear on the problem.

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