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On \(\ell\)-independence of algebraic monodromy groups in compatible systems of representations. (English) Zbl 0778.11036

The paper studies compatible systems of \(\ell\)-adic Galois representations (\(\ell\) a varying prime), as arise from the étale cohomologies of algebraic varieties. The purpose seems to be to derive as much as possible from general principles. The main results are as follows: Let \(G_ \ell\) denote the connected reductive part of \(\text{Im}(\rho_ \ell)\). Then there exists a finite Galois extension \(E\) of \(\mathbb{Q}\) such that, except for \(\ell\) in a set of density zero, \(G_ \ell\) splits over \(E\mathbb{Q}_ \ell\) and its Weyl group depends only on the conjugacy class of \(\text{Frob}_ \ell\) in \(\text{Gal}(E/\mathbb{Q})\). Furthermore, if all \(\rho_ \ell\) are absolutely irreducible, the root- datum and the representation of \(G_ \ell\) also depend only on the conjugacy class of \(\text{Frob}_ \ell\).

MSC:

11G35 Varieties over global fields
22E46 Semisimple Lie groups and their representations
14F20 Étale and other Grothendieck topologies and (co)homologies
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References:

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