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Residual irreducibility of compatible systems. (English) Zbl 1446.11104

In the present paper, the authors prove that in a compatible system of absolutely irreducible Galois representations, a density \(1\) set of residual representations are absolutely irreducible. In the special case of Hodge-Tate regular compatible systems, this gives a new proof of Proposition 5.3.2 in [T. Barnet-Lamb et al., Ann. Math. (2) 179, No. 2, 501–609 (2014; Zbl 1310.11060)]. In the special case of abelian varieties, this gives that let \(A\) be an abelian variety over a number field \(F\) with \(\text{End}(A)=\mathbb{Z}\), then the representation \(\Gamma_F\) of the absolute Galois group of \(F\) on \(A[\ell]\) is absolutely irreducible for \(\ell\) in a set of primes of density \(1\).
Reviewer: Lei Yang (Beijing)

MSC:

11F80 Galois representations

Citations:

Zbl 1310.11060
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