Patrikis, Stefan T.; Snowden, Andrew W.; Wiles, Andrew J. Residual irreducibility of compatible systems. (English) Zbl 1446.11104 Int. Math. Res. Not. 2018, No. 2, 571-587 (2018). 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) Cited in 3 Documents MSC: 11F80 Galois representations Citations:Zbl 1310.11060 PDFBibTeX XMLCite \textit{S. T. Patrikis} et al., Int. Math. Res. Not. 2018, No. 2, 571--587 (2018; Zbl 1446.11104) Full Text: DOI arXiv