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Compatible systems of \(\text{mod } p\) Galois representations. (English. Abridged French version) Zbl 0868.11052
Let \(G=\text{Gal} (\overline{\mathbb{Q}}/\mathbb{Q})\) and \({\mathcal O}\) be the ring of integers of a number field \(K\). Compatibility, for a system of (semi-simple) representations \(\rho_{\mathfrak l}:G\to GL_n({\mathcal O}/{\mathfrak l})\) indexed by a cofinite set of prime ideals \({\mathfrak l}\) of \({\mathcal O}\), is defined by analogy with strict compatibility of \(\ell\)-adic representations [J.-P. Serre, Abelian \(\ell\)-adic representations and elliptic curves, Benjamin (1968; Zbl 0186.25701)]. The author proves that a compatible one-dimensional system of \((\text{mod }l)\) representations with the prime-to-\(l\) part of the Artin conductor bounded above must be the Tate twist of a globally defined character of finite order.

11R32 Galois theory
11R37 Class field theory