# zbMATH — the first resource for mathematics

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.

##### MSC:
 11R32 Galois theory 11R37 Class field theory