In number theory, a compatible system of ℓ-adic representations is an abstraction of certain important families of ℓ-adic Galois representations, indexed by prime numbers ℓ, that have compatibility properties for almost all ℓ.

Prototypical examples include the cyclotomic character and the Tate module of an abelian variety.

A slightly more restrictive notion is that of a strictly compatible system of ℓ-adic representations which offers more control on the compatibility properties. More recently, some authors^[1] have started requiring more compatibility related to p-adic Hodge theory.

Compatible systems of ℓ-adic representations are a fundamental concept in contemporary algebraic number theory.

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• Taylor, Richard (2004), "Galois representations", Annales de la Faculté des Sciences de Toulouse, 6, 13 (1): 73–119, arXiv:math/0212403, CiteSeerX 10.1.1.363.4678, doi:10.5802/afst.1065, MR 2060030, S2CID 16064051, Zbl 1074.11030