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.
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Examples
Prototypical examples include the cyclotomic character and the Tate module of an abelian variety.
Variations
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 have started requiring more compatibility related to p-adic Hodge theory.
Importance
Compatible systems of ℓ-adic representations are a fundamental concept in contemporary algebraic number theory.
References
Compatible system of ℓ-adic representations Wikipedia(Text) CC BY-SA