Compatible system of ℓ-adic representations

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.

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

• Serre, Jean-Pierre (1998) [1968], Abelian l-adic representations and elliptic curves, Research Notes in Mathematics, vol. 7, with the collaboration of Willem Kuyk and John Labute, Wellesley, MA: A K Peters, ISBN 978-1-56881-077-5, MR 1484415

• 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