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[1] 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.

Notes
  1. Such as Taylor 2004
gollark: ?news
gollark: Consequentialist-ly speaking (yes, I am aware you don't subscribe to this) a technological development could be "bad", if the majority of the possible uses for it are negative, or it's most likely to be used for negative things. To what extent any technology actually falls into that is a separate issue though.
gollark: You can show that 2 + 2 = 4 follows from axioms, and that the system allows you to define useful mathematical tools to model reality.
gollark: If you're going to say something along the lines of "see how it deals with [SCENARIO] and rate that by [OTHER STANDARD]", this doesn't work because it sneaks in [OTHER STANDARD] as a more fundamental underlying ethical system.
gollark: I don't see how you can empirically test your ethics like you can a scientific theory.

References
  • Serre, Jean-Pierre (1998) [1968], Abelian l-adic representations and elliptic curves, Research Notes in Mathematics, 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, MR 2060030
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