Compatible system of ℓ-adic representations

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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.


  1. ^ Such as Taylor 2004


  • 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