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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.

Examples

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

Variations

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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

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

Notes

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  1. ^ Such as Taylor 2004

References

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  • 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