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
- Such as Taylor 2004
gollark: I mean, they're more useful there.
gollark: For the second thing, it does seem... pretty much fine... to ship emergency-use goods from places without natural disasters going on to places with them.
gollark: Apparently yggdrasil gets around issues with memory using some sort of strange algorithm involving trees and by dropping the requirement to always find the best available path.
gollark: There are some experiments like yggdrasil and cjdns, but I don't know how well they scale beyond the few thousand random people testing it.
gollark: Apparently doing not-much-configuration mesh routing is a very hard problem, and it seems like the existing protocols are designed in ways which make it annoying too.
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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