In mathematics, an Euler system is a collection of compatible elements of Galois cohomology groups indexed by fields.
[1] For every square-free positive integer n pick an n-th root ζn of 1, with ζmn = ζmζn for m,n coprime.
Kolyvagin used this Euler system to give an elementary proof of the Gras conjecture.
Kolyvagin constructed an Euler system from the Heegner points of an elliptic curve, and used this to show that in some cases the Tate-Shafarevich group is finite.
Kato's Euler system consists of certain elements occurring in the algebraic K-theory of modular curves.