Editing Euler system (section)

{{Short description|Mathematical concept}}
In [[mathematics]], particularly [[number theory]], an '''Euler system''' is a collection of compatible elements of [[Galois cohomology]] groups indexed by [[Field extension|fields]].  They were introduced by {{harvs|txt|authorlink=Victor Kolyvagin|last=Kolyvagin|year=1990}} in his work on [[Heegner point]]s on [[modular elliptic curve]]s, which was motivated by his earlier paper {{harvtxt|Kolyvagin|1988}} and  the work of {{harvtxt|Thaine|1988}}.  Euler systems are named after [[Leonhard Euler]] because  the factors relating different elements of an Euler system resemble the [[Euler factor]]s of an [[Euler product]].

Euler systems can be used to construct annihilators of [[ideal class group]]s or [[Selmer group]]s, thus giving bounds on their orders, which in turn has led to deep theorems such as the finiteness of some [[Tate-Shafarevich group]]s. This led to [[Karl Rubin]]'s new proof of the [[main conjecture of Iwasawa theory]], considered simpler than the original proof due to [[Barry Mazur]] and [[Andrew Wiles]].