TITLE: what is an Euler system and the motivation for it?
QUESTION [22 upvotes]: I tried to read the definition of it on Rubin's book "Euler systems" but it looks highly technical. Can anyone shed some light on it? In particular, is there some starting examples?
The wiki entry is too short and does not contain much useful information.
And what is the motivation of such objects? In particular, why is it called "Euler"?
And if possible, can someone say something about how it is used in number theory? such as Iwasawa theory?

REPLY [5 votes]: These are good answers.  I would just like to add that while the applications of Euler systems have been mainly p-adic, they are actually motivic (ie units if the motive is h0(Spec F)).  One might hope that if one is able to attach an L-function to a geometric object, there is also an Euler system living in an appropriate motivic cohomology.  A sort of cohomological Euler product if you will.  At least that is motivation for the sweeping special values conjectures such as the Tamagawa Number Conjecture.