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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?

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3 Answers 3

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My understanding is that they're named "Euler systems" because that "Frobenius acting on T" in the definition (line 4, p. 22 of Rubin's book) is an "Euler factor" as in Euler's product decomposition of the Riemann zeta function.

The two easiest examples of Euler systems are the so-called cyclotomic units (not the roots of unity, but slightly more complicated, but still classical, beasts built out of them) and the elliptic units (built out of torsion points on CM elliptic curves). Not coincidentally, these are related to the only two types of global fields that we know how to explicitly construct abelian extensions of. Then there is Kato's more complicated Euler system of Heegner points [EDIT - This phrase was wrong- please see post below!], and Kolyvagin's Euler system of Stickelberger elements. All these are described in Rubin's book, but if you haven't seen them before, it might help to have more references.

If group cohomology is still new to you, the cyclotomic units are the best Euler systems to start with, since you don't need Galois cohomology to define them. (Norm-coherent units map to corestriction-coherent cohomology classes under the Kummer map, which is why these global units form an Euler system in the sense of Rubin's book). Rubin's appendix in Lang's republished books Cyclotomic Fields I and II is easier reading for this. Rubin's Inventiones paper on the main conjecture for CM elliptic curves also contains a nice introduction to the technique.

The application of Euler systems to number theory is the following: Euler systems allow us to bound Selmer groups of p-adic Galois representations. These generalize the Selmer group attached to an abelian variety, the ideal class group, and other objects of arithmetic interest. Bounding them is good because it allows us to prove Iwasawa Main Conjectures, which link the behavior of Selmer groups to p-adic L-functions and encompass basically every classical arithmetic tool for computing p-parts of special values of L-functions.

Washington's book on cyclotomic fields explains this in the cyclotomic example, and the spirit and flavor of the general theory is the same. Coates and Sujatha's book Cyclotomic Fields and Zeta Values is also an excellent introduction to the Euler system technique in the cyclotomic setting.

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Thank you, Hunters! Your answer is very informative. (You could probably put some of them on wikipedia!) I am comfortable with Galois cohomology and Selmer groups (of p-adic representations), so if you'd like, you can talk more details about the application of Euler system! Thank you again! –  natura Jan 11 '10 at 5:06

A small correction to Hunter Brooks's answer: Kato's Euler system is unrelated to Heegner points but comes from Beilinson's elements on modular curves. The fact that Heegner points form an Euler system for the Galois representation attached to an elliptic curve (or more generally to a modular abelian variety) is due to Kolyvagin, who also coined the name, and thus predates the construction of Kato's Euler system. For technical (but important) reasons, the Euler system of Heegner points is not discussed in Rubin's book. It would also be wrong to conclude that the examples given by Hunter Brooks and I are the only examples known of Euler systems: sticking to inherently different objects from those already mentioned, one should also mention the Euler system of Stark units and various Euler systems coming from special cycles on Shimura curves or varieties.

As is suggested in Hunter Brooks's answer, Euler systems are useful to bound Selmer groups, so they are linked with p-adic L-functions via the Iwasawa Main Conjecture (which relates the p-adic L-function to the characteristic ideal of some Selmer group). Interestingly, it is sometimes known how to relate Euler systems with p-adic L-functions directly using (generalization of) the so-called Coleman map and explicit reciprocity law. In plain terms, once an Euler system is known, it is sometimes possible to construct the p-adic L-function as the image of this Euler system via some well-chosen maps. This construction is known to be possible in particular for the Kubota-Leopoldt p-adic L-function (using the Euler system of p-adic units) and for modular forms (using Kato's Euler system).

One last comment directed to the original poster: if you wish to learn about how Euler systems can be used to bound Selmer groups, a useful reference beside Rubin's book is the book Kolyvagin systems by B.Mazur and K.Rubin. The treatment of Iwasawa theory, in particular, is in my opinion easier to understand at first reading.

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Perhaps one should mention that Euler system of Stark units is conjectural. Also, that the Euler system of Heegner points is not directly related to p-adic L-functions. –  Mahesh Kakde Dec 9 '10 at 11:12

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.

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