Euler system

Euler systems can be used to construct annihilators of ideal class groups or Selmer groups, thus giving bounds on their orders, which in turn has led to deep theorems such as the finiteness of some Tate-Shafarevich groups. 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.

Definition

Although there are several definitions of special sorts of Euler system, there seems to be no published definition of an Euler system that covers all known cases. But it is possible to say roughly what an Euler system is, as follows:

• An Euler system is given by collection of elements c_F. These elements are often indexed by certain number fields F containing some fixed number field K, or by something closely related such as square-free integers. The elements c_F are typically elements of some Galois cohomology group such as H¹(F, T) where T is a p-adic representation of the absolute Galois group of K.
• The most important condition is that the elements c_F and c_G for two different fields F ⊆ G are related by a simple formula, such as

${\displaystyle {\mathrm{c}\mathrm{o}\mathrm{r}}_{G/F}(c_G)=\prod _{q\in \mathrm{\Sigma }(G/F)}P({\mathrm{F}\mathrm{r}}_q^{-1}|{\mathrm{H}\mathrm{o}\mathrm{m}}_O(T,O(1));{\mathrm{F}\mathrm{r}}_q^{-1})c_F}$

Here the "Euler factor" P(τ|B;x) is defined to be the element det(1-τx|B) considered as an element of O[x], which when x happens to act on B is not the same as det(1-τx|B) considered as an element of O.

• There may be other conditions that the c_F have to satisfy, such as congruence conditions.

Kazuya Kato refers to the elements in an Euler system as "arithmetic incarnations of zeta" and describes the property of being an Euler system as "an arithmetic reflection of the fact that these incarnations are related to special values of Euler products".