Editing Euler system (section)

==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''<sub>''F''</sub>. These elements are often indexed by certain [[number field]]s ''F'' containing some fixed number field ''K'', or by something closely related such as square-free integers. The elements ''c''<sub>''F''</sub> are typically elements of  some Galois cohomology group such as H<sup>1</sup>(''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''<sub>''F''</sub> and ''c''<sub>''G''</sub> for two different fields ''F''&nbsp;⊆&nbsp;''G'' are related by a simple formula, such as
:<math> {\rm cor}_{G/F}(c_G) = \prod_{q\in \Sigma(G/F)}P(\mathrm{Fr}_q^{-1}|{\rm Hom}_O(T,O(1));\mathrm{Fr}_q^{-1})c_F</math>
:Here the "Euler factor" ''P''(&tau;|''B'';''x'') is defined to be the element det(1-&tau;''x''|''B'') considered as an element of O[''x''], which when ''x'' happens to act on ''B'' is not the same as det(1-&tau;''x''|''B'') considered as an element of O.
*There may be other conditions that the ''c''<sub>''F''</sub> 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".<ref>{{harvnb|Kato|2007|loc=§2.5.1}}</ref>