Euler system

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In mathematics, an Euler system is a technical device in the theory of Galois modules, first noticed as such in the work around 1990 by Victor Kolyvagin on Heegner points on modular elliptic curves. This concept has since undergone an axiomatic development, in particular by Barry Mazur and Karl Rubin.

There is a general motivation for the use of Euler systems, which is that they are supposed to be essentially derived from group cohomology, and to have the capability to 'control' or bound Selmer groups, in different contexts. According to generally accepted ideas, such control is a feature of L-functions, through their values at particular points. The virtue of Euler systems is that they may function as a 'middle term', lying between knowledge of L-functions that apparently lies deep, and the Selmer groups that are the object of direct study in diophantine geometry. The theory is still under development; in essence it is expected to apply to abelian extensions, organised in infinite towers, and their pro-finite Galois groups. The Euler system concept is supposed to pin down an idea of coherent system of cohomology classes in such a tower, with respect to some level-changing maps of the general field norm type, in the presence of a local-global principle.

The Euler system idea made a celebrated but abortive entry in the Andrew Wiles proof of Fermat's last theorem. The use of an Euler system was Wiles's original approach, but failed to deliver in that case.