- Euler system
In
mathematics , an Euler system is a technical device in the theory ofGalois module s, first noticed as such in the work around 1990 byVictor Kolyvagin onHeegner point s onmodular elliptic curve s. This concept has since undergone anaxiomatic development, in particular byBarry Mazur andKarl 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 boundSelmer group s, in different contexts. According to generally accepted ideas, such control is a feature ofL-function s, 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 indiophantine geometry . The theory is still under development; in essence it is expected to apply toabelian extension s, organised in infinite towers, and theirpro-finite Galois group s. 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 generalfield norm type, in the presence of alocal-global principle .The Euler system idea made a celebrated but abortive entry in the
Andrew Wiles proof ofFermat's last theorem . The use of an Euler system was Wiles's original approach, but failed to deliver in that case.References
* "Euler Systems" (Annals of Mathematics Studies 147), Karl Rubin, Princeton University Press, 2000.
External links
* Several papers on Kolyvagin systems are available at [http://abel.math.harvard.edu/~mazur/projects.html Barry Mazur's web page] (as of July 2005).
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