Euler system

Euler system

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.

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).


Wikimedia Foundation. 2010.

Игры ⚽ Поможем сделать НИР

Look at other dictionaries:

  • Euler [2] — Euler, 1) Leonhard, Mathematiker und Physiker, geb. 15. April 1707 in Basel, gest. 18. Sept. 1783 in Petersburg, studierte in Basel Mathematik und erwarb schon 1723 den Magistergrad, studierte dann noch Theologie, orientalische Sprachen und… …   Meyers Großes Konversations-Lexikon

  • Euler angles — The Euler angles were developed by Leonhard Euler to describe the orientation of a rigid body (a body in which the relative position of all its points is constant) in 3 dimensional Euclidean space. To give an object a specific orientation it may… …   Wikipedia

  • Euler–Lagrange equation — In calculus of variations, the Euler–Lagrange equation, or Lagrange s equation is a differential equation whose solutions are the functions for which a given functional is stationary. It was developed by Swiss mathematician Leonhard Euler and… …   Wikipedia

  • Euler's three-body problem — In physics and astronomy, Euler s three body problem is to solve for the motion of a particle that is acted upon by the gravitational field of two other point masses that are either fixed in space or move in circular coplanar orbits about their… …   Wikipedia

  • Euler-Lagrange-Gleichung — Der Lagrange Formalismus ist eine 1788 von Joseph Louis Lagrange eingeführte Formulierung der klassischen Mechanik, in der die Dynamik eines Systems durch eine einzige skalare Funktion, die Lagrangefunktion, beschrieben wird. Dadurch wird… …   Deutsch Wikipedia

  • Euler-Lagrange-Gleichungen — Der Lagrange Formalismus ist eine 1788 von Joseph Louis Lagrange eingeführte Formulierung der klassischen Mechanik, in der die Dynamik eines Systems durch eine einzige skalare Funktion, die Lagrangefunktion, beschrieben wird. Dadurch wird… …   Deutsch Wikipedia

  • Euler equations (fluid dynamics) — In fluid dynamics, the Euler equations govern inviscid flow. They correspond to the Navier Stokes equations with zero viscosity and heat conduction terms. They are usually written in the conservation form shown below to emphasize that they… …   Wikipedia

  • Euler's totient function — For other functions named after Euler, see List of topics named after Leonhard Euler. The first thousand values of φ(n) In number theory, the totient φ(n) of a positive integer n is defined to be the number of positive integers less than or equal …   Wikipedia

  • Euler method — In mathematics and computational science, the Euler method, named after Leonhard Euler, is a first order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic kind of explicit… …   Wikipedia

  • Euler brick — In mathematics, an Euler brick, named after the famous mathematician Leonhard Euler, is a cuboid with integer edges and also integer face diagonals. A primitive Euler brick is an Euler brick with its edges relatively prime.Alternatively stated,… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”