relations modulo all primes above l where l is a prime not dividing n and Fl is a Frobenius automorphism with Fl(ζn) = ζ. Kolyvagin used this Euler system to give an elementary proof of the Gras conjecture. Gauss sums Elliptic units Heegner points Kolyvagin constructed an Euler system from the Heegner points of an elliptic curve, and used this to show that in some cases the Tate-Shafarevich group is finite. Kato's Euler system Kato's Euler system consists of certain elements occurring in the algebraic K-theory of modular curves. These elements—named Beilinson elements after Alexander Beilinson who introduced them in —were used by Kazuya Kato in to prove one divisibility in Barry Mazur's main conjecture of Iwasawa theory for elliptic curves. Notes References . Proceedings of the congress held in Madrid, August 22–30, 2006 External links Several papers on Kolyvagin systems are available at Barry Mazur's web page (as of July 2005). Algebraic number theory