The Heegner point Kolyvagin system

TL;DR

This paper proves a key divisibility in Perrin-Riou's Iwasawa main conjecture relating Heegner points to Selmer groups, providing an upper bound on the rank of elliptic curves over quadratic imaginary fields.

Contribution

It establishes one divisibility of Perrin-Riou's conjecture using Kolyvagin's Euler systems, advancing understanding of the connection between Heegner points and Selmer groups.

Findings

Proved one divisibility in Perrin-Riou's conjecture.

Derived an upper bound on the Mordell-Weil rank.

Connected Heegner points to the structure of Selmer groups.

Abstract

Perrin-Riou has formulated a form of the Iwasawa main conjecture, which relates Heegner points to the Selmer group of an elliptic curve as one goes up the anticyclotomic Z_p extension of a quadratic imaginary field K. Building on the earlier work of Bertolini on this conjecture, and making use of the recent work of Mazur and Rubin on Kolyvagin's theory of Euler systems, we prove one divisibility of Perrin-Riou's conjectured equality. As a consequence, one obtains an upper bound on the rank of the Mordell-Weil group E(K) in terms of Heegner points.