Non-vanishing of Kolyvagin systems and Iwasawa theory

TL;DR

This paper proves Kolyvagin's conjecture for elliptic curves over Q with certain primes p, showing the non-triviality of Heegner point Kolyvagin systems and relating divisibility indices to Tamagawa numbers.

Contribution

It establishes Kolyvagin's conjecture for a broad class of primes p and elliptic curves, and refines the conjecture by linking divisibility indices to Tamagawa numbers.

Findings

Proves non-vanishing of Heegner point Kolyvagin systems for good ordinary primes p.

Relates divisibility index of Kolyvagin systems to Tamagawa numbers of E.

Provides analogous results for Kato's Euler system.

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve and $p$ an odd prime. In 1991 Kolyvagin conjectured that the system of cohomology classes for torsion quotients of the $p$-adic Tate module of $E$ derived from Heegner points over ring class fields of a suitable imaginary quadratic field $K$ (i.e., the Heegner point Kolyvagin system of $E/K$) is non-trivial. In this paper we prove Kolyvagin's conjecture when $p$ is a prime of good ordinary reduction for $E$ that splits in $K$. In particular, our results cover many cases where $p$ is an Eisenstein prime for $E$, complementing Wei Zhang's earlier results on the conjecture by a different approach. Our methods also yield a proof of a refinement of Kolyvagin's conjecture expressing the divisibility index of the Heegner point Kolyvagin system in terms of the Tamagawa numbers of $E$, as conjectured by Wei Zhang in 2014, as well as proofs of analogous results for the Kolyvagin system obtained from Kato's Euler system.