A proof of Kolyvagin's Conjecture via the BDP main conjecture

TL;DR

This paper proves Kolyvagin's conjecture for modular abelian varieties over  using the BDP main conjecture, offering a new approach that extends to totally real fields.

Contribution

It adapts Zhang's proof by replacing the cyclotomic main conjecture with the BDP main conjecture, enabling broader applicability.

Findings

Proof of Kolyvagin's conjecture for modular abelian varieties over 

Reduction to a BDP-conjecture-tractable case

Method suitable for extension to totally real fields

Abstract

We adapt Wei Zhang's proof of Kolyvagin's conjecture for modular abelian varieties over $\mathbb{Q}$ to rely on the BDP main conjecture instead of on the cyclotomic main conjecture. The main ingredient is a reduction to a case that is tractable by the BDP main conjecture, in a similar spirit to Zhang's reduction to the rank one case. By using the BDP main conjecture instead of the cyclotomic main conjecture, our approach is more suitable than Zhang's to extend to modular abelian varieties over totally real fields.