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

2019/09/17 by Murilo Zanarella, Zanarella, Murilo
Computer Science · Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1909.07835

openalex publication_date 2019/09/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We adapt Wei Zhang's proof of Kolyvagin's conjecture for modular abelian varieties over 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.

Citations

• On the Iwasawa main conjectures for modular forms at non-ordinary primes

Related

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• On the Iwasawa theory of CM fields for supersingular primes
• Complete verification of strong BSD for many modular abelian surfaces over Q
• The Iwasawa main conjecture for semistable abelian varieties over function fields
• Kolyvagin's conjecture for modular forms