2023/12/12 by Timo Keller, Michael Stoll, Keller, Timo +1
Computer Science · Mathematics · #11G40 (Primary) 11G10 #14G05 (Secondary) #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2312.07307
openalex publication_date 2023/12/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We develop the theory and algorithms necessary to be able to verify the strong Birch--Swinnerton-Dyer Conjecture for absolutely simple modular abelian varieties over Q. We apply our methods to all 28 Atkin--Lehner quotients of X0(N) of genus 2, all 97 genus 2 curves from the LMFDB whose Jacobian is of this type and six further curves originally found by Wang. We are able to verify the strong BSD Conjecture unconditionally and exactly in all these cases; this is the first time that strong BSD has been confirmed for absolutely simple abelian varieties of dimension at least 2. We also give an example where we verify that the order of the Tate--Shafarevich group is 72 and agrees with the order predicted by the BSD Conjecture.