Elliptic curves with exceptionally large analytic order of the\n Tate-Shafarevich groups

2021/03/19 by Andrzej Dąbrowski, Dąbrowski, Andrzej, Lucjan Szymaszkiewicz +1
Mathematics · #11G40 #11Y50 #2010 Mathematics Subject Classification: 11G05 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Geometric Analysis and Curvature Flows #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2103.11001

openalex publication_date 2021/03/19 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

We exhibit 88 examples of rank zero elliptic curves over the rationals with\n| sza(E)| > 63408², which was the largest previously known value for any\nexplicit curve. Our record is an elliptic curve E with | sza(E)| = 1029212²\n= 2⁴\⋅ 79² \⋅ 3257². We can use deep results by Kolyvagin, Kato,\nSkinner-Urban and Skinner to prove that, in some cases, these orders are the\ntrue orders of sza. For instance, 410536² is the true order of sza(E)\nfor E= E₄(21,-233) from the table in section 2.3.\n

Citations

• A majority of elliptic curves over \mathbb Q satisfy the Birch and Swinnerton-Dyer conjecture

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