On several families of elliptic curves with arbitrary large Selmer groups

2009/11/02 by Fei Li, Derong Qiu
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Combinatorics #Elliptic curve #Mathematics #Prime (order theory) #Pure mathematics #math.AG #math.NT #msc:11G05 #msc:14H52

paper · pdf · doi:10.1007/s11425-010-4035-2

published as Science China, Mathematics, vol. 53, 9 (2010), 2329-2340 · 22 pages

arxiv created 2009/11/02 · openalex publication_date 2010/08/09 · arxiv updated 2012/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

In this paper, we calculate the ϕ(ϕ)-Selmer groups S^(ϕ) (E / \Q) and S^(φ) (E^′ / \Q) of elliptic curves y² = x (x + εp D) (x + εq D) via descent theory (see [S, Chapter X]), in particular, we obtain that the Selmer groups of several families of such elliptic curves can be arbitrary large.

Citations

• On the equation y^m = P(x)
• Mordell-Weil groups and Selmer groups of two types of elliptic curves
• Algebra
• The Arithmetic of Elliptic Curves