3-Selmer group, ideal class groups and cube sum problem

2022/07/25 by Somnath Jha, Jha, Somnath, Dipramit Majumdar +3
Mathematics · #11R29 #11R34 #11S25 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT) #Primary 11G05 #Secondary 11G40

paper · pdf · doi:10.48550/arxiv.2207.12487

openalex publication_date 2022/07/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Consider a Mordell curve E_a:y²=x³+a with a ∈ \mathbb Z. These curves have a rational 3-isogeny, say φ. We give an upper and a lower bound on the rank of the φ-Selmer group of E_a over \mathbb Q(ζ₃) in terms of the 3-part of the ideal class group of certain quadratic extension of \mathbb Q(ζ₃). Using our bounds on the Selmer groups, we prove some cases of the rational cube sum problem. Further, using these bounds, we give explicit families of the Mordell curves to show that for a positive proportion of E_a, \rm Sel³(E_a/\mathbb Q)=0 (respectively \rm Sel³(E_a/\mathbb Q) has \mathbb F₃-rank 1).

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