√(-3)-Selmer groups, ideal class groups and large 3-Selmer ranks

2025/02/03 by Somnath Jha, Jha, Somnath, Dipramit Majumdar +3
Mathematics · #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2502.01069

openalex publication_date 2025/02/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the family of elliptic curves E_a,b:y²=x³+a(x-b)² with a,b ∈ ℤ. These elliptic 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,b over K:=ℚ(ζ₃) in terms of the 3-part of the ideal class group of certain quadratic extension of K. Using our bounds on the Selmer groups, we construct infinitely many curves in this family with arbitrary large 3-Selmer rank over K and no non-trivial K-rational point of order 3. We also show that for a positive proportion of natural numbers n, the curve E_n,n/ℚ has root number -1 and 3-Selmer rank =1.

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