On the Selmer group and rank of a family of elliptic curves and curves of genus one violating the Hasse principle

2021/11/05 by Eleni Agathocleous, Agathocleous, Eleni
Mathematics · #11G05 #11R29 #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2111.03723

openalex publication_date 2021/11/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study an infinite family of j-invariant zero elliptic curves E_D:y²=x³+16D and their λ-isogenous curves E_D':y²=x³-27⋅16D, where D and D' = -3D are fundamental discriminants of a specific form, and λ is an isogeny of degree 3. A result of Honda guarantees that for our discriminants D, the quadratic number field K_D = ℚ(√(D)) always has non-trivial 3-class group. We prove a series of results related to the set of rational points E_D'(ℚ) ∖ λ(E_D(ℚ)), and the SL(2,ℤ)-equivalence classes of irreducible integral binary cubic forms of discriminant D. By assuming finiteness of the Tate-Shafarevich group, we derive a parity result between the rank of E_D and the rank of its 3-Selmer group, and we establish lower and upper bounds for the rank of our elliptic curves. Finally, we give explicit classes of genus-1 curves that correspond to irreducible integral binary cubic forms of discriminant D=48035713, and we show that every curve in these classes violates the Hasse Principle.

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