Selmer companion curves

2012/03/03 by Barry Mazur, Karl Rubin, Mazur, Barry +1 · 1 citation
Computer Science · Mathematics · #11G05 (Primary) 11G40 #11G07 (Secondary) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1203.0620

openalex publication_date 2012/03/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We say that two elliptic curves E₁, E₂ over a number field K are n-Selmer companions for a positive integer n if for every quadratic character χof K, there is an isomorphism between the n-Selmer groups Sel_n(E₁^χ/K) and Sel_n(E₂^χ/K) of the quadratic twists E₁^χ, E₂^χ. We give sufficient conditions for two elliptic curves to be n-Selmer companions, and give a number of examples of non-isogenous pairs of companions.

Citations

• Good Reduction of Abelian Varieties

Cited by

• 2-Selmer companion modular forms

Related

• Constructing families of 3-Selmer companions
• p-twisted Selmer near-companion curves
• 2-Selmer near-companion curves
• 2-Selmer near-companion curves
• p^r-Selmer companion modular forms