p-twisted Selmer near-companion curves

2025/04/15 by Minseok Kim, Kim, Minseok
Mathematics · #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2504.10896

Abstract

Let E₁ and E₂ be elliptic curves over a number field K. In \citescc, Mazur and Rubin define the concept of n-Selmer near-companions and conjecture that if E₁ and E₂ are n-Selmer near-companions over K, then E₁[n] is G_K-isomorphic to E₂[n]. Yu proves the conjecture on n-Selmer near-companion curves in the case n=2. We we introduce the notion of p-twisted Selmer near-companions (p-TSNC) over K and prove that if E₁ and E₂ are p-TSNC over K, then K(E₁[p])=K(E₂[p]) under certain conditions.

Citations

• Strong Selmer companion elliptic curves
• 2-Selmer near-companion curves
• A Markov model for Selmer ranks in families of twists
• Selmer companion curves
• Disparity in Selmer ranks of quadratic twists of elliptic curves
• Twisting commutative algebraic groups
• Finding large Selmer rank via an arithmetic theory of local constants
• Propri�t�s galoisiennes des points d'ordre fini des courbes elliptiques

Related

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