Faltings elliptic curves in twisted \mathbb Q-isogeny classes

2025/09/27 by Enrique González–Jiménez, González-Jiménez, Enrique, Joan-C. Lario +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2509.23283

openalex publication_date 2025/09/27 · openalex created_date 2025/10/19 · openalex updated_date 2026/07/28

Abstract

Let G be the graph attached to the \mathbb Q-isogeny class of an elliptic curve defined over \mathbb Q: that is, a vertex for every elliptic curve defined over \mathbb Q in the isogeny class, and edges in correspondence with the prime degree rational isogenies between them. Stevens shows that there is a unique elliptic curve in G with minimal Faltings height. We call this curve the Faltings elliptic curve in G. For every square-free integer d, we consider the graph~G^d attached to the twisted elliptic curves in G by the quadratic character of \mathbb Q(√(d)). It turns out that G and G^d are canonically isomorphic as abstract graphs (the isomorphism identifies the vertices with equal j-invariant). In this paper we determine which vertex is the Faltings elliptic curve in G^d. We also obtain the probability of a vertex in G to be the Faltings elliptic curve in~G^d. It turns out that this probability depends on the p-adic valuations of rational values of certain modular functions.

Citations

• The Natural Density of Some Sets of Square-free Numbers
• Periods of Quadratic Twists of Elliptic Curves

Related

• Exact counts of elliptic curves of bounded height over \mathbb F_q(t) in characteristics 2 and 3
• Mazur's isogeny theorem
• Infinite Families Of Isogeny-Torsion Graphs
• Computing Isogenies at Singular Points of the Modular Polynomial
• Counting elliptic curves of bounded Faltings height