2024/11/20 by Adrian Barquero-Sanchez, Barquero-Sanchez, Adrian, Jimmy Calvo-Monge +1
Mathematics · Social Sciences · #11G05 (Primary) #11G15 #11N45 (Secondary) #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Vietnamese History and Culture Studies
paper · pdf · doi:10.48550/arxiv.2411.13526
openalex publication_date 2024/11/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we study the density and distribution of CM elliptic curves over ℚ. In particular, we prove that the natural density of CM elliptic curves over ℚ, when ordered by naive height, is zero. Furthermore, we analyze the distribution of these curves among the thirteen possible CM orders of class number one. Our results show that asymptotically, 100% of them have complex multiplication by the order ℤ[(-1 + √(-3))/(2) ], that is, have j-invariant 0. We conduct this analysis within two different families of representatives for the ℚ-isomorphism classes of CM elliptic curves: one commonly used in the literature and another constructed using the theory of twists. As part of our proofs, we give asymptotic formulas for the number of elliptic curves with a given j-invariant and bounded naive height.