2024/06/11 by Yuri G. Zarhin, Zarhin, Yuri G.
Computer Science · Mathematics · #11G07 #14H52 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2406.07240
openalex publication_date 2024/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We call an order O in a quadratic field K odd (resp. even) if its discriminant is an odd (resp. even) integer. We call an elliptic curve E over the field C of complex numbers with CM odd (resp. even) if its endomorphism ring End(E) is an odd (resp. even) order in the corresponding imaginary quadratic field. Suppose that j(E) is a real number and let us consider the set J(R,E) of all j(E') where E' is any elliptic curve that enjoys the following properties. 1) E' is isogenous to E; 2) j(E') is a real number; 3) E' has the same parity as E. We prove that the closure of J(R,E) in the set R of real numbers is the closed semi-infinite interval (-∞,1728] (resp. the whole R) if E is odd (resp. even). This paper was inspired by a question of Jean-Louis Colliot-Thélène and Alena Pirutka about the distribution of j-invariants of certain elliptic curves of CM type.