CM Elliptic Curves: Volcanoes, Reality and Applications, Part II

2022/12/27 by Clark, Pete L., Saia, Frederick
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2212.13327

Abstract

Let M | N be positive integers, and let Δ be the discriminant of an order in an imaginary quadratic field K. When Δ_K < -4, the first author determined the fiber of the morphism X₀(M,N) → X(1) over the closed point J_Δ corresponding to Δ and showed that all fibers of the map X₁(M,N) → X₀(M,N) over J_Δ were connected. Here we complement this prior work by addressing the most difficult cases Δ_K ∈ \-3,-4\. These works provide all the information needed to compute, for each positive integer d, all subgroups of E(F)[tors], where F is a number field of degree d and E_/F is an elliptic curve with complex multiplication.

Related

• CM Elliptic Curves: Volcanoes, Reality and Applications
• Elliptic curves with complex multiplication and abelian division fields
• Computing the cardinality of CM elliptic curves using torsion points
• Growth of Torsion Groups of Elliptic Curves Over Number Fields without Rationally Defined CM
• Odd and Even Elliptic Curves with Complex Multiplication