Torsion points and Galois representations on CM elliptic curves
Abbey Bourdon, Pete L. Clark Clark
Abstract
We prove several results on torsion points and Galois representations for complex multiplication (CM) elliptic curves over a number field containing the CM field. One result computes the degree in which such an elliptic curve has a rational point of order $N$, refining results of Silverberg. Another result bounds the size of the torsion subgroup of an elliptic curve with CM by a nonmaximal order in terms of the torsion subgroup of an elliptic curve with CM by the maximal order. Our techniques also yield a complete classification of both the possible torsion subgroups and the rational cyclic isogenies of a $K$-CM elliptic curve $E$ defined over $K(j(E))$.
Topics & Concepts
MathematicsSchoof's algorithmElliptic curveTorsion subgroupTorsion (gastropod)Hessian form of an elliptic curveComplex multiplicationTwists of curvesPure mathematicsSupersingular elliptic curveDivision polynomialsModular elliptic curveElliptic curve point multiplicationTripling-oriented Doche–Icart–Kohel curveGalois moduleEdwards curveJacobian curveSato–Tate conjectureOrder (exchange)Discrete mathematicsGalois extensionMathematical analysisAlgebraic Geometry and Number TheoryCryptography and Residue ArithmeticAdvanced Algebra and Geometry