Torsion points and Galois representations on CM elliptic curves

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 $K(j(E))$.

31 pages; updated version has new examples in addition to expository changes

Cited by in corpus (10)

• $\ell$-adic images of Galois for elliptic curves over $\mathbb{Q}$
• Galois representations attached to elliptic curves with complex multiplication
• Entanglement in the family of division fields of elliptic curves with complex multiplication
• Torsion points and isogenies on CM elliptic curves
• A Local-global principle for isogenies of composite degree
• An algorithm for determining torsion growth of elliptic curves
• Torsion points on isogenous abelian varieties
• How big is the image of the Galois representations attached to CM elliptic curves?
• Ray class groups and ray class fields for orders of number fields
• CM points on Shimura curves via QM-equivariant isogeny volcanoes