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Torsion points and Galois representations on CM elliptic curves

### Abbey Bourdon and Pete L. Clark

Vol. 305 (2020), No. 1, 43–88
##### 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 (Compositio Math.68:3 (1988), 241–249; Contemp. Math. 133 (1992)). 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 $K\left(j\left(E\right)\right)$.

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##### Keywords
torsion points, elliptic curves, Galois representations, complex multiplication
##### Mathematical Subject Classification 2010
Primary: 11G05, 11G15
##### Milestones
Received: 1 March 2019
Revised: 15 September 2019
Accepted: 27 October 2019
Published: 17 March 2020
##### Authors
 Abbey Bourdon Department of Mathematics and Statistics Wake Forest University Winston-Salem, NC United States Pete L. Clark Department of Mathematics University of Georgia Athens, GA United States