Vol. 305, No. 1, 2020

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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(j(E)).

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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