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arXiv · 1612.03229

Torsion points and Galois representations on CM elliptic curves

Abbey Bourdon·Pete L. Clark
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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 $K(j(E))$.

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Abbey Bourdon, Pete L. Clark. 2016-12-10. Torsion points and Galois representations on CM elliptic curves. https://doi.org/10.2140/pjm.2020.305.43

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