On 5-torsion of CM elliptic curves
Laura Paladino

TL;DR
This paper classifies the fields generated by 5-torsion points of CM elliptic curves over number fields, detailing their structure, degrees, and Galois groups, with applications to divisibility problems and modular/ Shimura curves.
Contribution
It provides a complete classification of the fields $K_5$ for CM elliptic curves with specific Weierstrass forms, including generators, degrees, and Galois groups, and explores their applications.
Findings
Classified $K_5$ fields for CM elliptic curves with given Weierstrass forms.
Determined degrees and Galois groups of these fields.
Applied results to divisibility problems and modular/Shimura curves.
Abstract
Let be an elliptic curve defined over a number field . Let be a positive integer. We denote by the -torsion subgroup of and by the number field obtained by adding to the coordinates of the points of . We describe the fields , when is a CM elliptic curve defined over , with Weiestrass form either or . In particular we classify the fields in terms of generators, degrees and Galois groups. Furthermore we show some applications of those results to the Local-Global Divisibility Problem, to modular curves and to Shimura curves.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Algebra and Geometry · Vietnamese History and Culture Studies
