On 7-division fields of CM elliptic curves
Jessica Alessandr\`i, Laura Paladino

TL;DR
This paper classifies the 7-division fields of CM elliptic curves over number fields, providing explicit generators, Galois groups, and applications to divisibility problems and modular curves.
Contribution
It offers a complete classification of the 7-division fields for CM elliptic curves, including explicit generators and Galois groups, advancing understanding of their arithmetic properties.
Findings
Explicit generators for K_7/K are given.
All Galois groups of K_7/K are determined.
Applications to Local-Global Divisibility and modular curves are demonstrated.
Abstract
Let be a CM elliptic curve defined over a number field , with Weiestrass form or . For every positive integer , we denote by the -torsion subgroup of and by the -th division field, i.e. the extension of generated by the coordinates of the points in . We classify all fields . In particular we give explicit generators for and produce all Galois groups . We also show some applications to the Local-Global Divisibility Problem and to modular curves.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Vietnamese History and Culture Studies · Historical Studies and Socio-cultural Analysis
