TL;DR
This paper revises and clarifies Kenku's method for proving the nonexistence of 27-torsion points on elliptic curves over quadratic fields, and explains its limitations for other torsion cases over cubic fields.
Contribution
It provides a complete, corrected version of Kenku's method and analyzes its applicability to different torsion cases over various number fields.
Findings
Kenku's method is effective for 27-torsion over quadratic fields.
The method cannot be extended to 32- and 24-torsion over cubic fields.
Clarifies the limitations of Kenku's approach.
Abstract
In this note, we present a complete and corrected version of Kenku's method in the proof of the nonexistence of 27-torsion on elliptic curves over quadratic fields and show why this method can not be applied to the cases of 32- and 24-torsion on elliptic curves over cubic fields.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Cryptography and Residue Arithmetic · Polynomial and algebraic computation