# A note on the Kenku's method

## 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.

## Key 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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Source: https://tomesphere.com/paper/1812.11032