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Torsion of elliptic curves over cyclic cubic fields

Maarten Derickx, Filip Najman

Abstract

We determine all the possible torsion groups of elliptic curves over cyclic cubic fields, over non-cyclic totally real cubic fields and over complex cubic fields.

Torsion of elliptic curves over cyclic cubic fields

Abstract

We determine all the possible torsion groups of elliptic curves over cyclic cubic fields, over non-cyclic totally real cubic fields and over complex cubic fields.

Paper Structure

This paper contains 8 sections, 17 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Strategy for determining the Galois groups of degree $d$ points on a curve
  3. Previously known results
  4. Torsion groups over cyclic cubic fields
  5. Explicit rational point computations
  6. Mordell-Weil sieving with partial Mordell-Weil information
  7. Rational points on $C_2(16), C_3(16)$ and $C_2(20)$
  8. Torsion groups over complex and over totally real and non-Galois cubic fields

Key Result

Theorem 3.1

bbdn

Theorems & Definitions (36)

  • Remark 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Theorem 4.1
  • Definition 4.2
  • Remark 4.3
  • Remark 4.4
  • ...and 26 more