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Torsion Packet Envelope and Rational Points of Algebraic Curves

Ryo Ichikawa

Abstract

In this paper, we give an elementary new method for determining the rational points on algebraic curves using torsion packets. We also provide examples of curves for which all rational points can be completely determined by our method.

Torsion Packet Envelope and Rational Points of Algebraic Curves

Abstract

In this paper, we give an elementary new method for determining the rational points on algebraic curves using torsion packets. We also provide examples of curves for which all rational points can be completely determined by our method.
Paper Structure (11 sections, 3 theorems, 36 equations, 1 table)

This paper contains 11 sections, 3 theorems, 36 equations, 1 table.

Table of Contents

  1. Introduction
  2. Proof of Main Theorem
  3. Examples
  4. The hyperelliptic curve $y^2=x^5+d$
  5. The case $\overline{d}=\overline{7}$
  6. The case $\overline{d}=\overline{9}$
  7. The case $\overline{d}=\overline{1}$
  8. The remaining case
  9. The hyperelliptic curve $y^2=x^{p-1}+dx^{\frac{p-1}{2}}-1$
  10. The hyperelliptic curve $y^2=x^p-x$
  11. The hyperelliptic curve $y^2=(x^2-1)(x^2-4)(x+3)$

Key Result

Theorem 1.2

The notation being as before. Assume that there exists a torsion packet envelope $(F,T,w)$ for $C(\mathbb{Q})$. Then, $C(\mathbb{Q})\cap J(\mathbb{Q})_{\mathrm{tor}}\coloneqq \iota^{-1}(\iota(C(\mathbb{Q}))\cap J(\mathbb{Q})_{\mathrm{tor}})\subseteq T$.

Theorems & Definitions (10)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Corollary 1.5
  • proof
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Remark 3.4