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Cubic and quartic points on modular curves using generalised symmetric Chabauty

Josha Box, Stevan Gajović, Pip Goodman

Abstract

Answering a question of Zureick-Brown, we determine the cubic points on the modular curves $X_0(N)$ for $N \in \{53,57,61,65,67,73\}$ as well as the quartic points on $X_0(65)$. To do so, we develop a "partially relative" symmetric Chabauty method. Our results generalise current symmetric Chabauty theorems, and improve upon them by lowering the involved prime bound. For our curves a number of novelties occur. We prove a "higher order" Chabauty theorem to deal with these cases. Finally, to study the isolated quartic points on $X_0(65)$, we rigorously compute the full rational Mordell--Weil group of its Jacobian.

Cubic and quartic points on modular curves using generalised symmetric Chabauty

Abstract

Answering a question of Zureick-Brown, we determine the cubic points on the modular curves for as well as the quartic points on . To do so, we develop a "partially relative" symmetric Chabauty method. Our results generalise current symmetric Chabauty theorems, and improve upon them by lowering the involved prime bound. For our curves a number of novelties occur. We prove a "higher order" Chabauty theorem to deal with these cases. Finally, to study the isolated quartic points on , we rigorously compute the full rational Mordell--Weil group of its Jacobian.

Paper Structure

This paper contains 22 sections, 20 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. A general symmetric Chabauty theorem
  3. Uniformisers and differentials
  4. Coleman integration
  5. An overview of the symmetric Chabauty--Coleman method
  6. An overview of our relative Chabauty--Coleman method
  7. The main theorem
  8. Trace maps and ramification
  9. Lemmas to control the prime bound
  10. Proof of Theorem \ref{['bigtheorem']}
  11. Chabauty using the higher order terms
  12. The Mordell--Weil sieve
  13. A formal description
  14. Implementing the sieve efficiently
  15. Applying Chabauty to modular curves
  16. ...and 7 more sections

Key Result

Theorem 1.1

The set of cubic points on each of the curves is finite and listed in Section section_results. The quartic points on $X_0(65)$ form an infinite set. This infinite set consists of inverse images of quadratic points on the quotient curve $X^+_0(65)$ and a finite number of isolated points. The isolated points are listed in § subsection_quarticpoin

Theorems & Definitions (50)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • ...and 40 more