QuadraticPoints

Magma code to compute quadratic points on X_0(N).

The code is associated to the following paper: http://arxiv.org/abs/2303.12566 "Computing quadratic points on modular curves X_0(N)" by Nikola Adzaga, Timo Keller, Philippe Michaud-Jacobs, Filip Najman, Ekin Ozman, and Borna Vukorepa.

The repository consists of the following files and folders:

• Additional_examples (folder). This folder contains additional example computations that do not form part of the paper.
• Classical_Chabauty (folder). This folder contains code for perfoming classical Chabauty computations.
• Output_files (folder). Contains all output files for tha 'rank 0' and 'Atkin-Lehner sieve' levels as well as the output of the verifications file.
• AL_sieve.m Main file for the 'Atkin-Lehner sieve' method. Performs the computations.
• AL_sieve_auxiliary.m. Contains functions for the 'Atkin-Lehner sieve' method.
• going_down.m. Main file for the 'going-down' method. Performs the computations.
• models_and_maps.m. This file contains code to compute models for X_0(N), quotient curves, quotient maps, j-maps and more.
• pullbacks.m. This file contains a function to search for quadratic points on a model of X_0(N) which arise as pullbacks of rational points on AL quotients. Also provides code to compute j-invariants and CM discriminants.
• rank_0.m. Main file for the 'rank 0' method. Performs the computations.
• rank_0_auxiliary.m. Contains functions for the 'rank 0' method. This code is taken from the paper: Ozman-Siksek, Quadratic Points on Modular curves.
• rank_calcs.m. This file contains functions for computing the ranks over Q of J_0(N) and its quotients.
• symm_chab.m. This file contains functions for verifying the symmetric Chabauty criterion we use in the Atkin-Lehner sieve.
• verifications.m. This file checks (and prints most of) the data presented in the tables of the paper, it also provides coordinates for each quadratic point we consider.