CIRM project on isolated points

Code associated to the paper "Towards a classification of isolated j-invariants" by Abbey Bourdon, Sachi Hashimoto, Timo Keller, Zev Klagsbrun, David Lowry-Duda, Travis Morrison, Filip Najman, Himanshu Shukla

System requirements: Magma version at least 2.27 is required to read the data files.

Input:

On a specific curve: j-invariant On a data set of curves: txt file with list of j-invariants delimited by newline character, e.g. adelicgenusgt0curves.txt

Output:

On a specific curve: [j-invariant, empty list] or [j-invariant, set of pairs (level and degree)]

The list [< n_1,d_1>, ..., < n_k,d_k>] contains all points such that any isolated point $x \in X_1(N)$ with $j(x)= j$ maps down under the natural projection map to an isolated point of degree $d_i$ on $X_1(n_i)$ for $1 \leq i \leq k$.

INSTRUCTIONS

1. This repository depends David Roe's modified version of David Zywina's repository found here https://github.com/roed314/OpenImage.git as well as https://github.com/AndrewVSutherland/ell-adic-galois-images by by Jeremy Rouse, Andrew V. Sutherland, and David Zureick-Brown. To clone these repos, run

   git clone https://github.com/roed314/OpenImage.git
   git clone https://github.com/AndrewVSutherland/ell-adic-galois-images.git
   

   To run for a specific curve given by a j-invariant j, do the following.

2. Open magma.

3. Attach the required files for the above repositories, by running

   AttachSpec("path/to/OpenImage/OpenImage.spec");
   Attach("path/to/ell-adic-galois-images/groups/gl2.m");
   

4. Attach the spec file for this repository by running

   AttachSpec("/path/to/isolated.spec");
   

5. Define

   path := OpenImageContext("path/to/data-files")
   

   where the path leads to the data files in the OpenImage repository.

6. Run the command

   NotIsolated(j, path);
   

Advanced instructions

To run for the entire list of curves in the file adelicgenusgt0curves.txt you also need to install GNU parallel. Then execute the following command

parallel --joblog JOB_LOG_FILE_NAME --shuf --timeout TIMEOUT \
  --eta -jNUMBER_OF_CORES \
  magma -b seq:={} doline.m ::: {LINE_NUMBERS_OF_INTEREST} >> OUTPUT_FILE_NAME.out

Details of the last command:

• TIMEOUT: Set it to the amount of time in seconds for which each instance of NotIsolated.m has to run e.g. 200

• NUMBER_OF_CORES: Set it to the number of cores which have to engaged during parallel execution. Check the number of cores available or the machine beforehand.

• JOB_LOG_FILE_NAME: Set it to the desired file name for job logs. An exitval value of -1 for a particular instance in this file will indicate if a particular instance was terminated because it required more time than TIMEOUT.

• LINE_NUMBERS_OF_INTEREST: Set it to the line numbers corresponding to the curves in the file Curves.txt for which NotSporadic.m has to be executed. E.g. 11, 13, 19,... or one can use a continuous range as in magma like 1..100.

• OUTPUT_FILE_NAME: Set it to the desired file name for the output file. This file contains output of every successful execution i.e. all the executions which were not timed out.

Note: Users with a Bash terminal can run install_dependencies.sh using the command ./install_dependencies.sh. This will install GNU parallel 2023 version and also the OpenImage github repo.

License Information

Copyright (c) 2024, held by the isolated points team: Bourdon, Hashimoto, Keller, Klagsbrun, Lowry-Duda, Morrison, Najman, and Shukla.

The magma files

• appendix.m
• Code_for_X1(17)_and_X1(24).m
• doline.m
• extrafilters.m
• isolatedpoints.m
• testlozanorobledo.m

are made available as documentation, code, and educational materials under the Creative Commons Attribution 4.0 (CC-BY-4.0) license. See https://creativecommons.org/licenses/by/4.0/ for more details, or LICENSE for a copy of the license text.

We provide these in the hope that they will be useful. Please contact us if there are licensing reasons preventing you from making use of this code. We note that this license is compatible with the GPL as long as you make proper attribution and follow the other guidelines of CC-BY-4.0.

The paper Towards a classification of isolated j-invariants (including early versions of the tex files) is not covered by this license.