quadratic-torsion

Code to compute the torsion subgroups that occur over a fixed quadratic field. This accompanies the paper Torsion subgroups of elliptic curves over quadratic fields and a conjecture of Granville.

The paper itself explains what parts of this repo are used where. Here we give an overview of the repo according to each folder.

python_scripts

This directory contains a single file positive_rank_twists.py. The code in this file uses SageMath's modular symbol methods to compute which twists of the relevant modular curves have positive analytic rank. The output of these computations are automatically written to the positive_rank_lists folder. To reproduce these computations simply do:

cd python_scripts
sage positive_rank_twists.py

this computation should take between 1 to 2 days. After the above computations finished the commands

cd positive_rank_lists
git diff .

can be used to check if the output of the run is the same as the output we obtained initially. git diff . will show any changes in the output. So if that command doesn't print anything then the computations were successfully reproduced.

magma_scripts

This contains the meat of our code. Most of the computations here rely on the results of the output of positive_rank_twists.py in the python_scripts directory, since as described in the paper we only need to consider the twist of positive algebraic rank. Since algebraic rank > 0 implies analytic rank > 0 we restrict all computations to the twists in the positive_rank_lists folder.

To reproduce the computations in the file example.m in this directory either do

cd magma_scripts
magma example.m

or copy paste the contents of example.m into an interactive Magma session.

The genus 1 modular curves

For the 5 genus 1 modular curves, we only need to verify that the curves that have positive analytic rank also have positive algebraic rank as predicted by the BSD conjecture.

We initially attempted this in Magma, using the script positive_rank_twists.m. This script was not able to verify all of the ranks; see that file for the issues encountered. We then subsequently re-did the verification in Sage, using the script verify_positive_ranks_genus_1.py.

$X_1(13)$ and $X_1(18)$

For these two modular curves we first do a two descent and a point search. These code for these computations can be found in x1_13.m and x1_18.m. These computations should finish in several minutes, and deal with all but a few cases. For the remaining case we use the Mordell-Weil sieve the code for this is in MWSieve-x1_13.m and MWSieve-x1_18.m. These computations should finish in about a week. Most of the time here is actually spend in searching for generators of the Mordell-Weil group. And the cases we cannot deal with are because we were unable to find these generators despite the fact that BSD predicts they should exist.

The verification of the torsion computations of Lemma's 3.3, 3.5 and 4.1 can be found in torsionVerifications.m. These computations should be finished within a minute.

$X_1(16)$

For $X_1(16)$ we do a point search which can be found in x1_16_point_search.m. This computation should be done in under a minute. We also combine two cover descent together with elliptic curve Chabauty in order to determine all points in the relevant twists in x1_16_chabauty.m. This computation should finish in 1 to 2 hours.

Additionally, Granville's work on points on twists of genus 2 curves predicts that one should be able to produce abc-triples form the j-invariants of small points on quadratic twists of $X_1(16)$. The code to compute these triples can be found in x1_16_abc_triples.m and the list of triples themselves can be found in x1_16_abc_triples_list.txt. These computations should be finished within a minute.

granville

Contains code to compute the constants of Granville in Section 5. It also contains the graphs - as well as the code to produce them - that compare our computations to Granville's conjecture.

positive_rank_lists

For each of the genus 1 and 2 curves this lists which twists have positive analytic rank. For the five genus 1 curves we have also verified that these actually have positive algebraic rank as expected. From this one can directly deduce over which quadratic fields these five genus 1 modular curves have extra points, as described in section 2 of the paper.

logs

These contains log files of several magma sessions. These can be used to check if the results from the magma_scripts agree with the results we obtained when we originally ran these computations.

Project layout

.
├── granville
├── logs
├── magma_scripts
├── positive_rank_lists
├── python_scripts

Copyright

####  Copyright (C) 2024 Barinder S. Banwait and Maarten Derickx

Quadratic Torsion is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or any later version.

This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU General Public License for more details.

You should have received a copy of the GNU General Public License
along with this program.  If not, see <https://www.gnu.org/licenses/>.

The authors can be reached at: barinder.s.banwait@gmail.com and
maarten@mderickx.nl.