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Description

This repository contains a mix of Magma, Pari and SageMath code for calculating the endomorphism algebras and rings of Jacobian varieties of curves over number fields and finite fields.

PLEASE NOTE: The current version of the repository uses Magma version 2.25, which includes the period algorithms by Molin--Neurohr mentioned below. If you have an earlier version of Magma, please try checking out commit 7433e5 and following the instructions on including the period functionality for old versions that is described below.

Prerequisites

An installation of Magma, Pari and (optionally) SageMath, so that all of these are available on the command line, is required to run all of the code. Most of the algorithms are written in Magma, whose algebro-geometric and numerical capabilities are essential. Some of the heavy lifting in the creation of number fields and the recognition of complex numbers as algebraic numbers is outsourced to Pari, whose performance when working with number fields is better than that of Magma. Finally, SageMath is mainly used for the calculation of Frobenius endomorphisms and as a wrapper, because its Python substrate allows for easier creation and manipulation of data structures.

For optimal results, set your Pari stack size to a decent size by for example adding

parisize = "4096M"

to your ~/.gprc file. This is an optional improvement.

In order to decompose Jacobians, you will need to similarly install and attach JRSijsling/curve_reconstruction and its dependencies.

If your Magma version predates 2.25, then you should also install pascalmolin/hcperiods and include the path to its spec file in your .magmarc file, using AttachSpec in the same way as in the section on the Magma installation below (but with a different target spec file). This is not an optimal improvement; the package will not run without it. At any rate Molin--Neurohr's code makes numerical integration on curves far more stable and reliable, so you will want to install it.

The same holds for code by Christian Neurohr that enables the computation of period matrices of plane curves. It is available via the dependency JRSijsling/RiemannSurfaces, a fork containing small modifications of the magnificent original version at christianneurohr/RiemannSurfaces.

Additional prerequisite for older Magma versions

In older version of Magma the file magma/package/Algebra/AlgQuat/interface.m had the following as line 145:

c := [Trace(theta), Norm(theta)];

This should be replaced by

cpol := MinimalPolynomial(theta);  
assert Degree(cpol) eq 2;  
c := [Coefficient(cpol,1), Coefficient(cpol, 0)];

Magma installation

The subdirectory endomorphisms/magma/ includes code that can be run purely within Magma. You can enable the functionality of this code in Magma by attaching the endomorphisms/endomorphisms/magma/spec file with AttachSpec. To make this independent of the directory in which you find yourself, and to active this on startup by default, you may want to indicate the relative path in your ~/.magmarc file, by adding the line

AttachSpec("~/Programs/endomorphisms/endomorphisms/magma/spec");

SageMath installation

To install the package in SageMath, first clone the repository via

git clone https://github.com/edgarcosta/endomorphisms.git

then go to the newly created directory and type

sage -pip install --user --upgrade .

After that, a new package called endomorphisms will be available for import in SageMath. Once the package is updated on GitHub, pulling the new changes and running the same command will update your installation.

While delegating commands to Magma, the same prerequisites are needed as those mentioned above. For this reason, please create the file ~/.sage/init.sage if it does not yet exist or add the following line to it:

magma.load('~/.magmarc')

This ensures that all relevant packages are loaded when outsourcing to Magma.

Usage

Examples, both in Magma and in SageMath, are given in the directory examples/. The creation of database files, as well as interaction with the LMFDB, is described in the directory database/.

Verbose comments are enabled by

SetVerbose("EndoFind", m);
SetVerbose("EndoCheck", n);

where m and n are either 1, 2, or 3. A higher value gives more comments.

More detailed information

A further description of the data structures and return values used is given in the directory documentation/.

Credits

The fast calculation of period matrices of hyperelliptic curves in pascalmolin/hperiods is based on:

Pascal Molin and Christian Neurohr
Computing period matrices and the Abel-Jacobi map of superelliptic curves
Mathematics of Computation, 88 (316) (2017)

The calculation of period matrices of plane quartic curves uses the following work:

Christian Neurohr
Efficient integration on Riemann surfaces & applications
Ph.D. thesis, Carl-von-Ossietzky-Universität Oldenburg (2018)

Citing this code

Please cite the following preprint, as well as the two mentioned above, if this code has been helpful in your research.

Edgar Costa, Nicolas Mascot, Jeroen Sijsling, and John Voight
Rigorous computation of the endomorphism ring of a Jacobian Mathematics of Computation, 88 (2019), 1303-1339

DOI: https://doi.org/10.1090/mcom/3373

Preprint available at arXiv:1705.09248

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