Rigorous computation of the endomorphism ring of a Jacobian
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README.md

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.

Prerequisites

An installation of Magma, Pari and 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 used for the calculation of Frobenius 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.

You should also install pascalmolin/hperiods and include the path to its spec file in your .magmarc 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.

Finally, in order to decompose Jacobians, you will need curve_reconstruction.

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 load all the Magma specific files by attaching the endomorphisms/magma/spec file with AttachSpec. For example, if you start your session of Magma inside the git directory, you can do this by typing

AttachSpec("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.

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/.

More detailed information

A description of the data structures used in the SageMath wrapper can be found in the files Dicts.md and Descs.md in the directory documentation/.