CM SageMath code

## Project Description

=========================

REpository of Complex multiplication SageMath code

=========================

.. image:: https://travis-ci.org/mstreng/recip.svg?branch=master

:target: https://travis-ci.org/mstreng/recip

Once the Travis CI set-up has been completed, the documentation for the package can be found at https://mstreng.github.io/recip/doc/html/

Installation

------------

Local install from source

^^^^^^^^^^^^^^^^^^^^^^^^^

Download the source from the git repository::

$ git clone https://github.com/mstreng/recip.git

Change to the root directory and run::

$ sage -pip install --upgrade --no-index -v .

For convenience this package contains a [makefile](makefile) with this

and other often used commands. Should you wish too, you can use the

shorthand::

$ make install

Usage

-----

Once the package is installed, you can use it in Sage with::

sage: from recip import *

sage: CM_Field([5,5,5])

CM Number Field in alpha with defining polynomial x^4 + 5*x^2 + 5

Source code

-----------

All source code is stored in the folder ``recip`` using the same name as the

package. This is not mandatory but highly recommended for clarity. All source folder

must contain a ``__init__.py`` file with needed includes.

Tests

-----

This package is configured for tests written in the documentation

strings, also known as ``doctests``. For examples, see this

[source file](recip/ultimate_question.py). See also

[SageMath's coding conventions and best practices document](http://doc.sagemath.org/html/en/developer/coding_basics.html#writing-testable-examples).

With additional configuration, it would be possible to include unit

tests as well.

Once the package is installed, one can use the SageMath test system

configured in ``setup.py`` to run the tests::

$ sage setup.py test

This is just calling ``sage -t`` with appropriate flags.

Shorthand::

$ make test

Documentation

-------------

The documentation of the package can be generated using Sage's

``Sphinx`` installation::

$ cd docs

$ sage -sh -c "make html"

Shorthand::

$ make doc

#*****************************************************************************

# Copyright (C) 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017 Marco Streng

# <marco.streng@gmail.com>

#

# This program 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 2 of the License, or

# (at your option) 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, write to the Free Software Foundation, Inc.,

# 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA.

#*****************************************************************************

RECIP -- REpository of Complex multIPlication SageMath code.

This started out as code meant for computing with Shimura's RECIProcity law,

but grew into a collection of much of the SageMath code written by me for my

research.

Version 0.2.5 (tested with SageMath 8.0, short tests only).

When using this package in a publication, it is highly likely that it is

approprate certain publications. Please CITE the relevant JOURNAL publications,

as well as giving the URL of this repository.

Here is a list of functionalities of this repository, together with the

publications that should be cited when you use them, and the name of the file

that has examples.

* Igusa class polynomials (proven correct)

See both "Igusa class polynomials (not proven correct)" and

"Denominators of Igusa class polynomials" below.

* Non-maximal orders of CM-fields and their polarized ideal classes and Igusa

class polynomials.

cite [BissonStreng] (code is written for, part of, and based on, this publication)

see orders.sage for examples

* (n,n)-isogenies between polarized ideal classes

cite [BLS]

see bls.sage for examples

* Computations related to Shimura's reciprocity law

cite [Streng12] (code is written for, part of, and based on, this publication)

see article.sage for examples

* Igusa class polynomials (not proven correct)

cite [Streng14], [vWamelen], [Weng] (code is based on these publications)

* Denominators of Igusa class polynomials

cite [BouyerStreng] (code is written for, and hence part of, this publication)

and depending on how the code is used, and on the kind of quartic CM-field,

also cite one or more of:

[BouyerStreng], [GL], [LV], [Yang] (large parts of the code are based on these)

see denominators.sage for examples

Here is a list of SageMath programs written by my students and me that is not part

of this repository.

* Height reduction of binary forms and hyperelliptic curves.

(with Florian Bouyer)

https://bitbucket.org/mstreng/reduce

cite [BouyerS] (code is written for, part of, and based on, this publication)

* Solving conics and Mestre's algorithm

(with Florian Bouyer)

now part of the standard SageMath functionality

* Hilbert modular polynomials

(by Chloe Martindale)

contact her if you are interested

* CM class number one for genus 2 and 3

(by Pınar Kılıçer)

contact her if you are interested

To use the latest version of this package directly from the web, start SageMath

and type::

sage: load("https://bitbucket.org/mstreng/recip/raw/master/recip_online.sage")

To use this package offline, download it first and extract it to some

directory, say "somewhere_on_my_drive/recip", then start SageMath and type::

sage: load_attach_path("somewhere_on_my_drive/recip")

sage: load("recip.sage")

[ABLPV] - Comparing arithmetic intersection formulas for denominators of

Igusa class polynomials -- Jacqueline Anderson, Jennifer S.

Balakrishnan, Kristin Lauter, Jennifer Park, and Bianca Viray

[BissonS] - On polarised class groups of orders in quartic CM fields --

Gaetan Bisson and Marco Streng

http://arxiv.org/abs/1302.3756

[BLS] - Abelian surfaces admitting an (l,l)-endomorphism -- Reinier Broker,

Kristin Lauter, and Marco Streng

accepted for publication by Journal of Algebra, 2013

http://arxiv.org/abs/1106.1884

[BouyerS] - Examples of CM curves of genus 2 defined over the reflex field --

Florian Bouyer and Marco Streng

http://arxiv.org/abs/1307.0486

[GJLSVW] - Igusa class polynomials, embeddings of quartic CM fields, and

arithmetic intersection theory -- Helen Grundman, Jennifer

Johnson-Leung, Kristin Lauter, Adriana Salerno, Bianca Viray, and

Erika Wittenborn

http://arxiv.org/abs/1006.0208

[GL] - Genus 2 curves with complex multiplication -- Eyal Goren and

Kristin Lauter

[LV] - An arithmetic intersection formula for denominators of Igusa class

polynomials -- Kristin Lauter and Bianca Viray

arXiv:1210.7841v1

[Yang] - Arithmetic interseciton on a Hilbert modular surface and the

Faltings height -- Tonghai Yang

http://www.math.wisc.edu/~thyang/general4L.pdf

[recip] - recip, SageMath package for explicit complex multiplication -- Marco

Streng

https://bitbucket.org/mstreng/recip/

[Streng12]- An explicit version of Shimura's reciprocity law for Siegel

modular functions -- Marco Streng

arXiv:1201.0020

[Streng14]- Computing Igusa Class Polynomials

Mathematics of Computation, Vol. 83 (2014), pp 275--309

REpository of Complex multiplication SageMath code

=========================

.. image:: https://travis-ci.org/mstreng/recip.svg?branch=master

:target: https://travis-ci.org/mstreng/recip

Once the Travis CI set-up has been completed, the documentation for the package can be found at https://mstreng.github.io/recip/doc/html/

Installation

------------

Local install from source

^^^^^^^^^^^^^^^^^^^^^^^^^

Download the source from the git repository::

$ git clone https://github.com/mstreng/recip.git

Change to the root directory and run::

$ sage -pip install --upgrade --no-index -v .

For convenience this package contains a [makefile](makefile) with this

and other often used commands. Should you wish too, you can use the

shorthand::

$ make install

Usage

-----

Once the package is installed, you can use it in Sage with::

sage: from recip import *

sage: CM_Field([5,5,5])

CM Number Field in alpha with defining polynomial x^4 + 5*x^2 + 5

Source code

-----------

All source code is stored in the folder ``recip`` using the same name as the

package. This is not mandatory but highly recommended for clarity. All source folder

must contain a ``__init__.py`` file with needed includes.

Tests

-----

This package is configured for tests written in the documentation

strings, also known as ``doctests``. For examples, see this

[source file](recip/ultimate_question.py). See also

[SageMath's coding conventions and best practices document](http://doc.sagemath.org/html/en/developer/coding_basics.html#writing-testable-examples).

With additional configuration, it would be possible to include unit

tests as well.

Once the package is installed, one can use the SageMath test system

configured in ``setup.py`` to run the tests::

$ sage setup.py test

This is just calling ``sage -t`` with appropriate flags.

Shorthand::

$ make test

Documentation

-------------

The documentation of the package can be generated using Sage's

``Sphinx`` installation::

$ cd docs

$ sage -sh -c "make html"

Shorthand::

$ make doc

#*****************************************************************************

# Copyright (C) 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017 Marco Streng

# <marco.streng@gmail.com>

#

# This program 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 2 of the License, or

# (at your option) 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, write to the Free Software Foundation, Inc.,

# 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA.

#*****************************************************************************

RECIP -- REpository of Complex multIPlication SageMath code.

This started out as code meant for computing with Shimura's RECIProcity law,

but grew into a collection of much of the SageMath code written by me for my

research.

Version 0.2.5 (tested with SageMath 8.0, short tests only).

When using this package in a publication, it is highly likely that it is

approprate certain publications. Please CITE the relevant JOURNAL publications,

as well as giving the URL of this repository.

Here is a list of functionalities of this repository, together with the

publications that should be cited when you use them, and the name of the file

that has examples.

* Igusa class polynomials (proven correct)

See both "Igusa class polynomials (not proven correct)" and

"Denominators of Igusa class polynomials" below.

* Non-maximal orders of CM-fields and their polarized ideal classes and Igusa

class polynomials.

cite [BissonStreng] (code is written for, part of, and based on, this publication)

see orders.sage for examples

* (n,n)-isogenies between polarized ideal classes

cite [BLS]

see bls.sage for examples

* Computations related to Shimura's reciprocity law

cite [Streng12] (code is written for, part of, and based on, this publication)

see article.sage for examples

* Igusa class polynomials (not proven correct)

cite [Streng14], [vWamelen], [Weng] (code is based on these publications)

* Denominators of Igusa class polynomials

cite [BouyerStreng] (code is written for, and hence part of, this publication)

and depending on how the code is used, and on the kind of quartic CM-field,

also cite one or more of:

[BouyerStreng], [GL], [LV], [Yang] (large parts of the code are based on these)

see denominators.sage for examples

Here is a list of SageMath programs written by my students and me that is not part

of this repository.

* Height reduction of binary forms and hyperelliptic curves.

(with Florian Bouyer)

https://bitbucket.org/mstreng/reduce

cite [BouyerS] (code is written for, part of, and based on, this publication)

* Solving conics and Mestre's algorithm

(with Florian Bouyer)

now part of the standard SageMath functionality

* Hilbert modular polynomials

(by Chloe Martindale)

contact her if you are interested

* CM class number one for genus 2 and 3

(by Pınar Kılıçer)

contact her if you are interested

To use the latest version of this package directly from the web, start SageMath

and type::

sage: load("https://bitbucket.org/mstreng/recip/raw/master/recip_online.sage")

To use this package offline, download it first and extract it to some

directory, say "somewhere_on_my_drive/recip", then start SageMath and type::

sage: load_attach_path("somewhere_on_my_drive/recip")

sage: load("recip.sage")

[ABLPV] - Comparing arithmetic intersection formulas for denominators of

Igusa class polynomials -- Jacqueline Anderson, Jennifer S.

Balakrishnan, Kristin Lauter, Jennifer Park, and Bianca Viray

[BissonS] - On polarised class groups of orders in quartic CM fields --

Gaetan Bisson and Marco Streng

http://arxiv.org/abs/1302.3756

[BLS] - Abelian surfaces admitting an (l,l)-endomorphism -- Reinier Broker,

Kristin Lauter, and Marco Streng

accepted for publication by Journal of Algebra, 2013

http://arxiv.org/abs/1106.1884

[BouyerS] - Examples of CM curves of genus 2 defined over the reflex field --

Florian Bouyer and Marco Streng

http://arxiv.org/abs/1307.0486

[GJLSVW] - Igusa class polynomials, embeddings of quartic CM fields, and

arithmetic intersection theory -- Helen Grundman, Jennifer

Johnson-Leung, Kristin Lauter, Adriana Salerno, Bianca Viray, and

Erika Wittenborn

http://arxiv.org/abs/1006.0208

[GL] - Genus 2 curves with complex multiplication -- Eyal Goren and

Kristin Lauter

[LV] - An arithmetic intersection formula for denominators of Igusa class

polynomials -- Kristin Lauter and Bianca Viray

arXiv:1210.7841v1

[Yang] - Arithmetic interseciton on a Hilbert modular surface and the

Faltings height -- Tonghai Yang

http://www.math.wisc.edu/~thyang/general4L.pdf

[recip] - recip, SageMath package for explicit complex multiplication -- Marco

Streng

https://bitbucket.org/mstreng/recip/

[Streng12]- An explicit version of Shimura's reciprocity law for Siegel

modular functions -- Marco Streng

arXiv:1201.0020

[Streng14]- Computing Igusa Class Polynomials

Mathematics of Computation, Vol. 83 (2014), pp 275--309

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