REpository of Complex multiplication SageMath code

The documentation for the package can be found at https://mstreng.github.io/recip/doc/html/

Installation

This package was last tested with SageMath 10.2.

Local install from source

You can install the package into SageMath using with few easy commands if you have standard linux tools installed.

Download the source from the git repository:

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

Change to the main directory of what was just installed and run:

$ make install

To update to the latest version:

$ git pull

And then do make install again.

Once the package is installed, you can use it in SageMath 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

Using it directly from the web

If your copy of SageMath is built with ssh support, then whenever you have an internet connection, you can do the following inside SageMath to use the package without installing anything:

sage: load("https://raw.githubusercontent.com/mstreng/recip/master/recip/recip_online.sage")
sage: CM_Field([5,5,5])
CM Number Field in alpha with defining polynomial x^4 + 5*x^2 + 5

#***************************************************************************** # Copyright (C) 2010 -- 2024 Marco Streng # <[email protected]> # # 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.

See the file VERSION for the current version.

When using this package in a publication, it is highly likely that it is appropriate to cite 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://raw.githubusercontent.com/mstreng/recip/master/recip/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")

Alternatively, download the package and use python import commands. For this, it is recommended to add the following to your .sage/sagerc file:

export PYTHONPATH=$PYTHONPATH:somewhere_on_my_drive/recip/

[ABLPV] - Comparing arithmetic intersection formulas for denominators of

Igusa class polynomials -- Jacqueline Anderson, Jennifer S. Balakrishnan, Kristin Lauter, Jennifer Park, and Bianca Viray Women in numbers 2: research directions in number theory, 65–82, Contemp. Math., 606, Centre Rech. Math. Proc., Amer. Math. Soc., Providence, RI, 2013

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

Gaetan Bisson and Marco Streng Math. Res. Lett., Vol. 24 (2017), number 2, pp 247 - 270 http://arxiv.org/abs/1302.3756

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

Kristin Lauter, and Marco Streng Journal of Algebra, Vol. 394 (2013), pp 374--396 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 LMS Journal of Computation and Mathematics, Vol. 18 (2015), issue 01, pp 507-538

[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 WIN—women in numbers, 35–60, Fields Inst. Commun., 60, Amer. Math. Soc., Providence, RI, 2011

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

Kristin Lauter Int. Math. Res. Not. IMRN 2012, no. 5, 1068–1142.

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

polynomials -- Kristin Lauter and Bianca Viray arXiv:1210.7841v1 Amer. J. Math. 137 (2015), no. 2, 497–533

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

Faltings height -- Tonghai Yang http://www.math.wisc.edu/~thyang/general4L.pdf Asian J. Math. 17 (2013), no. 2, 335–381

[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