Q Quadratic

A PARI/GP package for integral binary quadratic forms and quaternion algebras over Q, with an emphasis on indefinite quadratic forms and indefinite quaternion algebras.

This package is not regularly maintained, and with updates to PARI/GP some things may break. If they do, please let me know, and I will try to fix them.

Installation Instructions

Downloading the code

Call git clone https://github.com/JamesRickards-Canada/Q-Quadratic.git. If you are on Windows, be sure to git clone from WSL (see below), as Windows line endings (carriage returns) may be added to files, causing issues.

Prerequisites

• PARI/GP, but not the downloaded ready-to-go binary. The PARI/GP website has binaries for Windows and Mac avaliable, but these will not work with the package. See below for OS specific instructions.
• Version at least 2.13.1, though the more up-to-date the better.
• You should have a guess as to the location of the pari.cfg file for the version of PARI/GP you are running. Suggestions on how to do this can be found below.

Operating systems

• Linux - No further requirements
• Windows - You need to use Windows Subsytem for Linux. See the guide I wrote for additional instructions.
• Mac - You need to have Homebrew installed. This is also an easy way to install PARI/GP: brew install pari

Where is pari.cfg?

• The configuration file will search for this, but it is preferrable to not search your entire hard drive (as this can be very slow). So, you should at least supply a guess as to the location of pari.cfg. Often only the top-level folder (e.g. /usr or /opt) suffices.
• On Linux or WSL, if you build PARI/GP from source, it should be located in /usr/local/lib/pari/pari.cfg, or at least somewhere in the /usr folder.
• On a Mac, if you install PARI/GP with Homebrew, it may be found in a folder like /opt/homebrew/Cellar/pari/VERSION/lib/pari. Searching /opt/homebrew should be fine.
• If you are obtaining it through SageMath, it might be found where the library files of SageMath are
• Assuming you open PARI/GP with the command gp, try type -a gp, which will display where this command lives. The corresponding file(s) are likely symbolic links, and you can call readlink -f LOCATION on each of them to see where it lives. This can provide a clue as to the place to search for pari.cfg.
• Another clue comes from gp itself. Open gp, and type default(). Look for the entries datadir and help. It is often the case that datadir is in X/share/pari, help is in X/bin/gphelp, and pari.cfg is in X/lib/pari/pari.cfg.

Configuring and building the package

• From inside the project folder, call ./configure to initialize the project. This helps you search for pari.cfg, and stores the location to a file. You should supply it with a folder to search in!
• The script displays the corresponding versions of the found files, so if you have multiple versions, you can choose the correct one. This can be useful if you keep multiple copies of PARI/GP around.
• If the location of the installation of PARI/GP does not change, you do not need to reconfigure. If when you update PARI/GP there is a new location (e.g. if the version number is in the file path of pari.cfg), you should call ./configure again.
• Call make to build the project, and make clean to remove all .o object files. If you update to a new version of PARI/GP, you must remake the project.
• Once this is done, a call to gp qquadratic starts gp with the package installed!
• Call ?qq or consult the User's Manual to access further help for this package.

Documentation

The file QQuadratic_GP_guide.pdf is a guide to the GP-accessible methods. It may be slightly out of date.

The file QQuadratic_PARI_guide.pdf is a guide to using this package in library mode. It too may be slightly out of date.

Papers

All relevant computations from my thesis were made with this package. The same can be said for the 3 papers that emanated from this thesis:

• Computing intersections of closed geodesics on the modular curve
• Counting intersection numbers of closed geodesics on Shimura curves
• Hecke operators acting on optimal embeddings in indefinite quaternion algebras

If you want to test the examples of intersection series from "Hecke operators acting on optimal embeddings in indefinite quaternion algebras", use the file "intersectionseries.gp".

Warnings

This was the first package I wrote in PARI, and there are some poor syntax/decision choices, code that is somewhat redundant with more up-to-date PARI/GP versions (when I began coding in gp, the version was 2.7.6), and code that can be significantly sped up. I do not plan to come back here and do those modifications. Instead, they are slowly working their way into other projects. For example, if you are interested in fundamental domains, then you should look at the infinitely improved Fundamental-domains-for-Shimura-curves. If most of this package ends up improved elsewhere, then I might come back and update this repository (or make a new one).