CMH: Computation of genus 2 class polynomials

This software package computes Igusa (genus 2) class polynomials, which parameterise the CM points in the moduli space of 2-dimensional abelian varieties, i.e. Jacobians of hyperelliptic curves.

This program is also able to compute theta constants at arbitrary precision (but the interface for this is still to be documented more clearly).

This documentation consists of several chapters.

Introduction

CMH computes Igusa class polynomials.

See the download page for releases.

The main authors are:

A code base by Régis Dupont is at the origin of this work.

In March 2014, we announced the computation of class polynomials for Shimura class number 20,016. See the separate announcement text for this computation.

License

    cmh -- computation of genus 2 class polynomials and theta constants.
    Copyright (C) 2006, 2010, 2011, 2012, 2013, 2014, 2015 Régis Dupont, Andreas
    Enge, Emmanuel Thomé

    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 3 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, see <http://www.gnu.org/licenses/>.

Prerequisites

The following software libraries are required to compile and use cmh; we usually recommend to use the latest version and not the absolutely minimally required one:

The following libraries are optional

The development platforms are recent GNU Guix and Debian GNU/Linux distributions, and most testing has been done in these environments. As a general rule of thumb, if things bomb out, a reasonable explanation could be subtle distribution differences, which are not that hard to fix, but terribly annoying indeed.

A convenience script for fetching and building all the software prerequisites is in ./config/build-dependencies.sh ; this builds all the needed packages in $topdir/cmh-deps, but can also be tuned to build elsewhere (see the script for more information).

Compiling

Distributed tarballs should ship with a ./configure shell script. Development checkouts do not. For generating ./configure, please run the ./config/autogen.sh script, which requires the usual autotools dependencies (autoconf, automake, libtool)

To compile and install, type:

./configure [[all relevant flags]]
make
make install

It might be very relevant to pass an appropriate --prefix option to the configure script, so that installation is done to a directory writable by the user.

out-of-source build is supported.

The recognised flags are

  --enable-mpi            use mpi to compile a parallel version [default=no]
  --with-gmp=DIR          GMP installation directory
  --with-mpfr=DIR         MPFR installation directory
  --with-mpc=DIR          MPC installation directory
  --with-mpfrcx=DIR       MPFRCX installation directory
  --with-fplll=DIR        FPLLL installation directory
  --with-pari=DIR         PARI installation directory

For MPI, see the section later in this document.

Note that make install is not optional, since making the program run without doing make install first is difficult.

Using

The input to this program is a defining equation for a (for the moment, only dihedral) quartic CM field, in the form of two nonnegative integers $A$,$B$ for a corresponding defining equation $X^4+A X^2+B$. The pair $(A,B)$ must be so that $A^2-4B$ is a positive integer which is not a perfect square, and $(A,B)$ minimal in the sense that no smaller pair defines the same field (the script complains if this is the case). The special case where the Galois group of $X^4+A X^2+B$ is $Z/2 \times Z/2$ is also forbidden.

The main entry point is the script placed in the following location by make install:

    $prefix/bin/cmh-classpol.sh

Its syntax is:

    $prefix/bin/cmh-classpol.sh -p -f A B

where $A$ and $B$ are the integers defining the CM field as discussed above. This, in effects, does the two out of three possible steps.

Additionally, adding the -c flag does a third step, which checks the computed class polynomial for correctness as follows (see also BUGS).

Some other flags are mostly for interal use. Noteworthy ones are -N, which disables the temporary checkpoint data creation, and -b xxxx, which modifies the starting precision.

Output

All output of the program goes to the data/ directory (which may be a symbolic link to auxiliary storage).

All files are created with a common prefix D_A_B, where $D$,$A$,$B$ are three integers. $A$ and $B$ are the integers discussed above, while $D$ is the discriminant of the real quadratic subfield (this is the fundamental discriminant of $\mathbb{Q}(\sqrt{A^2-4B})$.

| file name | description | | ----------------- | ---------------------------------------------------------- | D_A_B.in | description of the set of period matrices describing the different irreducible factors of the class polynomials. The format of this file is used internally, but its details are discussed in the "internal details" section. | | D_A_B.pol | the different irreducible factors of the class polynomials (more precisely of the CM variety in the moduli space). This is given in triangular form (H1,H2hat,H3hat), and discussed in the "internal details" section. These polynomials are defined over the real quadratic subfield of the reflex field. | | D_A_B.gp.log | output (terse) of the pari/gp program which computes the .in file from (A,B) | | D_A_B.out | output of the C code for computing .pol from .in | | D_A_B.check.log | output of the pari/gp program which computes a hyperelliptic curve whose Jacobian has CM by the desired field, and checks its cardinality for consistency with the expected value. [ only if -c was provided on the command line of cmh-classpol.sh ] | | D_A_B/ | temporary checkpointing and restart data. The precise meaning and format of these files is not documented, and subject to incompatible change without notice. |

Caveats

See BUGS

Internal documentation

The .in and .pol file formats are discussed in README.format

Advanced usage, including (but not limited to) MPI

The main C binary which is used to compute class polynomials (the .pol file) from orbits of period matrices (the .in file) has an MPI version. This version can be compiled by passing --enable-mpi to configure.

The non-parallel binary is called cm2, and the binary is created in the $builddir/src. make install installs cm2 in $prefix/bin/cm2

The parallel binary is called cm2-mpi, and the binary is created in the $builddir/src. Obviously, cm2-mpi is created only if --enable-mpi is passed to configure. make install installs cm2-mpi in $prefix/bin/cm2-mpi

If you intend to use MPI for computing the class polynomials, this very likely means that you are well beyond the intended scope for the cmh-classpol.sh script. For this reason, cmh-classpol.sh has no provision for calling cm2-mpi directly, and this binary must be called manually.

We assume that you have successfully created a .in file (using cmh-classpol.sh -p). Then, the syntax for cm2, or cm2-mpi, is:

    cm2 -i D_A_B.in -o D_A_B.pol [other flags]

    mpirun -n #jobs [other mpi options] cm2-mpi -i D_A_B.in -o D_A_B.pol [other flags]

This will eventually write the result in D_A_B.pol ; the code has provision for resuming interrupted computations, as intermediate computation checkpoints are saved in a subdirectory called D_A_B/ ; checkpoints are enabled by default, but they may be disabled using the --no-checkpoints command-line option.