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.
- Internal documentation
- Advanced usage, including (but not limited to) MPI
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.
cmh -- computation of genus 2 class polynomials and theta constants. Copyright (C) 2006, 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020, 2021, 2022 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/>.
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:
- GNU MP version 4.3.2 or above
- GNU MPFR version 2.4.2 or above
- GNU MPC version 1.0 or above
- MPFRCX version 0.4.2 or above
- FPLLL version 4.0.4 or above, but not version 5 or above
- PARI/GP version 2.7.0 or above
The following libraries are optional
- MPI implementation: any implementation can be used, e.g. openmpi, mpich, mvapich2, ...
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
$topdir/cmh-deps, but can also be tuned to build
elsewhere (see the script for more information).
Distributed tarballs should ship with a
./configure shell script.
Development checkouts do not. For generating
./configure, please run the
./config/autogen.sh script, which requires the usual autotools
To compile and install, type:
./configure [[all relevant flags]] make make install
It might be very relevant to pass an appropriate
--prefix option to the
configure script, so that installation is done to a directory writable by
out-of-source build is supported.
The recognised flags are
--enable-mpi use mpi to compile a parallel version [default=no] --with-gmp=DIR GMP installation directory --with-mpfr=DIR MPFR installation directory --with-mpc=DIR MPC installation directory --with-mpfrcx=DIR MPFRCX installation directory --with-fplll=DIR FPLLL installation directory --with-pari=DIR PARI installation directory
For MPI, see the section later in this document.
make install is not optional, since making the program run
make install first is difficult.
The input to this program is a defining equation for a (for the moment,
only dihedral) quartic CM field, in the form of two nonnegative integers
B$ for a corresponding defining equation $
X^4+A X^2+B$. The
(A,B)$ 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
Its syntax is:
$prefix/bin/cmh-classpol.sh -p -f A B
A$ and $
B$ are the integers defining the CM field as discussed
above. This, in effects, does the two out of three possible steps.
-poption does the "preparation", which consists in computing a list of period matrices corresponding to one orbit under complex multiplication. This list ends up in the
.infile (see next section for location and naming).
-foption does the big part of the computation, which is the computation of class polynomials from the config files. This is programmed in C (here and there, when we refer to "the C code", this designates the code which gets run for this step). An optional feature is provided for doing this step in parallel over several processors or nodes. The output ends up in the
.polfile (again see below).
Additionally, adding the
-c flag does a third step, which checks the
computed class polynomial for correctness as follows (see also
- find a Weil number $
- find a triple of Igusa invariants corresponding to roots of the class
polynomials mod $
- use Mestre's algorithm to reconstruct a hyperelliptic curve from its invariants
- check that the Jacobian of the curve above has cardinal
Some other flags are mostly for interal use. Noteworthy ones are
which disables the temporary checkpoint data creation, and
-b xxxx, which
modifies the starting precision.
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 $
B$ are three
A$ and $
B$ are the integers discussed above, while $
D$ is the
discriminant of the real quadratic subfield (this is the fundamental
discriminant of $
| 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
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. |
.pol file formats are discussed in
Advanced usage, including (but not limited to) MPI
The main C binary which is used to compute class polynomials (the
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
make install installs
The parallel binary is called
cm2-mpi, and the binary is created in the
cm2-mpi is created only if
passed to configure.
make install installs
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
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 -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
checkpoints are enabled by default, but they may be disabled using the
--no-checkpoints command-line option.