Emmanuel Thomé committed Jun 23, 2020 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29  cmh: computation of genus 2 class polynomials. ============================================== This software package computes Igusa (genus 2) class polynomials, which parameterize 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](#introduction) - [License](#license) - [Prerequisites](#prerequisites) - [Compiling](#compiling) - [Using](#using) - [Output](#output) - [Caveats](#caveats) - [Internal documentation](#internal-documentation) - [Advanced usage, including (but not limited to) MPI](#advanced-usage-including-but-not-limited-to-mpi) Introduction ------------ CMH computes Igusa class polynomials.  Emmanuel Thomé committed Jun 23, 2020 30 31 32 33 See the [releases](../../releases) page for releases. Note that much of the development of CMH has happened only in the git repository om the recent years.  Emmanuel Thomé committed Jun 23, 2020 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 The main authors are: - [Andreas Enge](http://www.math.u-bordeaux1.fr/~enge/) - [Emmanuel Thomé](http://www.loria.fr/~thome/) A code base by [Régis Dupont](http://www.lix.polytechnique.fr/Labo/Regis.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](announcement-record-computation-cm2.txt) 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 .  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: - [GNU MP](http://gmplib.org/) version 4.3.2 or above - [GNU MPFR](http://mpfr.org/) version 2.4.2 or above - [GNU MPC](http://mpc.multiprecision.org/) version 1.0 or above - [MPFRCX](http://mpfrcx.multiprecision.org/) version 0.4.2 or above - [FPLLL](http://perso.ens-lyon.fr/damien.stehle/fplll/) version 4.0.4 or above - [PARI/GP](http://pari.math.u-bordeaux.fr/) 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 platform is recent Debian GNU/Linux, and most testing has been done in this environment. 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 $topdir/cmh-deps, but can also be tuned to build elsewhere (see the script for more information). Compiling --------- Distributed tarballs should ship with a ./configure shell script. Development checkouts do not. For generating ./configure, please run the ./config/autogen.sh script, which requires the usual autotools dependencies (autoconf, automake, libtool) To compile and install, type:  ./configure [[all relevant flags]] make make install  It might be very relevant to pass an appropriate --prefix option to the configure script, so that installation is done to a directory writable by the user. out-of-source build is supported. The recognized flags are  --enable-mpi use mpi to compile a parallel version [default=no] --with-gmp=DIR GMP installation directory --with-mpfr=DIR MPFR installation directory --with-mpc=DIR MPC installation directory --with-mpfrcx=DIR MPFRCX installation directory --with-fplll=DIR FPLLL installation directory --with-pari=DIR PARI installation directory  For MPI, see the section later in this document. Note that make install is *not* optional, since making the program run without doing make install first is difficult. Using ----- The input to this program is a defining equation for a (for the moment, only dihedral) quartic CM field, in the form of two nonnegative integers$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$are the integers defining the CM field as discussed above. This, in effects, does the two out of three possible steps. - The -p option does the "preparation", which consists in computing a list of period matrices corresponding to one orbit under complex multiplication. This list ends up in the .in file (see next section for location and naming). - The -f option does the big part of the computation, which is the computation of class polynomials from the config files. This is programmed in C (here and there, when we refer to "the C code", this designates the code which gets run for this step). An optional feature is provided for doing this step in parallel over several processors or nodes. The output ends up in the .pol file (again see below). Additionally, adding the -c flag does a third step, which checks the computed class polynomial for correctness as follows (see also [BUGS](BUGS)). - find a Weil number$p$- find a triple of Igusa invariants corresponding to roots of the class polynomials mod$p$- use Mestre's algorithm to reconstruct a hyperelliptic curve from its invariants - check that the Jacobian of the curve above has cardinal $\operatorname{Norm}(1\pm pi)$. 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](BUGS) Internal documentation ---------------------- The .in and .pol file formats are discussed in [README.format](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.