SYMmetric POWer elliptic curve L-functions

SYMPOW is a mathematical program to compute special values of symmetric power elliptic curve L-functions; it can compute up to about 64 digits of precision.

Installation

Sage

Note that SYMPOW is a dependency of SageMath: if SageMath is installed on your computer, SYMPOW must be already present.

Distributions

Otherwise, when applicable, it is highly recommended to install SYMPOW through the package manager system of your Operating System.

# Debian derivatives
sudo apt install sympow

By hand

As a (very) last resort, you can always build and install SYMPOW as you are used to do with any other classical Open Source Software that comes with its own building-installing machinery.

Build-essential GNU tools along PARI/GP must be present.

# overview
sh Configure
make all
make install
make clean
make distclean

By default SYMPOW is installed in the /usr/local directory wrt to FHS. More control is possible.

# HOME installation example
PREFIX=${HOME} VARPREFIX=${HOME}/var sh Configure
make all
make install
make distclean

# example for package or system maintainers
rm -rf _install
ADDBINPATH=yes PREFIX=/usr VARPREFIX=/var sh Configure
make build-arch
make build-indep
make install-arch  DESTDIR=_install/ARCHX
make install-indep DESTDIR=_install/INDEP
make distclean

Maintainers want also read the SYMPOW Data setup section.

Usage examples

basic usages

sympow -sp 2p16 -curve "[1,2,3,4,5]"

will compute L(Sym^2 E,edge) for E=[1,2,3,4,5] to 16 digits of precision. The result

 2n0: 8.370510845377639E-01
 2w0: 8.370510845377586E-01

consists of two calculations using different parameters to test the functional equation (to see if these are sufficiently close).

# heavy computation
sympow -sp 3p12d2,4p8 -curve "[1,2,3,4,5]"

will compute the 0th-2nd derivatives of L(Sym^3 E,center) to 12 digits and L(Sym^4 E,edge) to 8 digits.

# light alternatives
sympow -sp 3p12d2,4p8 -curve "[0,1,2,3,4]"
sympow -sp 3p12d2,4p8 -curve "[1,1,0,1,0]"

Special cases: When a curve has CM, Hecke power can be used instead

sympow -sp 7p12d1 -hecke -curve "[0,0,1,-16758,835805]"

will compute the 0th-1st derivatives of L(Sym^7 psi,special) to 12 digits.

Bloch-Kato numbers can be obtained for powers not congruent to 0 mod 4:

sympow -sp 2bp16 -curve "[1,2,3,4,5]"

should return

 2n0: 4.640000000000006E+02
 2w0: 4.639999999999976E+02

which can be seen to be very close to the integer 464.

Modular degrees can be computed with the -moddeg option.

sympow -curve "[1,2,3,4,5]" -moddeg

should return

Modular Degree is 464

Analytic ranks can be computed with the -analrank option:

sympow -curve "[1,2,3,4,5]" -analrank

should return

Analytic Rank is 1 : L'-value 3.51873e+00

and (if the mesh file for the fifth derivative is present)

sympow -curve "[0,0,1,-79,342]" -analrank

should return

Analytic Rank is 5 : leading L-term 3.02857e+01

add new symmetric powers

SYMPOW keeps data for symmetric powers in the datafiles directories If a pre-computed mesh of inverse Mellin transform values is not available for a given symmetric power, SYMPOW will fail. If GP is available, the command

sympow -new_data 2

will add the data for the 2nd symmetric power, while

sympow -new_data 3d2

will add the data for the 2nd derivative and 3rd symmetric power,

sympow -new_data 6d0h

will add the data for the 0th derivative of the 6th Hecke translate, and

sympow -new_data 4c

will add data for the 4th symmetric power for curves with CM (these need to be done separately for powers divisible by 4).

The mesh files are stored in binary form, and thus endian-ness is a worry when moving from one platform to another.

To enable modular degree computations, the 2nd symmetric powers must be extant, and analytic rank requires derivatives of the 1st power. There are lower-precision datafiles in the sympow.tar distribution to expedite the computation of analytic rank and modular degree. These latter can also be re-generated in about 5-10 minutes via running armd.sh as a shell script.

For higher precision, here is a test case; it gives 6.0e+2 to almost 64 digits:

sympow -sp 2bp64 -curve '[0,0,0,0,5]'

For Hecke powers, the special value for the (2N-1)st power is computed using the datafile for the Nth translate, and the (2N)th power requires both the Nth and (N+1)st.

Computation notes

• Calculation of periods should not be trusted to the full 64 digits.
• Analytic rank does not save the ap when more than 10^9 terms are needed: in such a case, it is probably easier to run sympow as

    sympow -sp 1w0p6D0,1w0p6D2,1w0p6D4,1w0p6D6,1w0p6D8

[for even signed curves, and however many derivatives are deemed relevant, and to a desired precision] and parse the output.

SYMPOW Data setup

SYMPOW, as forked, runs as any regular executable, that is to say, not in an esoteric manner: intermediate computing steps that were meant to be executed by hand by end-users are now launched on the fly. It also allows one to use a larger set of precomputed data which are now managed system wide rather than locally.

The user-computed plain data mesh files and their binary counter parts, are placed by default in the cache directory HOME/.sympow . This default/ historical cache folder can be overridden through the environment variable SYMPOW_CACHEDIR: the basename of the so passed path is assumed to begin with sympow, otherwise the effective path is assumed to be SYMPOW_CACHEDIR/sympow.

If the cache folder SYMPOW_CACHEDIR (or SYMPOW_CACHEDIR/sympow) does not exists, then it is created on the fly with respect to permissions and privileges whenever the following conditions are satisfied:

1. it terminates with at least three directory separators (the creation is group-centric when the number of separators is greater or equal to six, user-centric otherwise);
2. its parent directory exists. Along the same vain, the default/historical cache directory HOME/.sympow will be created on the fly if it is nonexistent (the creation is user-centric).

The precomputed plain data mesh files are looked in /usr/local/share/sympow; this default directory can be overridden through the environment variable SYMPOW_PKGDATADIR. Their binary counter parts, the binary data mesh files, are created on the fly and stored in the cache directory /var/cache/sympow; this default directory can be overrident through the environment variable SYMPOW_PKGCACHEDIR. The scripts effectivelly employed to compute data can be found in /usr/local/libexec/sympow; this default directory can be overridden through the environment variable SYMPOW_PKGLIBEXECDIR. Please note that the environment variables introduced here (SYMPOW_PKGDATADIR, SYMPOW_PKGLIBEXECDIR, and SYMPOW_PKGCACHEDIR) are meant for advanced usage or debugging.

The data in SYMPOW_CACHEDIR and SYMPOW_CACHEDIR/sympow are authoritative.

For minutes details, you want to peruse the associated patches as imported from Debian (package version 1.023-9) in release 2.023.1: system wide scheme and on fly new data, commits f9add3 and c6f6cf, respectively.

Note that the /usr/local and /var prefixes employed above are the defaults for the environment variables PREFIX and VARPREFIX, respectively, used by Configure at configuration time.

Bug reports, feature requests and ideas

Please submit bug reports, feature requests or ideas through the GitLab website https://gitlab.com/rezozer/forks/sympow .

History

The quad-double package was modified from David Bailey's package: crd.lbl.gov/~dhbailey/mpdist/

The squfof implementation was modified from Allan Steel's version of Arjen Lenstra's original LIP-based code.

The ec_ap code was originally written for the kernel of MAGMA, but was modified to use small integers when possible.

SYMPOW was originally developed using PARI/GP, but due to licensing difficulties, this was eliminated. SYMPOW also does not use the standard math libraries, thus eliminating possible license conflicts. However, the user should ensure that there are no C library license issues.

SYMPOW still can use GP to compute the meshes of inverse Mellin transforms (this is done when a new symmetric power is added to datafiles), but the datafiles can alternatively be downloaded from the SYMPOW website. The current license for GP appears to allow this usage. Please ensure that the GP you have allows this.

Versioning

The forked source versioning is based on the original source versioning as follows:

• the major version number is incremented by one
• the minor version number remains unchanged
• the micro version number (or patch version number) numbers the fork version

Notice that the two first forked versions are special:

• sympow forked version 2.023.0 is the original source itself (modulo this README.md file)
• sympow forked version 2.023.1 is the source as patched by Debian on June 2018 (Debian version 1.023-9)

Legal mumbo-jumbo

original code

Copyright: 2005-2018 Mark Watkins [email protected]

All rights reserved.

Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:

• Redistribution of source code must retain the above copyright notice, this list of conditions and the following disclaimer.
• Redistribution in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution.
• If redistribution is done as a part of a compilation that has a more restrictive license (such as the GPL), then the fact that SYMPOW has a less restrictive license must be made clear to the recipient. For example, a line like (include bracketed text if SYMPOW is modified): "This compilation includes [a modification of] SYMPOW whose [original] code has a less-restrictive license than the entire compilation." should appear in a suitable place in the COPYING and/or LICENSE file.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

forked code

Copyright: 2018-2024 Jerome G. M. Benoit [email protected]