SYMPOW is a package to compute special values of symmetric power elliptic
curve L-functions. It can compute up to about 64 digits of precision.
========================================================================
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, but due to licensing
difficulties, this was eliminated. SYMPOW also does not use the
standard math libraries, thus eliminating possible license conflicts.
However, ths 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.
========================================================================
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).
./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
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
========================================================================
Adding new symmetric powers:
SYMPOW keeps data for symmetric powers in the directory datafiles
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 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 power, 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.
For Hecke powers, the (2N-1)st power is computed using the Nth files,
and the (2N)th power requires the Nth and (N+1)st.
========================================================================
INSTALLATION:
You will need to have sh and make, or their equivalents
and rm, mkdir, tar, uname, cp, touch, and cc
[or equivalents, but these must be modified in Configure]
to build sympow. The shell commands cd, echo and which are also needed.
The usage should just be:
sh Configure
make
sympow will create a directory 'datafiles' of data for symmetric powers.
New symmetric powers can be added as above, if grep, gp, and sed were
available during the Configure.
Needed header files are: stdio.h, stdlib.h, string.h, strings.h, unistd.h
========================================================================
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.
========================================================================
ChangeLog/BugFixes:
* VERSION 1.000 --- released [30 Nov 2005]
* VERSION 1.001 --- release version with data included, so change generate.c
compute moddeg with Hecke in CM case [5 Dec 2005]
* VERSION 1.002 --- add multiplier to squfof (fix oo loop) [7 Dec 2005]
* VERSION 1.003 --- moved some Configure stuff to config.h [10 Dec 2005]
* VERSION 1.004 --- bug: QD_atan (mul by cos^2) & llint in binom [14 Dec 2005]
* VERSION 1.005 --- binom and sbinom improved, *.gp's changed [16 Dec 2005]
* VERSION 1.006 --- *.gp fixes, get_weight(llint), rootno.c [21 Dec 2005]
* VERSION 1.007 --- fix TACKS bug for Hecke, change BK for Hecke [22 Dec 2005]
* VERSION 1.008 --- removed mathlib support to try to help protect user
from possible license violations [13 Jan 2006]
* VERSION 1.009 --- added (conjectural) root numbers for higher powers,
fixed analrank bug for precision of 9 [16 Jan 2006]
* VERSION 1.010 --- moddeg bugs {v3(N)=4, M=32,64,128, 3-isog} [19 Jan 2006]
* VERSION 1.011 --- reduce size of ar/md tables, fix kronll [23 Jan 2006]
* VERSION 1.012 --- Fix new_data typo, add higher Hecke derivs,
add zero check option (undocumented) [20 Feb 2006]
* VERSION 1.013 --- Fix llint probs in ec_ap_large [23 Feb 2006]
* VERSION 1.014 --- Fix years in ChangeLog, allocatemem(2^28) in standard1.gp
(needed for 64-bit machines), add --help, etc. [1 Mar 2006]
* VERSION 1.018 --- speed up ec_ap via idea of Geoff Bailey, use lesser prec
in QD computations when possible, new fpu.c,
-mdspeed option, -noqdcheck option [24 Apr 2006]
* VERSION 1.019 --- getline -> getline0, 'which \cp' in config [11 May 2006]
