These scripts numerically verify Beilinson's conjecture on certain
quotients of Fermat Curves as described in my thesis "K-Theory of Fermat
Curves".
L-functions are computed using the ComputeL package by Tim Dokchitser.
To run the scripts you will need to have installed gcc and PARI/GP and
simply type
./make.sh
By default data is computed for all primes less than 1,000,000. This is
sufficient to allow ComputeL to calculate the L-values at the PARI/GP
default precision of 38 digits. The calculations complete in about 3
minutes on my machine.
The prime limit can be increased to somewhere near 36,000,000 before
breaking the scripts. This would be sufficient to achieve accuracy at
least 200 decimal digits on all of the relevant quotients.
If you really want to make such accurate computations it is probably
best to call the C programs fermat, fermat2 and fermat3 manually and
redirect their output into a file.
For example to compute the first 10,000,000 Dirichlet coefficients on
quotients C_{1,1,5} and C_{1,2,4} of F_7 run the function "go" in
get_prim.gp with n=10000000 and then run
./fermat 10000000 7 1 2 > fermat_7.txt
./fermat2 10000000 7 1 2 > fermat2_7.txt
./fermat3 10000000 7 1 2 > fermat3_7.txt
Now these files can be read in a PARI/GP session with
\r fermat.gp
\r fermat_7.txt
\r fermat2_7.txt
\r fermat3_7.txt
Choose a higher precision in PARI/GP with something like "\p 100" and
amend the function "get_L" in fermat.gp so that it calls the function
"akfast" with the argument "got_traces" being non-zero. If enough terms
have been computed then ComputeL should happily be able to compue the
L-values at the higher precision.
Also note that if you pass the known sign of the functional equation to
"get_L" then fewer coefficients will be required by ComputeL because it
will not be necessary to call "checkfeq".