Data and code related to "Vanishing of Quartic and Sextic Twists of $L$-functions" by Berg, Ryan and Young

Here you will find three .tgz files. The first, paper.tgz contains the images and .tex file used to create the paper. The second, data.tgz, contains the data we used to verify our conjectures and make the images in the paper. The third, code.tgz, contains the code we used to generate the data. The contents of each .tgz file is detailed below.

paper.tgz

The contents of this file include the images, .tex file, .bib file used in the paper.

code.tgz

The contents of this file are themselves .tgz files described here

notebooks.tgz Jupyter notebooks in which we use Python libraries like Pandas, Mathplotlib, etc. to analyze the data and make the histograms and plots in the paper.
osg.tgz Directory structure, submit files, executable, and character data we use to generate the data on the Open Science Grid. What is missing but would be required to replicate our calculations is a compiled version of a PARI/GP 2.13.4 executable store as pari.tgz. The workflow is (1) split the file of characters in the director into a number of smaller files (maybe with fewer than ten lines where each line contains an N and b that determines the character's Conrey label, (2) make a list of these smaller files of characters you just created and store it as something like file.list, (3) run condor_q job.submit. Inside each subdirectory you will find a job.submit file that contains, among other information, the amount of memory needed to calculate all the data for all the characters in that subdirectory. This ranged up to 192GB for a handful of twists by characters of large conductor. The vast majority required less than 16 GB.
totally-quartic-chars.tgz PARI/GP code that determines whether a character is totally quartic and related data.
totally-sextic-chars.tgz PARI/GP code that determines whether a character is totally sextic and related data.
uniformity-of-signs-of-FE.tgz SageMath scripts and bash scripts to use slurm to calculate characters of order $\ell$ in some range of conductors. Additionally it contains code to compute the Gauss sums using slurm to numerically check equidistribution as done in the paper.
test_quartic.gp A sample PARI/GP script that shows the code that is automatically generated on OSG to compute data related to a particular twisted L-function.

data.tgz

The contents of this file are the data calculated on OSG. The total wall time for computing all these data was more than 50 years. Two files related to whether the characters are totally order $\ell$ or not.

list_totally_quartic_chars.csv Each row in this file contains the conductor, the Conrey index, and a 1 if the character of that conductor and Conrey index is totally quartic and a 0 if not.
list_totally_sextic_chars.csv Each row in this file contains the conductor, the Conrey index, and a 1 if the character of that conductor and Conrey index is totally sextic and a 0 if not.

Six files computed about the twisted L-functions of particular elliptic curves. The columns of the data are (1) the conductor of the character being twisted by, (2) the Conrey index of the character, (3) the real part of L(E,1,chi), (4) the imaginary part of L(E,1,chi), (5) the imaginary part of the first zero, (6) the imaginary part of the second zero, (7) the real part of the sign of the functional equation, (8) the imaginary part of the sign of the functional equation, (9chi(-1), (10) the real part of chi(N_E), (11) the imaginary part of chi(N_E), (12) the eigenvalue of the Fricke involution, (13) a numerical approximation to the integer n_chi used in the discretization, (14) the previous approximation rounded to the nearest integer, (15) the divisors of the conductor, and (16) whether or not a naive generalization of a conjecture in David, Fearnley and Kisilevsky generalizes to our cases. The last two columns we do not use in this paper but include them in case they might be of interest.

e11a1-quartic-primes-final.csv Contains all the data in the preceding paragraph for the curve 11.a.1 for quartic characters of prime conductor not equal to 11 and less than 2000000.
e11a1-sextic-primes-final.csv Contains all the data in the preceding paragraph for the curve 11.a.1 for sextic characters of prime conductor not equal to 11 and less than 2000000.
e37a1-quartic-primes-final.csv Contains all the data in the preceding paragraph for the curve 37.a.1 for quartic characters of prime conductor not equal to 37 and less than 2000000.
e37a1-sextic-primes-final.csv Contains all the data in the preceding paragraph for the curve 37.a.1 for sextic characters of prime conductor not equal to 37 and less than 2000000.
e11a1-quartic-all-final.csv Contains all the data in the preceding paragraph for the curve 11.a.1 for quartic characters of conductor coprime to 11 and less than 700000.
e11a1-sextic-all-final.csv Contains all the data in the preceding paragraph for the curve 11.a.1 for sextic characters of conductor coprime to 11 and less than 300000.

License

The code and data is released under the Creative Commons Attribution-ShareAlike 4.0 International Public License.