Notes and References
The Sato-Tate conjecture appears in the mathematical literature for the first time in the article "Algebraic cycles and poles of zeta functions" by John Tate (in: Arithmetical Algebraic Geometry, proceedings of a conference held at Purdue University, 1963, Harper & Row, New York). On p. 107 of this article, Tate writes "I understand that M. Sato has found this sin2 distribution law experimentally with machine computations". Indeed, in 1962, Mikio Sato, a fellow researcher named Nagashima, and a graduate student named Kanji Namba, started carrying out calculations on the newly acquired Hitachi HIPAC103 computer at the department of applied mathematics of Tokyo University of Education (now Tsukuba University). In spring 1963 they arrived at the sin2 conjecture. Sato seemed to have gained confidence in the conjecture on April 3, the day he received some numerical tables from Nagashima (the date in the lower right corner of the first page means April 3, 1963). The conjecture is clearly spelled out in a letter from Sato to Namba, written at Osaka University and stamped April 13, 1963. Here is a graph used by Sato to illustrate the conjecture. After his return from Osaka, Sato explained the sin2 conjecture to several people on the large desk in the computer room of Tokyo University of Education, on this piece of paper. The numbers 1650 and 14000 on this sheet indicate that there are about 1650 primes below 14000 (the exact number is 1652); these were calculational limits forced upon the researchers by the memory capacity of their Hitachi computer. More details are given in the article Dedekind's η function and Sato's sin2 conjecture by Kanji Namba, Reports of the Institute for Mathematics and Computer Science 27, 16th Symposium on the History of Mathematics, Inst. of Math. and Compt. Sci. Tsuda Univ. (2006), 95-167 (in Japanese). We are very grateful to Kanji Namba for providing us with this material!
The Sato-Tate conjecture for many elliptic curves has recently been proved by Richard Taylor et al.; see Automorphy for some l-adic lifts of automorphic mod l representations. II. However, the more general conjecture for modular forms is still open.
How did we make the animations?
The histograms on the previous pages where made using the following Mathematica program.
This program accepts, as input, a list of Satake parameters, in the form of Frobenius angles, from any modular form. For example, here are the first 200 Satake parameters for the Ramanujan τ function.
The program generates a histograms showing the distribution of the Satake parameters, plotted against the (conjectured) limiting distribution of sin2(x) from the Sato-Tate conjecture. To execute, the following command should be run from the Mathematica directory: 'math.exe -initfile init.txt filename1 filename2 stepsize'. Filename1 should be the name of the file containing the Satake parameters, entered as decimal numbers, separated by commas. Filename2 should be the name of the desired image output file; file names with extension .gif, .jpg, or .pdf are acceptable. Stepsize indicates how many additional parameters are graphed in each successive image.
A sequence of gif files so produced can be combined to an animated gif. This can be done with a variety of software packages; we used the free and simple GIFfun program.
Finally, some numbers
- 5000 Satake parameters for the Ramanujan delta function
- 5000 Satake parameters for the weight 16 cusp form
- 5000 Satake parameters for the weight 18 cusp form
- 1000 Satake parameters for the first weight 24 eigen cusp form
- 1000 Satake parameters for the second weight 24 eigen cusp form
- 1000 Satake parameters for the Elliptic curve y2 = x3 - x
- 1000 Satake parameters for the Elliptic curve y2 + y = x3 - x