This page was created by Julian Rosen and Ralf Schmidt at the University of Oklahoma Mathematics Department.
Our goal is to give an easy introduction to the Sato-Tate Conjecture
in the theory of modular forms and elliptic curves.
What is this about?
The Sato-Tate Conjecture is a statement about the statistical distribution
of certain sequences of numbers. As an example,
consider the following formal product in the variable q:
If we multiply everything out, we get a formal power series:
The coefficient of qn in this power series is traditionally called τ(n),
and the function that maps n to τ(n) is called the Ramanujan τ-Function.
n
τ(n)
1
1
2
-24
3
252
4
-1472
5
4830
6
-6048
7
-16744
8
84480
9
-113643
10
-115920
997
-21400415987399554
The numbers τ(n) have several remarkable properties, one of which is
Another property is
for p a prime number and r a positive integer. It is therefore enough to know the
values τ(p) for prime numbers p. Ramanujan had conjectured in 1917, and Deligne
proved in 1970, that
p
|τ(p)|p-11/2
2
0.53033
3
0.598734
5
0.691213
7
0.376548
11
1.00087
13
0.431561
17
1.17965
19
0.987803
23
0.603975
29
1.16251
997
0.688016
for prime numbers p. Therefore, there exists a unique angle
&phip, the Frobenius angle,
between 0 and π such that
The Sato-Tate conjecture asserts that the angles φp
are distributed according to the function
The following displays allow you to explore this conjecture. The left graph below shows
the function (2/π)sin2(φ) and
the location of the first Satake parameter. You can view the distribution of more
parameters (up to 1000) by clicking on the buttons. On the right side we have an
animation running through these first 1000 graphs.
The following display is similar, except that we are working with the first 5000 primes
and a step size of 50.
