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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.
nτ(n)
11
2-24
3252
4-1472
54830
6-6048
7-16744
884480
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
20.53033
30.598734
50.691213
70.376548
111.00087
130.431561
171.17965
190.987803
230.603975
291.16251
9970.688016
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.
Sato-Tate Graphs for the Ramanujan τ-Function (1000 Parameters, Step 1)
Parameters:
The following display is similar, except that we are working with the first 5000 primes and a step size of 50.
Sato-Tate Graphs for the Ramanujan τ-Function (5000 Parameters, Step 50)
Parameters:
There is more - go to page 2.
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