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:^{n}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 |

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 |

_{p}, the Frobenius angle, between 0 and π such that

_{p}are distributed according to the function

^{2}(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.

There is more - go to page 2.

Or click on the red box on top of the page in order to navigate.