### Elliptic Curves

The original Sato-Tate conjecture wasn't about modular forms, but about a class of geometric objects called elliptic curves. These curves are defined by equations similar to the following:p | N(p) | a_{p} | |a_{p}|p^{-1/2} |

2 | 2 | 0 | 0 |

3 | 3 | 0 | 0 |

5 | 7 | -2 | 0.89 |

7 | 7 | 0 | 0 |

11 | 11 | 0 | 0 |

13 | 7 | 6 | 1.66 |

17 | 15 | 2 | 0.49 |

19 | 19 | 0 | 0 |

23 | 23 | 0 | 0 |

29 | 39 | -10 | 1.86 |

31 | 31 | 0 | 0 |

37 | 39 | -2 | 0.33 |

_{p}defined by

_{p}between 0 and π. Sato and Tate conjectured around 1960 that these angles satisfy the sin

^{2}law, that is, they are distributed according to the function

By the celebrated Modularity Theorem, the Sato-Tate conjecture
for elliptic curves is actually a consequence of the Sato-Tate conjecture for modular forms.
Namely, the Modularity Theorem states that the numbers a

_{p}defined above for an elliptic curve are in fact the Fourier coefficients of a modular form (a cuspidal eigen-newform of weight 2, to be precise). Hence, if we know the sin^{2}law for modular forms, we know it for elliptic curves.
We should remark that the modular forms in the Modularity Theorem are not exactly
of the type that we defined on page 2. It is necessary to consider modular forms
with ''level''. For example, the modular form corresponding to the elliptic curve
y

^{2}+y=x^{3}-x above is the essentially unique cusp form of weight 2 and level 11.
There is still more - go to page 4.