In 1846, Ernst Eduard Kummer conjectured a distribution of values of a cubic Gauss sum after computing a few values by hand. This was forgotten about for nearly 100 years until John von Neumann and Herman Goldstine attempted to verify the conjecture as a way to test the new ENIAC machine in 1953. They found evidence that the conjecture was false, but trusted Kummer more than they did their digital computer. The conjecture would hold until 1979, when Roger Heath-Brown and Samuel Patterson proved it to be false.
A few years earlier in 1965, Mikio Sato and John Tate independently came up with a conjecture which gave the correct distribution of these cubic Gauss sums — although it was expressed slightly differently in terms of counting points of elliptic curves over finite fields. In this talk, we give an overview of the Sato-Tate Conjecture, present an approach by Jean-Pierre Serre following his paper from 1967, then sketch the 2006 proof of the conjecture following the ideas of Laurent Clozel, Michael Harris, Nicholas Shepherd-Barron and Richard Taylor.
Here are the slides of this lecture: Edray Goins’ slides.