Number Theory @ SDU---Please Enjoy Here!

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FINDING MEANING IN ERROR TERMS
BARRY MAZUR
In memory of Serge Lang
Introduction

Four decades ago, Mikio Sato and John Tate predicted the shape of probability distributions to which certain “error terms” in number theory conform. Their prediction—known as the Sato-Tate Conjecture—has been verified for an important class of cases thanks to the recent work of Laurent Clozel, Michael Harris, and Richard Taylor [3], and of Michael Harris, Nicholas Shepherd-Barron, and Richard Taylor [18], combined with Richard Taylor’s most recent [54], which establishes this advance in our understanding.

Part of the beauty of this breakthrough is how it pulls together progress made over the past quarter century and work from significantly different viewpoints—from the theory of automorphic representations, from algebraic geometry, and from Galois deformation theory—a demonstration, yet again, of the intense unity of mathematical thought.

My aim is to discuss, in concrete terms, two “sample problems”—one still open, and one settled by the recent work—that give rise to error terms, about which the Sato-Tate Conjecture makes precise predictions.

http://www.ams.org/bull/2008-45-02/S0273-0979-08-01207-X/home.html