The University of Sydney
School of Mathematics and Statistics  
Number Theory Seminar  

University of Sydney Number Theory Seminar

John Voight

Thuesday 4 April 2006, 3:05-4:00PM, Carslaw 373

Shimura curves: A survey

Ever wondered what a Shimura curve is? They lie at the crossroads of many areas of mathematics: complex analysis, number theory, Diophantine equations, group theory, noncommutative algebra, algebraic geometry, Lie theory--even coding theory! The study of the first examples of these curves (the modular curves) can be traced back as far back as Gauss, and then later Klein and Fricke; recently, they have played an important role in the proof of Fermat's last theorem and in the solution of other number theoretic problems. In this survey, we introduce Shimura curves for the non-expert with an algebraic outlook and provide a brief exposition of their relationship to other areas of mathematics.