Shimura curves 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 a general audience with an outlook toward computational algebra and discuss some of the central algorithmic problems for Shimura curves.

This talk is an abbreviated and repeated version of talks #1 and #3 from April 4 and 11.