Three Lectures on Shimura Curves

These notes are taken from a lecture series in three parts given at the University of Sydney in April 2006. In the rst part, we introduce Shimura curves for the non-expert, avoiding technicalities. In the other two sections, we discuss the moduli interpretation of Shimura curves and treat some of the computational aspects. 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 rst 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 the rst section, 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. In the second section, we switch gears to treat a much more technically involved subject, intended for the reader who wishes to know more details. Over the complex numbers, a Shimura curve is simply a Riemann surface which is uniformized by an arithmetic Fuchsian group. Such curves in fact have a much richer structure: they are (coarse) moduli spaces for certain abelian varieties with \extra endomorphisms". We present an abbreviated account of this theory which will focus on the cases of curves over the rational numbers. In the nal section, we treat computational aspects of Shimura curves. The study of the classical modular curves has long proved rewarding for number theorists both theoretically and computationally, and an expanding list of conjectures have been naturally generalized from this setting to that of Shimura curves. Recently, computational aspects of these curves been explored in more depth and we discuss some of the central algorithmic problems in this area. These notes are in rough form and are very underdeveloped, so comments and requests are welcome! 1. Survey of Shimura curves In this section, we provide an extended and hopefully well-motivated introduction to Shimura curves. The expert reader should excuse a few white lies, which are made for the purposes of exposition. We will start at the very beginning and at each stage see how Shimura curves arise as a natural generalization. 1.1. The j-line. The most \famous" Shimura curve is the j-line, which over the complex numbers C is just the complex plane, with coordinate j; by stereographic Date: April 16, 2006. 1