Math 546: Modular and Shimura Curves

Modular and Shimura curves are analytic and algebraic objects which have seen renewed interest. The aim of this course will be to introduce modular and Shimura curves, first from the perspective of complex analysis and then from the perspective of algebraic geometry. I will not presume any knowledge of quaternion algebras, the previous semester's course was WK Chan's course on quaternion algebras and as such I may be a little bit brief when it comes to substantive questions on that topic.

For the first half of the course, we will discuss modular and Shimura curves over the complex numbers. I will assume a very good grasp of complex analysis as I will basically start off from the uniformization theorem on simply-connected complex domains. I will describe general quotients of the upper half-plane, Fuchsian groups, the cusp calculus and the horocycle topology. I will then introduce modular and Shimura curves, their cusps and elliptic points and will conclude the first half with their connection to complex tori, abelian varieties, and if possible modular forms.

For the second half of the course, we will further explore the connection between modular or Shimura curves and abelian varieties. In particular, we will introduce the concept of a moduli space, and explore these curves from a "modular" perspective. I will introduce concepts from algebraic geometry as I need them in the pursuit of modular and Shimura curves over number fields and their reduction modulo p. A full treatment of that topic may be out of our reach, but there is a lot of interesting mathematics to be exposed here.

Schedule for first six weeks

Riemann Surfaces/Uniformization
Fundamental Groups and Group Actions
Fuchsian Groups, Volumes and Measures
Siegel's Theorem
Quaternion Algebras
Traces and the Geometry of Shimura Curves
The horocycle topology and compactified Riemann surfaces
The Fundamental Equation and Embeddings
Embeddings, Involutions, and Normalizers
Complex Tori and Abelian Varieties
The connection to Shimura Curves and modular forms
Involutions and Hecke Correspondances
Some Rough notes from the complex-analytic section

Tentative Schedule for the last seven weeks

Complex Algebraic Curves and the Riemann Existence Theorem
Belyi's Theorem and non-complex algebraic curves
The Zariski Topology
Tensor Products
Base Change, More Tensors, Flatness
Sheaves, Schemes, Separatedness, Properness
A First Look at Rational Points, Projective space, and Proj
A p-adic upper half-plane
Another look at Rational Points
(A slightly advanced supplemental on when you have to sheafify a functor in the Zariski topology, written forever ago when I was studying group schemes for myself. This is unedited, so be gentle.)

References on Riemann Surfaces

Constantin Teleman's notes from a Cambridge course on Riemann surfaces, an excellent expose of the basics of Riemann surfaces.
Rick Miranda's book, which is a really excellent expose of some more difficult concepts in the study of Riemann surfaces. I personally used this book to teach myself much of this material.
Proofs of the Uniformization Theorem, a Harvard senior thesis by KT Chan.
The Uniformization Theorem, another set of proofs by W Abikoff. (JStor link: read only on campus)

References on Fuchsian Groups

Svetlana Katok's book
Moon Duchin's course on Fuchsian groups

References on modular curves

Shimura's 1974 book on Automorphic functions
Dick Hain's "Lectures on moduli spaces of elliptic curves", which works entirely in the complex category but is generally unafraid of non-free group actions.

References on Abelian Varieties

IHES lectures by Genestier and Ngo
Milne's Notes
Lang's Introduction to Algebraic and Abelian Functions