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.

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

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.)

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)

Svetlana Katok's book

Moon Duchin's course on Fuchsian groups

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.

IHES lectures by Genestier and Ngo

Milne's Notes

Lang's Introduction to Algebraic and Abelian Functions