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