Math 621, Spring 2015
Instructor:
Ching-Li Chai
Office: DRL 4N36, Ext. 8-8469.
Office Hours:
Email:chai@sas.upenn.edu
General Information
- Lectures: MW 12:00--1:30 PM, 4C8 DRL
First meeting: Wednesday, August 27, 2014.
- Topics:
Elliptic functions, elliptic curves and modular forms
- General plan:
The general goal here is to give a connected account of the
closely related ideas of elliptic functions, elliptic curves
and modular forms, along a somewhat classical line to
get to the interesting phenomena as directly as possible.
We will start with the definition of Weistrass elliptic functions
(as doubly periodic functions on the complex plane) and the
projective embedding of elliptic curves they provide.
We will also discuss the Weistrass zeta and sigma functions,
the quasi-period, and the Jacobi theta functions with half-integral
characteristics.
Modular functions and modular forms appear naturally as Fourier
coefficients of the Weistrass elliptic function, in the form
of Eisenstein series. Modular forms of higher levels appear
as values of elliptic functions at torsion points of elliptic curves
in this context. We will see that interesting arithmetic
consequences pour out from relations between modular forms
constructed from different sources.
The theory of complex multiplication of elliptic curves will be
introduced first from an analytic point of view. Later we will
see that they provide an systematic way to construct abelian
extensions of imaginary quadratic fields.
To get a better idea on the topics mentioned above, you may want to
take a look at Lang's book on elliptic functions in the reference
section below. Although there won't be an "official textbook",
the mix and balance of algebra and analysis found in Lang's book
is the closest in spirit.
- Prerequisite:
Basic notions in algebra and complex analysis,
as covered in math 602 and 608. No prior knowledge on
algebraic number theory will be assumed.
- Textbook: There is no official textbook for this course.
However most of the materials can be found in
Lang's Elliptic Functions in some form.
- Reference Books:
The following books are
recommended as references.
- Much of the above are also treated in the first five chapters of
Shimura's red book Introduction to the Arithmetic Theory of
Automorphic Functions.
- Zagier's article Elliptic modular forms and their applications in
The 1-2-3 of Modular Forms is fascinating and entertaining at
the same time.
- The book Rational points on elliptic curves by Silverman and
Tate, based on Tate's lectures at Haverford College, treats
elliptic curves from an algebraic perspective.
- For those interested in exploring classical texts, chapters 20-23
of Whittaker and Watson's A Course of Modern Analysis are highly
recommended. We are unlikely to go into ch. 21 (on Jacobi elliptic
functions) and ch. 22 (Lame functions), but they are interesting.
- Kobilitz's Introduction to elliptic curves and modular forms
not only introduces modular forms, but also its application to
the BSD conjecture and the congruence number problem.
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