Topics in Algebra - MATH 649J 001

Elliptic Curves

Instructor -- Pavel Guerzhoy

The class meets on Tuesdays and Thursdays, 10:30 - 11:45am at 313, Keller Hall.

Office: 501, Keller Hall (5-th floor)
tel: (808)-956-6533
e-mail: pavel(at)math(dot)hawaii(dot)edu
Usually, I respond to e-mail messages within a day. If I do not, that means that I do not have a reasonable answer, and am still thinking.
Office hours: Tuesdays and Thursdays 1 - 4pm, and by appointment, whenever the participants have questions to ask or discoveries to report.

Class goals and content
Elliptic curves is a huge domain of contemporary research in Mathematics. In this class, we do not intend to cover all (or even a significant part of) what people know about the subject, which is simply impossible. Among many impressive applications, the theory of elliptic curves was heavily involved into the celebrated proof of Fermat's Last Theorem. Specifically, the Theorem was derived from Taniyama - Weil conjecture stating (very roughly) that
Every rational elliptic curve is modular.
All you may learn about Wiles' proof of this conjecture in this class, is that (see the epigraph in the relevant chapter in Milne's book)
the proof of Fermat's Last Theorem is indeed very complicated.
However, you will gain some understanding of the claim made in the conjecture, which is already not an overly easy task.
The conjecture mentioned above was formulated after a series of results commonly referred to as Eichler - Shimura theory was obtained in early 1970-th. It then became natural to ask whether the converse was true, and the conjecture claims an affirmative answer. The ultimate goal of this class is to provide an overview of the Eichler - Shimura theory.

In this class, we primarily use the following textbooks:
  1. J. Silverman, The Arithmetic of Elliptic Curves , second edition, Graduate Texts in Mathematics, 106, Springer
  2. A. Knapp, Elliptic Curves , Mathematical notes, 40, Princeton University Press
There are tons of other books which may be very useful. I will bound myself by mentioning only two of them. Note. I have electronic versions of the above books (and some other relevant materials) and will share them with students upon request.

Specific class and homework description
The class splits naturally into two parts. Each part entails a homework assignment.