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 (5th floor)
tel: (808)9566533
email: pavel(at)math(dot)hawaii(dot)edu
Usually, I respond to email 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 1970th. 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.
Literature
In this class, we primarily use the following textbooks:
 J. Silverman, The Arithmetic of Elliptic Curves , second edition, Graduate Texts in Mathematics, 106, Springer
 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.
 J. S. Milne Elliptic Curves , BookSurge Publishers, 2006
 R. Miranda, Algebraic Curves and Riemann Surfaces, Graduate Studies in Mathematics, 5, American Mathematical Society
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.
 Part 1
We start with a (very condensed) introduction to elliptic curves as algebraic varieties of dimension one.
Here we follow (with very few small alterations) Chapters 1,2,3, and 5 from Silverman's book.
We rush through this material quickly; our principal goal here is to acquire a dictionary.
We want to get to elliptic curves and their basic properties as soon as possible.
In order to make sure that you indeed understand basic constructions, facts, and properties of the objects involved, you
do the following assignment.
Homework 1
Think of two interesting (in your opinion) questions related to the material discussed in
Chapters I and II in Silverman's book.
Answer your questions, write everything down, and submit your work as the homework assignment. If your question turns out to
be too difficult to answer (that happens: it is frequently mostly difficult to answer the questions which are easy to ask)
a description of your attempts to approach your question make it a homework assignment absolutely worthwhile to submit.
The due date for this assignment is a week after Chapter II from Silverman's book is covered.
Homework 2
Think of two interesting (in your opinion) questions related to the material discussed in
Chapters III and V in Silverman's book.
Answer your questions, write everything down, and submit your work as the homework assignment. If your question turns out to
be too difficult to answer (that happens: it is frequently mostly difficult to answer the questions which are easy to ask)
a description of your attempts to approach your question make it a homework assignment absolutely worthwhile to submit.
The due date for this assignment is a week after Chapter V from Silverman's book is covered.
For both Homework I and II assignments, you may and are encouraged to discuss your questions with me.
Some clarifications.
Doing this assignment, you may use everything (e.g. material from any book, any lecture notes, anything found on the web,
asking anyone, me included, etc.) Please, however, disclose your sources: that makes it easier for me to read your assignments.
You should have tons of questions: if an avalanche of constructions and facts left you without questions, you probably have
to read the textbook again. You just have to choose two of them. To make it easier, here are some examples, and you may think of more and better.

I am surprised about Theorem XX: I cannot understand why that may be true at all.
I thus decided to see how the statement holds true in a specific example. Here is a description of the example. [...]
The following calculation [...] now makes it clear that the claim of the Theorem, in this example, reduces to 2+2=4.

I am surprised about Theorem YY: its conclusion is false in the following example. [...]
After a bit of thinking, I realized that my example violates the condition that [...].

It looks like Theorem AA contradicts Theorem BB. The reason is roughly the following[...].
After a bit of thinking I realized that while Theorem AA applies when x < 0, Theorem BB applies when x > 1.
These conditions are not given explicitly, but they follow from the calculation [...].
It is now interesting, what happens when 0 < x < 1, and both theorems do not apply.
In the example which I found elsewhere on the web, it seems that x=0.5, and the situation is as follows[...]

Theorem ZZ claims that every elephant is a mammal. It is natural to inquire whether the converse is true.
An extensive search on the web allowed me to find a mouse. Here is how one can construct it. [...]
Here is a proof that a mouse is not an elephant. [...]
 Part 2
In Chapter 6 of Silverman's book, we find that elliptic curves over complex numbers are Riemann surfaces.
After a brief digression on Riemann surfaces (Knapp's book, XI.4; no homework here because the subject belongs to analysis while
this is a "topics in algebra" class) we introduce modular curves (in a way quite parallel to one of the possible ways
to introduce elliptic curves, Knapp's book XI.2) and modular forms (a special case Eisenstein series appears already in Chapter 6).
That naturally leads us to start a discussion of Hecke operators (Knapp's book, VIII.7 and IX.6). From this point on, we are
in a position to follow the proof of the main theorem (Theorem 11.74 in Knapp's book) as it is presented in Chapter XI of
Knapp's book.
Homework 3
Here are descriptions of two research projects:
Project1
It is possible that the principal question asked in Project1 may be attacked using the theory developed in the book
Glenn Stevens, Arithmetic on modular curves, Progress in Mathematics, Vol. 20.
However, it is not at all clear (for me) how to do that.
All I can say now is that this book may be quite useful for the project.
Project2.
You have to choose one (you may try both if you wish), and make some progress.
Collaborative work is allowed. It is encouraged to discuss with me any questions/observations/suggestions as soon as you
have just anything to share.
IMPORTANT! Please, ask questions if anything is unclear in the project descriptions.
There may be, and probably are vague points and irregularities which I will be happy to correct. That would help everyone.
The worst case scenario is you better understand related mathematics after additional discussions.
Remember that in Math there is no such thing as a stupid question.
These are indeed research questions (meaning that I really do not know answers.) In this case,
just any progress (a small observation, a relevant idea found in the literature, a numerical example calculated) would be highly appreciated.
Numerical experiments are particularly encouraged: this is a way to observe numbers alive, and to gain understanding.
For these, if you are enthusiastic to conduct such, I recommend GP/pari calculator which you can download for free from
this website. Also very good calculators (if you know how to use them; I do not)
are magma which has a free online inbrowser service
here and sage.