**Final paper information **

In lieu of a final examination, every student in the class will be required to submit a paper on a selected topic related to the material in this class as a final project. This page contains an overview of what's expected, schedules for various due dates, and a list of possible topics.

**Introduction**

There are several purposes for this project. First and foremost, it will give you an opportunity to explore a subject you might find interesting but which we did not have a lot of time to cover in class. By writing a paper on that topic, it is likely that you will learn the content of your paper rather thoroughly. Many topics can serve as starting points into longer and deeper investigations into the subject of elliptic curves. On a more practical level, if you find your topic interesting and you are a non-senior, there is a possibility for your paper topic to lead naturally to a senior thesis.

Besides the mathematical learning, another objective of this project is to give you a beginning experience in mathematical exposition. This is especially important for students who want to go to graduate school in mathematics, but even for those who are not, learning how to write about a technical subject accurately while still being comprehensible is a skill which is useful in other subjects or in many professions.

A final objective is to teach you how to use LaTeX, which has become the standard typesetting tool in not just mathematics, but also much of theoretical computer science and theoretical physics. Basic LaTeX is not hard to learn, but is much better than traditional word processors when using lots of mathematical notation and equations. Strictly speaking, you will not *need* to use LaTeX, but I strongly suggest that you do.

**Requirements**

The final product of this project should be a paper about five or so pages in length, on a specific topic that you select a little more than a month before the final due date. The paper does not need to contain complete mathematical proofs of whatever you are writing about - as a matter of fact, for certain topics, it may very well be impossible to write any proofs at all. Unlike homework assignments, where you need to carefully justify every step, this paper will be more like an expository assignment, where you survey results on the subject you are writing about and discuss how they might be related to what we've been learning in class. If proofs are accessible and of reasonable length, then by all means feel free to include them. Whether you should include proofs and/or detailed mathematical calculations is going to be something we will discuss, because it will heavily depend on the subject you are writing about.

When typesetting your paper, broadly speaking you will have three options available to you:

- Option 1: Use LaTeX to write your paper. This is by far the most strongly suggested method, both because it makes it easier for me to read, and more importantly, because this is a useful skill for anyone who intends on writing content with mathematical formulas to know. Basic LaTeX is not hard or time-consuming to learn, and links to resources on how to start with LaTeX will be provided on this page in the near future.
- Option 2: Use a traditional word processor. This is more feasible if your topic does not require a tremendous amount of mathematical notation or formulas.
- Option 3: Write your paper by hand. This is the least preferable method, for a variety of reasons.

**Due Dates**
Prior to the final due date of the paper, there will be two due dates for various steps of this project:

Due Date 1: May 5, 2010. This is the date at which you will need to select a topic to write about. Before making your final decision, you will need to schedule a meeting with me to briefly discuss your topic. We will talk about how feasible your prospective topic is, what I might expect in the paper, and perhaps most importantly, I can provide you a list of potential readings for your topic. You should email me no later than May 1, 2010 to set up a time for a short meeting, although you can email me as early as you want.

Due Date 2: June 1, 2010. This date is when you should submit a draft of your paper to me. You should pretend that this is your final submission, so everything should be at a polished and finished level. I will read your paper and make various comments; if necessary I will schedule a meeting with you sometime during the week of June 1 to go over those comments. You can submit your paper as early as you want; one advantage of doing so is that I will probably have more time to provide detailed comments or recommendations.

Due Date 3: June 7, 2010. Your final paper should be submitted on this day.

When you submit your draft and your final paper, you should do so in a physical form - that is, do not email me a pdf, give me a link to a webpage, or submit the TeX source file.

Since this project is probably going to take a substantial amount of time, it is likely that we will have shorter, or possibly no, homework assignments the last two weeks of class.

**Possible topics**

First and foremost, feel free to suggest any topic that interests you, as long as it is related to what we are learning in this class. Your topic can be a detailed overview of something which was glossed over in the text, or a survey of more advanced topics which are not covered in the book but perhaps would be covered in a hypothetical sequel class.

After each topic there is brief commentary on possible pre-requisites and the type of work you might have to do to research a topic.

*Weierstrass normal form and birational transformations*:

We glossed over the reason why Weierstrass normal form really is completely general, so in this paper you would fill in those gaps. You might also discuss the more general theory explaining why group laws on cubic curves are preserved by birational maps.*Primality testing using elliptic curves*:

This is in contrast to the factorization algorithm we will learn about in this class.*Complex points of an elliptic curve*:

In the class, we take lots of facts about complex points of an elliptic curve on faith, so in this paper you would explain why those facts are true. You will need to know some complex analysis (Math 43) for this topic.*Stepanov's elementary method for bounds on number of points of elliptic curves over finite fields*:

We do not prove the Hasse-Weil bounds in this class, except possibly in a very special situation. This paper would discuss one method (there are several) for proving such a bound. There isn't a lot of material to read, but you would probably need to learn a little bit of the language of algebraic geometry.*Various topics on Diophantine approximation*:

A crucial ingredient in the proof of Siegel's Theorem are results in the subject of Diophantine approximation, which involves questions of how well we can approximate real numbers using rational numbers. It seems unlikely that we will cover much of Chapter 5, if we even reach it, so this paper would either discuss the Diophantine results used in Chapter 5, or other results of a similar flavor in Diophantine approximation.*L-functions of elliptic curves*:

It is possible to define a function, known as an L-function, attached to an elliptic curve defined over the rationals. This is related to the Riemann zeta function, Dirichlet L-functions, and many other types of L-functions. L-functions have proven to be central in number theory, but they turn out to be rather difficult to define in certain situations. Such a paper would discuss the definition and properties (either proven, or unproven but believed to be true) of such functions, and perhaps discuss a few famous conjectures or results about these functions. You would probably need to know some complex analysis (Math 43) and be willing to read a lot of relatively difficult material for this important topic.*Ranks of elliptic curves*:

The rank of an elliptic curve is still one of the least understood quantities attached to an elliptic curve. In this paper you would survey various results on ranks of elliptic curves. There has been a lot of progress in this field in the recent past, so there's no shortage of material to write about.*The Sato-Tate conjecture for elliptic curves*:

The Sato-Tate conjecture is a statement (acutally proven in many cases now) about the distribution of the numbers in the Hasse-Weil bound. There is no chance that you would be able to write about the actual proof of the cases of the Sato-Tate conjecture that are known now, but you could describe the conjecture and perhaps provide some numerical calculations in certain concrete cases illustrating the conjecture.

Remember, your topic does not have to come from the above list! You can write about anything which interests you, as long as you discuss your idea with me so that we know that your topic is a feasible one for a paper!