May 12 2010 announcement: Exercise 3.2b on HW5 is now optional. You can turn it in for extra credit at the end of the term.

Homework Assignments here.

Information about the final paper assignment here.

Basic information about typesetting is available here.

**Introduction**

Math 75 is a class typically taught every other year in the spring, which covers a different topic each time it is taught. This year, the class will cover the introductory theory of elliptic curves. Students who took the class two years ago may take the class again, since the topic changes with each iteration of the class.

The official prerequisites for this class are either a class in algebra or a class in number theory. In reality, knowledge of algebra is probably more useful than number theory, though having taken both classes is probably best. At certain points in this class, knowledge of complex analysis (Math 43) and a little bit of topology (Math 54) might be useful. It is a hallmark of number theory to draw techniques from a wide range of disciplines, but in this class we will briefly discuss any background material which might be necessary, most likely without proof.

**Instructor Information**

Name: Andrew Yang

Office: Kemeny 316

Office Hours: Monday, Wednesday, Friday 1:30pm - 2:30pm

**Grading**

Grading will be based on weekly problem sets and brief presentations of solutions to those problems in class. The weekly sets will be posted on this website. The presentations will be assigned several days in advance and will take place at the beginning of classes. For at least the first few weeks, if you are assigned to present a problem, you should meet with me prior to your presentation so we can briefly discuss your solution and what you might be expected to do.

There may also be a final assignment in the form of a short paper on a topic of your choice. A decision on whether such a paper will be required will be made in the next few weeks.

**Homework Policy**

Homeworks will be posted on this website and usually due about a week after they are posted. Most problems will be from the text, so make sure you have access to a copy.

Collaboration is allowed, but as usual, you are expected to write up your own solutions in your own words. It is suggested that you at least try each problem on your own for a while before working with others, so that you get a good feel for the ideas behind each problem.

**Textbook**

The required book for this class is *Rational Points on Elliptic Curves*, by Joseph Silverman and John Tate, ISBN 978-0387978253.

At the conclusion of this class, you might be interested in reading the two books *The Arithmetic of Elliptic Curves* and *Advanced Topics in the Arithmetic of Elliptic Curves*, both by Joseph Silverman. These present the theory of elliptic curves from a much more modern point of view (algebraic geometry is used heavily) and are useful references if you are interested in learning more about elliptic curves.

In a few days I will provide a reference page which will contain a variety of different books and articles you may want to consult during and after the class. They will be of widely varying difficulty, so some references may not be so easy to read.

**Course description**

Our goal in this class is to cover as much of the textbook as possible in the term. After giving a quick overview of the background algebra and number theory you should know in this class (if this is necessary), we will start by giving a naive description of what an elliptic curve is and describe how the study of elliptic curves fit into the larger subject of Diophantine geometry.

One remarkable feature of elliptic curves is that it is possible to give a natural group law on the set of points of an elliptic curve. This additional algebraic structure is what makes the study of elliptic curves so interesting. We can ask what the set of points on an elliptic curve "looks like" under various restrictions on what sets those points must lie in - for example, from the perspective of number theory, we might be interested in the set of rational points on an elliptic curve. We will examine points on elliptic curves considered over complex numbers, real numbers, rational numbers, finite fields (perhaps only of prime order), and integers. Along the way, we will see various applications of elliptic curves to problems of practical interest, such as integer factorization.

One of the distinguishing features of number theory is the wide range of techniques that are used to solve problems. The subject of elliptic curves is no different, and in this class we will come across a wide range of different techniques to answer natural number-theoretic questions. Hopefully, by the end of the class, you will have a good foundation on which to explore the theory of elliptic curves, or other topics in number theory which use some of the techniques we encounter in this class.

** Weekly schedule **

This schedule is preliminary and will almost certainly be adjusted over the course of the term.

Week 1: Introduction, Background Material on algebra, number theory

- Algebra: review of basics on groups, abelian groups, cyclic groups, finite generation, fundamental theorem of finitely generated abelian groups, homomorphisms, isomorphisms, quotient groups
- Number theory: Integers mod n, Fermat's Little Theorem, rational root theorem
- Reading: Introduction (this isn't related to the above, but provides a good overview of what the textbook, and by extension, this class covers)

Week 2: Basics on projective space, Bezout's Theorem, number theory on conics

- Projective space: definitions, relation to affine space, specific examples in two dimensions
- Bezout's Theorem: intersection multiplicity, statement of Bezout's Theorem (no proof), examples
- Conics: rational points on conics, Pythagorean triples, brief discussion of the Hasse-Minkowski local-global principle
- Reading: Appendix A1 - A3 (read A4 if you want proofs), Chapter I.1

Week 3: Definition of group law on cubic curves

- Weierstrass normal form
- A geometric definition of the group law
- Explicit formulas for the group law, examples
- Reading: Chapter I.2 - I.4

Week 4: Points of finite order

- The complex points of an elliptic curve: uniformization (no proofs)
- The Nagell-Lutz Theorem, its proof
- Reading: Chapter II

Week 5: Rational points on elliptic curves, the Mordell-Weil theorem

- Height functions on rational numbers, height functions on rational points on elliptic curves
- Proofs of various lemmas on height functions
- Reading: Chapter III

Week 6: Rational points on elliptic curves, the Mordell-Weil theorem continued

- More proofs of various lemmas
- How to calculate the rank of elliptic curves in special cases
- Reading: Chapter III (continued)

Week 7: Points on elliptic curves mod p

- A special case of the Hasse-Weil theorem
- Using points mod p to determine points of finite order over rationals
- Reading: Chapter IV.1 - IV.3

Week 8: Using elliptic curves to help factor large composite numbers

- Fast algorithms for exponentiation mod n, finding gcds
- Pollard's p-1 algorithm
- Lenstra's elliptic curve algorithm
- Reading: Chapter IV.4

Week 9: Integer points of elliptic curves, Siegel's Theorem

- Statement of Siegel's Theorem, overview of ideas
- Lemmas on Diophantine approximation, proofs of lemmas
- There is no chance for us to cover all this in a week, so we will either start this topic in the previous week, and/or omit the proofs of certain lemmas
- Reading: Chapter V