Math 788 -- Elliptic Curves and Arithmetic Geometry, Spring 2016

Instructor: Frank Thorne, LeConte 317O, thorne [at] math [dot] sc [dot] edu.

Office hours: Mondays 9:00-12:00, immediately after class, or by request.

What is arithmetic geometry?

Arithmetic (or diophantine) geometry is, equivalently:

Course objectives and learning outcomes:

Successful students will:

Extraordinarily successful students will:

Course prerequisites:

Hard prerequisites: Abstract algebra (701/702 or equivalent; concurrent enrollment okay), and elementary number theory (780 or equivalent, or willingness to learn this material on the side).

Soft prerequisites: Occasionally other mathematical disciplines will be brought in, especially algebraic geometry and algebraic number theory. The student who has studied these topics before will get the most out of the course.

Many students will not have had these prerequisites. Occasionally the course will go over their heads (I hope not too badly!) but that is par for a topics graduate course in any case. Such students will be okay -- it is hoped that the course will motivate them to learn a little bit on the side and to study these topics in depth later.

Course Textbooks:

Large portions of the course will simultaneously follow two books on elliptic curves. These are: The student should purchase and follow along in one (or both) of these books, according to his or her background and aims.

Required Work:

There are two components to your grade. Each counts for half, and for the first there are three options:

Written Work:

The "exercises" refer to Option 1 and the "problems" to Option 2.

Seminar Participation and Reports:

The student should regularly attend one or more of the following seminars: the Algebraic Geometry, Arithmetic Geometry, and Commutative Algebra Seminar, the Number Theory Seminar, and the Department Colloquium.

Students should write reports on each of at least six talks. (These talks should be given by six different speakers, and USC students or faculty don't count.) Reports should be at least one typeset page, and might discuss the speaker's main results, background material which the speaker presented, and questions you have. In writing your reports, you are encouraged to supplement your lecture attendance by reading outside material.

To set a good example I'll do it with you! Please see here for my list, as well as a LaTeX template which you may use if you like.

This is great practice for mathematicians at any career stage. If you don't believe me, then you should believe Ravi Vakil, David Zureick-Brown, Bjorn Poonen, etc.

If you attend any conferences related to arithmetic geometry, you are also welcome to report on talks given there. Here is an incomplete lists of some conferences worth attending. I recommend especially the Arizona Winter School, which is intended specifically for graduate students. (But that won't stop me from showing up myself.)

Grading Scheme:

Students are guaranteed an A for 80%, B for 60%, C for 40%, D for 20%.

Lecture Notes

Warning: This is what I planned to say. It may be very different from what I actually said.

Sources: Large portions of these notes closely follow Silverman, Silverman-Tate, and my notes from Nigel Boston's course on the subject.

Other References

Arithmetic Geometry:

Two other great books on elliptic curves are Knapp, Elliptic curves and Washington, Elliptic curves: number theory and cryptography. These cover similar material at a level intermediate between Silverman-Tate and Silverman. In particular you can read them with little or no knowledge of algebraic number theory. The Washington book (as may be inferred from the title) also covers cryptographic applications of elliptic curves (I haven't read this part).

You might also see McKean and Moll for an interesting approach emphasizing topology. Another good book is Koblitz's Introduction to elliptic curves and modular forms. It has the friendliest introduction to modular forms of half-integral weight of which I am aware.

A wonderful advanced book is Hindry and Silverman's Diophantine Geometry. (But Do Not Read Part A.) Their book is very much not limited to elliptic curves. There are also a wealth of outstanding, still more advanced books. See David Zureick-Brown's page for advice and further links.

There is also Sutherland's lecture notes, available free here from MIT OpenCourseWare.


A generally useful book is Dummit and Foote's Abstract Algebra. It has excellent brief introductions to subjects such as representation theory, Galois cohomology, etc. which will mostly suffice for this course. Lang's Algebra is also excellent, especially if you are not an absolute beginner. If you are using Aluffi, note that the categorical perspective won't be adopted heavily here.

Algebraic Geometry:

A good all-around (and inexpensive) book is Hulek's Elementary Algebraic Geometry. It contains pretty much all the algebraic geometry you'll need for this course.

Other excellent reads include Smith, Kahanpaa, Kekalainen, Traves's An Invitation to Algebraic Geometry and Harris's Algebraic Geometry: A First Course. Anyone wishing to seriously master the subject should master the theory of schemes: read Hartshorne, Algebraic Geometry, or Vakil, The Rising Sea: Foundations of Algebraic Geometry. (By "read" I mean, as usual, "do all the exercises".)

Elementary Number Theory:

I recommend Ireland and Rosen or Hardy and Wright, or the lecture notes from Matt Boylan's course. Another excellent resource is Filaseta's lecture notes.

Algebraic Number Theory:

The gold standard is Neukirch, Algebraic Number Theory. An excellent free alternative is Milne, Algebraic Number Theory.

Very Rough, Tentative List of Topics:

This will be refined based on student background and interest.