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:
• The study of solutions to polynomial equations, especially over the integers and over the rationals. For example a classical problem is to prove that the equation a^n + b^n = c^n has no nontrivial integer solutions when n is at least 3. This is known as "Fermat's Last Theorem" but its proof had to wait much later, for Andrew Wiles.
• Algebraic geometry over fields which are not algebraically closed. (Fermat's Last Theorem is much less interesting over C than it is over Z or Q.)

## Course objectives and learning outcomes:

Successful students will:

• Master a variety of techniques for finding rational and integral points on varieties when they do exist, and proving nonexistence when they don't. Students will, at least, be able to describe the set of rational points on conics and on elliptic curves.
• Master the basics of elliptic curves: what are they, what is the group law, how do you find the rational points, and what is the structure of the group of rational points. Be able to describe characteristics of elliptic curves over Q, over C, and over finite fields.
• Witness (and, if they put in significant additional effort, master) classical proofs in the subject, in particular that of the Mordell-Weil theorem.
• Understand the statement of the Birch and Swinnerton-Dyer Conjecture, one of the seven Millenium Problems (and one of the six unsolved Millenium Problems!), worth a bounty of \$1,000,000.
• Study the relevance of algebraic geometry for problems in number theory. Students who have studied algebraic geometry will learn how these techniques are applied by number theorists. Students who have not will see its beauty and power, and will hopefully be motivated to study algebraic geometry.
• Gain exposure to a wide variety of problems in arithmetic geometry, as well as the wide variety of mathematical tools (representation theory, harmonic and complex analysis, algebraic number theory, commutative algebra, scheme theory, modular forms, etc., etc., etc.) which go into its study. There is far too much material to master in a lifetime, let alone in a semester. The student will gain an appreciation for the subject in broad outline, sufficient to better understand conference talks and some of the research literature, and opening doors to the study of further special topics.

Extraordinarily successful students will:

• Prove the Birch and Swinnerton-Dyer conjecture. Be sure to say I was your thesis advisor.

## 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:
• Silverman and Tate, Rational Points on Elliptic Curves. This elementary book was written for advanced undergraduates. It is suitable for beginners, first-year students, or anyone whose thesis will not involve a heavy amount of algebraic machinery.

• Silverman, The Arithmetic of Elliptic Curves. This is the book on elliptic curves. Silverman works hard to be 'accessible' and 'friendly', while introducing the student to the highbrow perspective. In particular, Silverman illustrates the relevance of ideas from algebraic geometry, algebraic number theory, group cohomology, complex analysis, and a host of other algebraic topics.

Students wishing to master these topics, wanting to work in algebraic geometry, or whose theses will involve a heavy algebraic component should invest the time to closely study Silverman's book.

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:

• (Option 1) A large number of homework exercises will be posted throughout the term which will be largely computational in nature. Many will be taken from Silverman-Tate; this option is well suited to beginners, or to students not aiming to learn any algebraic geometry.
• (Option 2) At least ten homework problems will be posted throughout the term, many with multiple parts. The student is required to finish at least eight. Some will come from Silverman's book. These will be more difficult, and may require algebraic geometry, algebraic number theory, analysis, or other subjects. I will be happy to provide hints and suggested reading material for students who want to fill in their background.
• (Option 3. For students admitted to Ph.D. candidacy only) The student may confer with the instructor and/or his/her thesis advisor to find a special topic of interest, which is related to arithmetic geometry and to the student's research interests. The student should write a 5-10 page expository paper and give an hour-long lecture in a suitable venue such as the Number Theory Seminar or Student Algebra Seminar.

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.)

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.

• Week 1: Introduction, conics, projective space.
• Week 2: Applications of Bezout's theorem; introduction to elliptic curves.
• Week 3: Group law on elliptic curves; divisors and the Picard group.
• Week 4: 2- and 3-torsion; addition formulas; introduction to elliptic curves over C.
• Week 5: Elliptic curves over C; complex multiplication.
• Week 6: Elliptic curves over C; lattices and the j-invariant.
• Week 7: Arithmetic geometry over finite fields: introduction, zeta functions, Stepanov's method.
• Week 8: Elliptic curves over finite fields: partial proof of the Weil conjectures.

(The treatment is regrettably incomplete. After spring break I decided that I needed to make adequate time for elliptic curves over Q.)

• Week 9: Reduction of elliptic curves. (Jesse Kass gave the first two lectures; these notes are only for mine.)
• Week 10: Elliptic curves over Q; introduction to height functions.
• Week 11: Conclusion of the elementary proof of Mordell-Weil (with Q-rational 2-torsion).
• Week 12: Hilbert's Theorem 90 and Kummer theory; group cohomology.
• Weeks 12-13: The Selmer group (elementary perspective; twists, torsors, and the Weil-Chatalet group).

## 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.

### Algebra:

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.
• Rational Points on Conics (1-2 weeks)