Math 788 -- Elliptic Curves and Arithmetic Geometry, Spring 2020

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

Office hours: Tuesdays 9:00-10:30, Wednesdays 3:30-5:00, 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 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.

This is the clear choice for algebraists and for would-be algebraists.

The student should purchase and follow along in one (or both) of these books, according to his or her background and aims.

## Required Work:

Homework assignments, given every week or two.

Homework 1, due Monday, February 3.

Homework 2, due Monday, February 10.

Homework 3, due Wednesday, February 26.

Homework 4, due Friday, March 6.

Homework 5, due Monday, April 6.

(Yes, there was a Homework 6.)

Homework 7, due Tuesday, May 6.

### Lecture Notes

These are the lecture notes from Spring 2016. The relationship to what I said in class may be a bit ambiguous. Also, eventually, we will later deviate from the 2016 course.

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)
• Quadratic Forms, the p-adic numbers, and Local-to-Global (2 weeks)
• The geometry of curves (2 weeks - ??)
• Elliptic curves: the basics (2-4 weeks)
• Elliptic curves over C: Curves as complex tori (0-2 weeks)
• Elliptic curves over finite fields: The Weil conjectures (1-2 weeks)
• Elliptic curves over Q: The Mordell-Weil theorem (2-5 weeks?)
• Survey of Additional Topics (Faltings' theorem??) (????)