Arithmetic Geometry (18.782 Fall 2019)

Instructor: Junho Peter Whang
Email: jwhang [at] mit [dot] edu

Meeting time: TR 9:30-11 in Room 2-147
Office hours: M 10-12 or by appointment, in Room 2-238A

This is the course webpage for 18.782: Introduction to Arithmetic Geometry at MIT, taught in Fall 2019. This webpage will be updated regularly; be sure to refresh the page to access the most recent version.

Overview: Arithmetic geometry is an incredibly broad field of research at the interface of number theory and algebraic geometry. It has been particularly influential in the study of Diophantine equations (i.e. polynomials with integral coefficients, to be solved in integers or rationals). This course introduces some of the basic results and techniques in arithmetic geometry, with emphasis on Diophantine applications. Topics will include a subset of the following: the p-adic numbers, the local-global principle, algebraic geometry of curves, Mordell's theorem, Diophantine approximation, and Siegel's theorem.

Course policy and syllabus: The course policy and syllabus can be found here.

Prerequisites: 18.702 or equivalent (one year of abstract algebra)

Textbook: There is no required text; lecture notes and references will be provided. The following texts are excellent sources:

A course in arithmetic, by J.P. Serre.
The Arithmetic of elliptic curves, by J.H. Silverman.
Diophantine Geometry: an Introduction, by M. Hindry and J.H. Silverman.

Notes for this course from previous years by B. Poonen (here) and by A. Sutherland (here) are also available.

Exams: There will be no midterm or final exam in this course.

Grading: The final grade will be based on the following scheme.

Problem sets (60%)
Final paper (30%)
Final presentation (10%)

Problem sets

There will be 6 problem sets, each worth 10% of the final grade. Problem sets will be posted here. You should submit your problem set to the instructor by email in PDF form (LaTeX preferred) on the indicated due date before 11:59 pm.

Problem Set 1. [PDF] Due Thursday, September 19, 2019.
Problem Set 2. [PDF] Due Tuesday, October 8, 2019.
Problem Set 3. [PDF] Due Thursday, October 24, 2019.
Problem Set 4. [PDF] Due Tuesday, November 5, 2019.
Problem Set 5. [PDF] Due Thursday, November 21, 2019.
Problem Set 6. [PDF] Due Thursday, December 5, 2019.

Final project

Each student will write a short expository paper (approx. 5-10 pages) on a topic of their choice in arithmetic geometry, and give a short presentation (approx. 30 minutes) on the topic at the end of the semester. The collective goal is to highlight a variety of approaches to solving Diophantine equations developed in the early- to mid-twentieth century, many of which have formed the basis for current research. Some suggestions include, with varying levels of difficulty:

Runge's method
Integral points on Mordell curves
Skolem's method for Thue equations
Chabauty's method
Baker's theory and effective Thue
Schmidt's subspace theorem and Siegel's theorem
Gonality and rational points of bounded degree
Lang's G_m and torsion points on curves
Nevanlinna theory and Vojta's dictionary
Mazur's theorem on torsion points on elliptic curves
The abc conjecture
...

Students may also pursue other topics, provided that they fall within the scope of the course and are discussed with the instructor. The exposition should be aimed at fellow students learning the subject. Submitted articles will be compiled and made available to the class at the end of the semester.

Here are descriptions of some of the topics to help with choosing.

Lectures

Notes from the lectures will be uploaded here (last updated: December 1, 2019) as they are updated each weekend; be sure to clear the cache before reloading to get the most current version of the notes.

September 5. Introduction and overview. Rational parametrization of conics.
September 10. Absolute values on fields. p-adic absolute value. Completion and Q_p. Ostrowski's theorem.
September 12. Discrete valuations. p-adic integers and their properties. Inverse limit description of Z_p.
September 17. Solving equations in Z_p. Hensel's lemma. Structure of p-adic units.
September 19. Quadratic forms. Equivalence and representability. Hasse-Minkowski theorem.
September 24. Proof of Hasse-Minkowski. Sums of three squares, Gauss's theorem, Lagrange's theorem.
September 26. Description of some final project topics. Affine varieties. Hilbert Nullstellensatz.
October 1. Affine varieties: Zariski topology, coordinate ring, irreducibility, dimension.
October 3. By Ananth Shankar: Projective space. Projective varieties. Affine patches. Projective closure.
October 8. Hypersurfaces. Tangent spaces. Smoothness.
October 10. Morphisms of varieties. Rational maps.
October 15. No lecture (holiday following Columbus Day).
October 17. Interlude: first view of elliptic curves. Curves and valuations. Closed points.
October 22. Rational maps between curves. Divisors. Principal divisors.
October 24. Linear equivalence. Picard group. Genus of a curve.
October 29. Genus and Newton polygons. Riemann-Roch theorem.
October 31. Elliptic curves. Weierstrass equations. Group law on elliptic curves revisited.
November 5. Torsion on elliptic curves. Outline of Mordell's theorem. Proof of weak Mordell-Weil.
November 7. Reivew of Galois theory. Height and infinite descent. Heights of rational numbers.
November 12. Heights and rational maps. Height on elliptic curve. Conclusion of proof of Mordell's theorem.
November 14. Euclid's method. Solving x²-dy²=k and Pell equation. Dirichlet's lemma.
November 19. Diophantine approximation. Liouville's theorem. Thue's theorems.
November 21. Runge's method. Mordell's equation example.
November 26. Binary quadratic forms and cubic forms. Finiteness of integral solutions to Mordell's equation.
November 28. No lecture (Thanksgiving holiday).

Go to the webpage of Junho Peter Whang