Summer Graduate School

The Corvaja-Zannier proof of Siegel's theorem using subspaces. Illustrated by Sofía Pastén Vásquez.

The theory of Diophantine equations is understood today as the study of algebraic points in algebraic varieties, and it is often the case that algebraic points of arithmetic relevance are expected to be sparse.

This summer school will introduce the participants to two of the main techniques in the subject: (i) the filtration method to prove algebraic degeneracy of integral points by means of the subspace theorem, leading to special cases of conjectures by Bombieri, Lang, and Vojta, and (ii) unlikely intersections through o-minimality and bi-algebraic geometry, leading to results in the context of the Manin-Mumford conjecture, the André-Oort conjecture, and generalizations. This SGS should provide an entry point to a very active research area in modern number theory.

School structure

There will be two lectures per day. There will be a problem session in the morning after Lecture 1, and one in the afternoon after Lecture 2. These problem sessions will give the students the opportunity to work through some key aspects of the material presented in the lectures, thus deepening their understanding of the fundamental concepts and ideas, as well as allowing them to gain technical command of the methods.

The students will form smaller teams to work on the problems, and the lecturers and TA will supervise and facilitate their work. The problems will be carefully selected to be at the level of the material in the lectures, and at the same time, to offer some level of challenge so that the students experience the process of discovering new ideas by themselves.

After the afternoon problem sessions, there will be a “reports and discussions” session to allow the students to share their insights and to report on what they consider noteworthy; e.g. solutions to some of the problems, most difficult problems, practical tricks, some argument they feel proud about, etc.

Suggested prerequisites

Complex Analysis. Standard undergraduate background: Multivariable calculus, holomorphic functions of one complex variable.

Suggested literature:

• Stein-Shakarchi "Complex Analysis" Ch: 1,2,3,5,6,9

Algebra. Standard undergraduate background: Groups, rings, fields, Galois theory.

Suggested literature: 

• Dummit-Foote "Abstract Algebra" 3rd edition.
  Groups: Ch 1,2,3,4  Rings: Ch 7,8,9  Galois theory: Ch 13,14

Logic. No pre-requisites are required. Although the topic of o-minimality pertains to model theory, the necessary model-theoretical background for our diophantine applications is not too demanding, and it will be introduced as needed.

Suggested literature: 

• van den Dries "Tame Topology and O-minimal Structures" Ch: 1,2,3,4

Number theory. Basics of algebraic number theory: number fields, places, finiteness theorems (finiteness of the class group, finite generation of units). Arithmetic of elliptic curves over the rational numbers (e.g. heights, the Mordell-Weil theorem). Classification of complex elliptic curves via the action of SL2(Z) on the upper half plane.

Suggested literature:

• Chapters 2,3,4,5 of Marcus, Number Fields. (UTX, Springer)

• Chapters 1,2,3 of Silverman-Tate, Rational points on Elliptic curves. (UTM, Springer)

• Chapter VII, Sec. 1 and 2 of Serre, A course in Arithmetic. (GTM 7, Springer).

Algebraic geometry. The classical point of view for algebraic varieties over a field is enough (even simply over C). Affine and projective varieties. Divisors, rational functions. Curves (Riemann-Hurwitz, RiemannRoch). Some basic familiarity with abelian varieties, e.g. the Jacobian of a curve over C.

Suggested literature:

• Chapter 1 of Hartshorne, Algebraic Geometry. (GTM 52, Springer)

• Sections A4, A5, A6 of Hindry-Silverman, Diophantine Geometry, an introduction (GTM 201, Springer)

For eligibility and how to apply, see the Summer Graduate Schools homepage