Diophantine approximation 2019
Lecturers
Jan-Hendrik Evertse (Leiden), Damaris Schindler (Utrecht)
Exercise sessions are led by
Adelina Manzateanu (Leiden)
Carlo Verschoor (Utrecht)
Time/Place
Both the lectures and exercise classes will take place at the Vrije Universiteit in Amsterdam. Here is the schedule for the rooms
Week 37-42 room WN S607
Week 43 room WN C121
Week 44-50 room WN P624
Week 51 room WN C121 (room has been changed)
The lectures/exercise sessions take place on Thursday afternoons from 14:00 - 16:45 pm. We will each time start with 2x 45 minutes of lectures, with 15 minutes break in the middle and then have a 45 minutes exercise class.
During the course we have a one week break - there is no lecture/exercise class on Thursday 14th November 2019.
Note that there is also a mastermath course on algebraic number theory by Peter Stevenhagen and Bart de Smit on the same day at the VU. You can attend the Diophantine approximation course without taking the algebraic number theory course but contents wise they are a very nice fit.
Prerequisites
Algebra: basic group theory, rings, field extensions and Galois theory (see Chapter 1 - 14, 21, 22 in the Leiden Algebra course notes that can be found here http://websites.math.leidenuniv.nl/algebra/). Knowledge of Galois theory is convenient but can also be acquired during the course.
Aim of the course
Diophantine approximation deals with problems such as whether a given number is rational/irrational, algebraic/transcendental and more generally how well a given number can be approximated by rational numbers or algebraic numbers. Techniques from Diophantine approximation have been vastly generalized, and today they have many applications to Diophantine equations, Diophantine inequalities, and Diophantine geometry. Our present plan is to discuss the following topics (but this may be subject to changes):
- geometry of numbers and Minkowski's convex bodies theorems
- transcendence results
- approximation of algebraic numbers by rationals, including Roth's theorem and Schmidt's subspace theorem if time permits
Rules about Homework/Exam
The examination will consist of homework assignments and a written exam.
The average of the homework assignments will contribute 25% to the final grade, and the written exam for 75%.
The homework assignments also contribute to the grade in case of a resit.
The grade for the written exam must be at least 5.
Exam preparation
Here is a description of what is expected from you in the exam.
And an older exam from 2017
Exam and solutions
Here is a copy of the exam from 9th January 2020 as well as the solutions
solutions exam 9th January 2020
Weekly planning
Thursday 12th September: Chapter 1, Introduction and course overview
Exercise session on 12th September: Exercise 1.1, 1.2, 1.3 and 1.4 from Chapter 1
Thursday 19th September: We start with Chapter 2, especially section 2.1 and start to discuss Minkowski's first convex body theorem
Exercise session on 19th September: Exercise 1.5 from Chapter 1 and Exercise 2.1 and 2.2 from Chapter 2
Thursday 26th September: Chapter 2, proof of Minkowski's first convex body theorem and applications, successive minima
Exercise session on 26th September: Exercise 2.4, 2.5 and 2.6 from Chapter 2
Thursday 3rd October: Chapter 2, Minkowski's second convex body theorem
Exercise session on 3rd October: Exercise 2.7, 2.8 and 2.9 from Chapter 2
Thursday 10th October: Chapter 3, Siegel's lemma over the rational integers and any number field
Exercise session on 10th October: Exercise 2.15 and 2.16 from Chapter 2 and Exercise 3.8 from Chapter 3
Thursday 17th October: Chapter 4, transcendence of e and the Lindemann-Weierstrass theorem
Exercise session on 17th October: Exercise 4.1, 4.2 and 4.3 from Chapter 4
Thursday 24th October: Chapter 4, transcendence results, including Schanuel's conjecture
Exercise session on 24th October: Exercise 4.4, 4.6 and 4.7 from Chapter 4
Thursday 31st October: Chapter 5, Linear forms in logarithms and first applications; preparations for further applications.
Exercise session on 31st October: Exercise 4.5 from Chapter 4; Exercises 5.1 and 5.3 from Chapter 5.
Thursday 7th November: Chapter 5, Unit equations and Thue equations and maybe other topics if time permits.
Exercise session on 7th November: Exercises 5.5 and 5.8, if you have time you may try 5.2.
Remark: In Exercise 5.2 the condition that X^3-AX^2-BX-C is irreducible has to be added. The notes of Chapter 5 with the corrected Exercise 5.2 are now updated below.
Thursday 14th November: No course and exercise class.
Thursday 21st November: Chapter 6, Approximation of algebraic numbers by rationals, some first applications
Exercise session on 21st November: Exercises 6.1, 6.4, 6.5
Thursday 28th November: Continuation of Chapter 6
Exercise session on 28th November: 6.8, 6.10 (i)
Thursday 5th December: Chapter 7, The Subspace Theorem (a higher dimensional generalization of Roth's Theorem)
Exercise session on 5th December: Exercises 6.10 from Chapter 6 and 7.2 from Chapter 7
Thursday 12th December: Section 7.2, norm form equations.
Exercise session on 12th December: 7.3, 7.5 from Chapter 7
Thursday 19th December: Chapter 8: The p-adic Subspace Theorem
Exercise session on 19th December: 8.6 (i), (ii), 8.7
Hand-in exercises
First hand-in homework: Please do Exercise 1.6 from Chapter 1 and Exercise 2.3 from Chapter 2 and hand both of them in at latest during the lecture on Thursday 3rd October, or e-mail them by then to Adelina and Carlo.
Before you solve Exercise 2.3 from Chapter 2 we recommend reading the first proof of Theorem 2.3 in the lecture notes.
Secon hand-in homework: Please do Exercise 2.12 and 2.14 from Chapter 2 and hand both of them in at latest during the lecture on Thursday 17th October, or e-mail them by then to Adelina and Carlo.
Third hand-in homework: Please do Exercise 4.8 from Chapter 4 and Exercise 5.4 from Chapter 5 and hand them in at the latest during the lecture on 21st November or e-mail them by then to both Adelina and Carlo.
Don't forget to hand in your third assignment.
Fourth hand-in homework: Please do Exercises 5.7 from Chapter 5 (here you may use Exercise 5.6 without solving it) and Exercises 6.6, 6.7 from Chapter 6 and hand them in at latest 5th December or e-mail them by then to Adelina and Carlo.
Last hand-in homework: Please do exercises 6.11 from Chapter 6 and 7.1 from Chapter 7 and hand them in at the latest during the last lecture on 19th December or e-mail them by then to both Adelina and Carlo.
Lecture notes
In the course will follow the lecture notes written by Jan-Hendrik Evertse.
Literature
A. Baker, Transcendental Number Theory, Cambridge University Press, 1975.
Gives a broad but very concise introduction to Diophantine approximation. In particular, the book discusses linear forms in logarithms of algebraic numbers. ISBN 0-521-20461-5
E.B. Burger, R. Tubbs, Making transcendence transparent, Springer Verlag, 2004.
Gives a relaxed introduction to transcendence theory. ISBN 0-387-21444-5
J.W.S. Cassels, An Introduction to the Geometry of Numbers, Springer Verlag, 1997 (reprint of the 1971 edition)
This book gives a broad introduction to the geometry of numbers. ISBN 3-540-61788-4
W.M. Schmidt, Diophantine Approximation, Springer Verlag, Lecture Notes in Mathematics 785, 1980.
This book discusses among other things some basics of geometry of numbers, Roth's Theorem on the approximation of algebraic numbers by rational numbers, Schmidt's own Subspace Theorem, and several applications of the latter. ISBN 3-540-09762-7
C.L. Siegel, Lectures on the Geometry of Numbers, Springer Verlag, 1989.
This book contains lecture notes of a course of Siegel on the geometry of numbers, given in 1945/46 in New York. The main topics are a proof of Minkowski's 2nd convex body theorem, and a proof of Kronecker's approximation theorem. ISBN 3-540-50629-2