Part III Diophantine Analysis, Michaelmas 2024

This page contains information for students taking my lectures on Diophantine Analysis in Michaelmas 2024. The best way to contact me outside lectures is by email: pv270@dpmms.cam.ac.uk. (I do not monitor Moodle.)

What is this course about? A basic, but surprisingly useful fact is the following. Let \(a\neq b\in\mathbf{Z}\) be two distinct rational integers. Then their distance is at least \(1\): \(|a-b|\ge 1\). Can we give a better bound if we have more information about \( a\) and \( b\) ? What if we consider algebraic numbers instead of rational integers? We will discuss tools to answer such questions.

The course will be an introduction to the subspace theorem, linear forms in logarithms and their applications. In the first five lectures, I will introduce and state the main theorems and give some basic applications.

The next part of the course will give some insight into the proofs. I will prove some precursors of the main theorems, which already include many of the key ideas. I will prove Dyson's result on the irrationality exponent of algebraic numbers (but not via Dyson's lemma), and the Gelfond-Schneider theorem on the transcendence of numbers like \(\sqrt{2}^\sqrt{3}\) .

In the final part of the course, I will give more applications to various problems in number theory.

Prerequisites: Some knowledge of Galois theory, number fields and complex analysis will be assumed. If you are from Cambridge and took Part II Number Fields, then you probably also know enough Galois theory for this course. I wrote up a short summary of the definitions and facts about number fields that we use in the course. Chapters 10-12 in Baker's book offers a quick introduction to number fields, and covers more than what is needed for this course. A more thorough text is Marcus's book. My notes from last year are also available. Understanding the maximum modulus principle is the main prerequisite from complex analysis.

Lecture notes are available here. (Last updated: 20 December 2024.) The notes will be updated in the course of the term. Please check back, and see the changelog below. Non-examinable material is now clearly marked as such in the notes. It is explained at the beginning of the notes what this designation means.

Example sheets: The example sheets contain exercises to complement the course material. Solutions will be discussed at the example classes. The timetable will be published later (soon). You may submit your work for two questions that are designated on the sheet, where the deadline is also indicated. This is not mandatory and it will not count towards your mark at the exam, however, take up is strongly encouraged.

Lecture recordings are available via Panopto. However, I expect that most students will find the notes more useful. Example classes will not be recorded.

Changelogs

Notes

Example sheet 1

Example sheet 2

Example sheet 3

Example sheet 4