Galois Representations and Automorphic Forms (MasterMath)

Lecturers: Peter Bruin, Arno Kret. Course assistant: Maxim Mornev

The aim of this course is to give an introduction to the Langlands programme. This programme, first formulated by R. P. Langlands in the late 1960s and developed by many people, links two different areas of mathematics: number theory (in particular Galois theory) on the one hand, and representation theory (more precisely automorphic forms) on the other hand.

On the "number-theoretical" side, one takes algebraic varieties (defined by polynomial equations with rational coefficients) and associates Galois representations to them. Fundamental examples of these are Tate modules of elliptic curves. On the "automorphic" side, one studies certain highly symmetric functions on Lie groups known as automorphic forms. The classical examples of these are modular forms.

To objects on both sides, one can attach L-functions, a certain kind of analytic functions similar to the Riemann zeta function. The Langlands programme then predicts, roughly speaking, that there is a correspondence between Galois representation and automorphic forms, under which two objects on both sides correspond to each other if they give rise to the same L-function.

In this course, we will introduce both sides of this correspondence and give examples of the kind of objects that occur. We then explain Langlands's conjectures, as well as the cases of them that have been proved. We will strive to give many concrete examples to illustrate the general ideas and techniques.

Time and location

Thursdays during the weeks 37–51, hours: 10am to 1pm. Location: University of Amsterdam, Science Park 904, room SP G2.13.

Course materials

The course notes will be updated weekly.

Material treated and homework exercises:


The exam will be about the material discussed in class and the course notes. There will be an emphasis on the material from Section 2.5 up to and including Chapter 5. From the exercises, we especially recommend looking at

The questions of the practice exam and of the exam are available.


The mark for this course will be based for 40% on regular hand-in exercises and for 60% on a final exam (which will be written or oral, depending on the number of students), with the extra rule that in order to pass the course, the student needs to score at least a 5.0 on the exam.

Each week there will be a list of homework exercises from the list above. You get a grade by handing in a certain number of exercises, each one is worth 1, 2 or 3 points (depending on the difficulty). Your grade will be based on the first n exercises that you hand in, where n is maximal such that the first n exercises together are worth at most 12 points. Your grade will be Min(#points, 10).

For example: You can hand in 4 exercises of difficulty 3, and get at most 12 points. If you lose 2 points, you still get a perfect score. If you hand in 5 exercises of difficulty 3, the last solution may be checked by Maxim, but its score will be ignored in the computation for the mark.

The lowest homework mark (where a homework set not handed in counts as 0 points) is disregarded in the computation of the overall homework mark.

Homework should be handed in before the beginning of the lecture, either on paper or by e-mail to Maxim Mornev (email: m.lastname at Our policy for late homework is that your maximum grade will be 10 - 0.2*(days late).


In this course, we will assume some familiarity with the following topics:

Furthermore, knowledge of one or more of the following topics will be an advantage: