# 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:

• Week 1, Peter Bruin. Introduction; quadratic reciprocity; elliptic curves and modular forms; examples of L-functions
• 1 point: 1.2 and 1.6
• 2 points: 1.1, 1.9, and 1.10
• 3 points: 1.4, 1.5 (now with extra hint), 1.7 and 1.8 (beware, exercise 1.8 has 4 parts)
• Week 2, Arno Kret. Profinite groups; infinite Galois theory; local fields (to be continued next week).
• 1 point: 2.2, 2.16, 2.17
• 2 points: 2.5, 2.6, 2.7, 2.8, 2.8.1/2, 2.9, 2.10, 2.15, 2.18
• 3 points: 2.11, 2.12, 2.13, 2.14, 2.20, 2.24, 2.25
• Update: For 2.25 you need Hensel's lemma, which we have not treated yet. This exercise will also be homework next week, so I suggest you wait untill next week before making this question. If you do decide to make it, feel free to look up Hensel's lemma and use that result in your solution.
• Note: In the text please ignore question 2.21 and it's undefined reference.
• In question 2.8(c) the subgroup H of G is open, and a hint for 2.8 (second b): Try to find an open neighborhood of 1 in GL_n(CC) that contains no non-trivial subgroups. I separated question 2.8 (second c) into a separate question, to make the difficulty of the exercise more balanced in comparison to the other exercises.
• For question 2.7, if you are unfamiliar with schemes, you may assume k algebraically closed.
• In 2.16(b) the condition that n_p = 0 for allmost all p should be removed.
• I removed part 2.14(c) and made part (b) easier, to avoid complicated set theory arguments (this is not really the goal of the course). If you have an old version of the notes and still want to make the original parts (b) and (c), you still can of course, Maxim will give a bonus.
• Week 3: Arno. Local fields. Hensel's lemma. Eisenstein polynomials. (Higher) ramification.
• 10 points: Exercise 2.27
• 1 point: Exercise 2.28
• 1 point: Exercise 2.29 (but only if you do it for all primes p; this may be difficult, I have not done this exercise myself).
• In exercise 2.27(b), you need to assume 0 < v_p(c) < p
• Week 4: Peter. Algebraic number theory for infinite extensions. The adèle ring of Q.
• 1 point: 2.34, 2.35, 2.40, 2.43
• 2 points: 2.37, 2.38, 2.39, 2.42
• 3 points: 2.36, 2.41
• Fixed two typos in 2.41(b) and (d); and also in 2.35.
• In exercise 2.45(b), it was written "not closed"; this has been changed to "not open". In 2.41(e), the statement that the prime ell is inert is deleted (because it is false), what was meant that there is a unique prime above ell.
• Week 5: Peter. Adèles and idèles of number fields. Main statements of class field theory. Definition of weak and strong approximation.
• Important note: We would like to make the homework load somewhat lighter. This week, you therefore only have to hand in exercises worth at most six points (instead of twelve). Your grade will be Min(2 ⋅ #points, 10).
• Exercise numbers in the following list may differ from the previous version of the notes, so make sure you consult the current version.
• Exercises 2.64 and 2.65 are not in this list because they require more background on class field theory than what we have seen in the lecture.
• 1 point: 2.47, 2.52, 2.53, 2.55, 2.56, 2.62
• 2 points: 2.45, 2.48, 2.49, 2.51, 2.54, 2.58, 2.61, 2.66
• 3 points: 2.46, 2.50, 2.57, 2.59, 2.60, 2.63, 2.67, 2.68
• A typo in the map of Exercise 2.50(a) and a sign error in the map of Exercise 2.50(b) have been corrected.
• Week 6: Arno. Representation theory. Artin representations. Artin conductor.
• Note 1: As previous week, only need to hand in exercises worth at most six points.
• Note 2: Unfortunately I did not get to explaining ell-adic representations or Galois representation. Many of the exercises below rely on this, so before solving these exercises you are asked to read pages 68 and 69 of the course notes first. (I will also introduce ell-adic and Galois representations next week). Since there is plenty of choice, you can also restrict your attention to exercises that do not involve these concepts.
• Note 3: In two weeks Maxim will be traveling, and next week he will need time to prepare his visit. Therefore the corrections of the homework of week 5 will come after Maxim returns from his trip.
• 1 point: 3.1, 3.2, 3.3, 3.4, 3.5, 3.7, 3.8, 3.11, 3.13, 3.15, 3.18, 3.20, 3.21, 3.24, 3.30, 3.7.1/2
• 2 points: 3.6, 3.9, 3.10, 3.14, 3.16, 3.17, 3.19, 3.22, 3.23, 3.26, 3.32, 3.35, 3.37, 3.38
• 3 points: 3.12, 3.25, 3.28, 3.29, 3.33, 3.36
• Note 1: Last week's homework can be handed in one week late (so next week).
• Note 2: This week there is no homework
• Note 3: We have added 2 two point exercises to the list of last week, namely 3.37 and 3.38.
• Week 8: Arno. Elliptic curves and their Tate modules.
• Note: Only have to hand in exercises worth at most six points (instead of twelve). Your grade will be Min(2 ⋅ #points, 10).
• 1 point: 3.46, 3.50, 3.51,
• 2 points: 3.39, 3.40, 3.43, 3.45 3.47, 3.48, 3.52,
• 3 points: 3.41, 3.42, 3.49, 3.53, 3.54
• Note: There was a slight error in exercise 3.41(d) and 3.42 which has been corrected.
• Week 9: Peter. Introduction to étale cohomology.
• Note: As before, you only have to hand in exercises worth at most six points.
• 1 point: 3.60, 3.64, 3.65, 3.70
• 2 points: 3.61, 3.62, 3.63, 3.66, 3.67
• 3 points: 3.68, 3.69
• Week 10: Peter. Complex representations of p-adic groups.
• Note: As before, you only have to hand in exercises worth at most six points.
• 1 point: 4.1, 4.3, 4.5, 4.8
• 2 points: 4.2, 4.4, 4.6, 4.9, 4.11, 4.12
• 3 points: 4.7, 4.10
• Note: In Exercise 4.1, “locally constant” has been corrected to “continuous”.
• Note: In Exercise 4.6, a correction and a clarification have been made.
• Week 11: Peter. (g, K)-modules, automorphic forms and automorphic representations.
• Note: As before, you only have to hand in exercises worth at most six points.
• Exercise numbers in the following list may differ from the previous version of the notes, so make sure you consult the current version.
• 1 point: 4.14, 4.15, 5.2
• 2 points: 4.16, 5.1, 5.4, 5.5, 5.6
• 3 points: 4.17, 4.18, 5.3, 5.7
• Note: The notes on (g, K)-modules have been expanded slightly, and minor corrections have been made in the exercises.
• Note: A mistake in Definition 5.8 has been corrected (admissibility of (π, V) is not equivalent to simultaneous admissibility of the smooth representation of the locally profinite group on V and of the (g, K)-module structure on V).
• Note: Exercise 5.4 was incorrect; it has been replaced by a similar question with an easier part (a) and a harder part (b).
• Week 12: Arno. Unramified representations.
• The homework of this week will be combined with the homework of next week. The hand-in date is 15 December.
• Note: I made a small mistake in the normalization of the Satake transform.
• Week 13: Peter. Cuspidal representations, Weil-Deligne representations, local Langlands correspondence.
• As before, you only have to hand in exercises worth at most six points.
• Some of the exercises below refer to the Steinberg representation, which has not been discussed yet in the lectures. Arno will probably do this in the final lecture; you can also study the corresponding parts of the notes yourself.
• 1 point: 3.71, 3.72, 4.20, 4.21
• 2 points: 4.22, 4.23, 4.24, 4.25, 4.27, 4.31, 4.32, 4.34, 4.35, 4.37, 4.38, 4.39
• 3 points: 3.73, 4.26, 4.28, 4.29, 4.30, 4.33, 4.36, 4.40

## Exam

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

• 1.9, 1.10
• 2.9, 2.20(a), 2.39, 2.41, 2.48, 2.49, 2.50, 2.57, 2.59, 2.62
• 3.15, 3.32, 3.36, 3.37, 3.40, 3.45, 3.47, 3.48, 3.50, 3.51, 3.52, 3.53, 3.54, 3.61, 3.62
• 4.12, 4.13, 4.22, 4.23, 4.40
• 5.2, 5.5

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 math.leidenuniv.nl). Our policy for late homework is that your maximum grade will be 10 - 0.2*(days late).

## Prerequisites

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

• basic knowledge of complex analysis (holomorphic/meromorphic functions, Taylor/Laurent series)
• basic knowledge of group theory (groups, (normal) subgroups, cosets, quotients, group actions)
• algebraic number theory (including Galois theory)
• algebraic geometry

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

• elliptic curves (e.g. the course by Bright and Streng, autumn 2015)
• modular forms (e.g. the course by Bruin and Dahmen, spring 2016)
• p-adic numbers (e.g. the parallel course by Beukers and Dahmen, autumn 2016)
• representation theory