Modular Forms, Spring 2016


This is a Mastermath course.
Time and Place: Tuesday 14:00 - 16:45, Utrecht University, see the Program below for the rooms.
First lecture: February 9
Teachers: Peter Bruin <p.j.("last name")@math.leidenuniv.nl> and Sander Dahmen <s.r.("last name")@vu.nl>
Assistants: Raymond van Bommel <R.van.("last part of last name")@math.leidenuniv.nl> and Manolis Tzortzakis <e("last name")@gmail.com>
ECTS Credits: 8
Form of tuition: Generally, there will be 2 x 45 minutes of lecturing, followed by a one-hour exercise session.
Examination: The final mark will for 40% be based on regular hand-in exercises (only the ten highest homework marks are taken into account) and for 60% on a final written exam, with the extra rule that in order to pass the course the student needs to score at least a 5.0 on the final exam.
Prerequisites: Basic knowledge of group theory (groups, (normal) subgroups, cosets, quotients, first isomorphism theorem, group actions) and complex analysis (holomorphic/meromorphic functions, Taylor/Laurent series, path integrals, the residue theorem). These concepts are covered by most introductory courses in group theory and complex analysis, respectively, and can be found in almost any standard text on (abstract) algebra and complex analysis, respectively (see e.g. texts by Ahlfors or Lang).
Aim: The aim of this course is to familiarize students with basic concepts, techniques, and applications of modular form theory as well as with some modern (deep) results about modular forms and their applications. The students will also learn how to perform explicit calculations with modular forms using the (free open-source) mathematics software system SageMath.

Contents

This is an introductory course into the subject of modular forms and their applications. A modular form is a complex analytic function defined on the complex upper half plane which has a certain symmetry with respect to the action of SL(2,Z) (or some subgroup) on the upper half plane and which satisfies some growth condition. Near the turn of the 19th to 20th century it became clear that the coefficients of series expansions of modular forms are often integers of significant number-theoretical interest. This insight has only grown in the 20th century; nowadays modular forms are an indispensable tool in modern number theory. One of the great successes was their use in the proof of Fermat's Last Theorem by Andrew Wiles around 1994. Modular forms and their generalizations play a central role in the Langlands programme, which is the focus of much current research in number theory. Finally, they also play an important role in other subjects, e.g.mathematical physics.
Topics include: the modular group and congruence subgroups, definition of modular forms and first properties, Eisenstein series, theta series, valence formulas, Hecke operators, Atkin-Lehner-Li theory, L-functions, modular curves, modularity. In the final lecture(s) we will give a global outlook on the use of modular forms (and their associated Galois representations) in the solution of Diophantine equations, in particular Fermat's Last Theorem.

Course material

For background reading and more we can also recommend

Program

We use four different lecture rooms
  • Andro C101: Androclusgebouw (Yalelaan 1, Utrecht), room C101
  • Ruppert 040: Marinus Ruppertgebouw (Leuvenlaan 21, Utrecht), room 040
  • KBG Cosmos: Victor J.Koningsbergergebouw (Budapestlaan 4b, Utrecht), room Cosmos
  • Bestuurs Lier&Egg: Bestuursgebouw (Heidelberglaan 8, Utrecht), room Lier&Egg

    Exercises with a star (*) have to be handed in before/at the start of the next lecture. You may hand them in by email to the assistants. In this case, the solutions must be typeset (e.g. using latex) and in pdf format, unless you agreed in advance to some other form with the assistants.

  • February 9, Andro C101 (lecturer: Sander)
    Covered. 1.1, 1.2, 2.1 (mostly), 2.2, 2.3 (only results)
    Exercises. from the notes: 1.1*, 1.2, 2.2; Problem Sheet 1: 1, 2*, 3
  • February 16, Ruppert 040 (lecturer: Sander)
    Covered. 1.3, rest of 2.1, rest of 2.3, 2.4, 2.5 (delta)
    Exercises. from the notes: 2.1, 2.3, 2.4*; Problem Sheet 2: 1, 2, 3*
  • February 23, Ruppert 040 (lecturer: Sander)
    Covered. 2.5 (j-invariant), 2.6, 2.7, 2.8 (read Thm. 2.12 and Cor. 2.13 yourself)
    Exercises. Problem Sheet 3: 1, 2*, 3, 4, 5*, 6
  • March 1, Ruppert 040 (lecturer: Peter)
    Covered. 3.1, 3.2 (up to the definition following Exercise 3.6), 3.4 (up to Theorem 3.6)
    Exercises. from the notes: 3.1, 3.2, 3.3*, 3.4, 3.5, 3.6, 3.9; Problem sheet 4: 1, 2*, 3, 4
  • March 8, Ruppert 040 (lecturer: Raymond)
    Covered. rest of 3.2, 3.3, 3.5, 3.6
    Exercises. from the notes: 3.7, 3.8*, 3.10, 3.11; Problem sheet 5: 1*, 2
  • March 15, KBG Cosmos (lecturer: Peter)
    Covered. rest of 3.4, 3.7, 3.8
    Exercises. from the notes: 3.12, 3.13; Problem sheet 6: 1, 2*, 3*, 4
  • No class on March 22

  • March 29, Andro C101 (lecturer: Sander)
    Covered. 4.1, 4.2
    Exercises. from the notes: 4.3; Problem sheet 7: 1*, 2, 3, 4*
  • No class on April 5

  • April 12, Ruppert 040 (lecturer: Sander)
    Covered. 4.3, 4.4 (other proof that Tp commutes with diamond operator), 4.5
    Exercises. Problem sheet 8: 1*, 2, 3*, 4, 5
  • April 19, Ruppert 040 (lecturer: Sander)
    Covered. 4.6, intro SageMath (see Course material above)
    Exercises. Problem sheet 9: 1, 2*, 4, 5*
  • April 26, Bestuurs Lier&Egg (lecturer: Peter)
    Covered. 5.1, 5.2 (up to Proposition 5.9), 5.3 (definition of old/new subspaces)
    Exercises. Problem sheet 10: 1, 2*, 3, 4, 5*
  • May 3, Bestuurs Lier&Egg (lecturer: Peter)
    Covered. rest of 5.2, 5.3 (except the proof of Theorem 5.13)
    Exercises. Problem sheet 11: 1*, 2, 3, 4, 5*
  • May 10, Bestuurs Lier&Egg (lecturer: Peter)
    Covered. 5.3 (proof of Theorem 5.13), Chapter 6
    Exercises. from the notes: 6.1*; Problem sheet 12: 1, 2, 3*
  • May 17, Bestuurs Lier&Egg (lecturer: Peter)
    Covered. Elliptic curves, modularity, and the conjecture of Birch and Swinnerton-Dyer
    Exercises. Practice Exam: This may be handed in, and will count as one of the ten highest homework marks, replacing the lowest of these if the mark for the practice exam is higher, if handed in individually (no collaboration allowed) before the start of the next lecture. You can find solutions here.
  • May 24, Bestuurs Lier&Egg (lecturer: Sander)
    Covered. Fermat's Last Theorem (this is not exam material, notes containing more background, amongst other things, can be found here)
    Exercises. none

    Exams

    Exam material consists of everything covered in class, except the material from the last class (on May 24) and SageMath. See the lecture notes, extra notes, as well as all the (non-SageMath) exercises above. A good impression should be given by the practice exam; you can find solutions here.
  • Exam: June 7, 10:00 - 13:00, Educatorium, room Beta
  • Retake: June 28, 10:00 - 13:00, Buys Ballotgebouw, room 020