Modular Forms, Spring 2015
This is a Masters course in mathematics at the VU University Amsterdam.
Time and Place: Wednesday 10:00 - 12:45 in room P624 of the
Mathematics and Science (W&N) Building.
First lecture: February 4
Teacher: Sander Dahmen, s.r.("last name")@vu.nl
Assistant: Abhijit Laskar, a.("last name")@vu.nl
ECTS Credits: 6
Form of tuition: Two hours of lecturing (with breaks), followed by a one-hour exercise session.
Examination: Homework exercises and a final oral exam on May 27.
Prerequisites: Basics of linear algebra, group theory, and complex analysis.
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
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 the
series expansions are often integers with an important number
theoretical interest. This insight has only grown in the 20th century
and nowadays modular forms are an indispensable tool in modern number
theory (but also play an important role in other subjects, e.g.
physics). One of the great successes was their use in the proof of
Fermat's Last Theorem by Andrew Wiles around 1994.
Topics include: the modular group and congruence subgroups, definition
of modular forms and first properties, Eisenstein series, theta series,
dimension of modular form spaces, Hecke operators, Petersson inner
product. In the final part of the course 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
- We will cover parts of the book "A first course in modular forms" by Fred Diamond and Jerry Shurman, Springer Graduate Texts in Mathematics Volume 228.
- Some additional handouts will be provided later in the course.
- The free open-source mathematics software system Sage will be used to perform explicit modular forms computations.
For background reading and more we can also recommend
References below are to the book by Diamond and Shurman, unless explicitly stated otherwise.
Exercises with a star (*) have to be handed in before/at the start of the next lecture.
- February 4
Exercises: 1.1: 1, 4, 7*, 8* (for definitions of Delta and j see the bottom of p.6 and top of p.7)
- February 11
Covered: 1.2 until (not including) Prop. 1.2.4, skipped theta series
Exercises: 1.2: 2*, 3, 6 (first read Prop. 1.2.4), 11*
- February 18
Covered: Poison summation, theta series (including proof, modulo last exercise below, that theta^2 is a modular form of weight 1 w.r.t. Gamma_1(4)), remainder of 1.2 except Dedekind eta function;
Exercises: 1.2: 8*, 9, 4 + also show that only taking the two generators with the plus-sign gives Gamma_1(4)
- Februari 25
Covered: Eta function, these notes (read the details of Section 2 yourself)
Exercises: 1, 2*, 3*, 4 (from the notes)
- March 4
Covered: summary of 1.3, 1.4, 1.5, a little bit of Chapter 2, and a very little bit of Chapter 3
Exercises: read your favorite details in 1.3, 1.4, 1.5 (no hand-in exercises)
- March 11
Covered: 5.1 and 4.3
Exercises: 4.3: 1, 4* (only a and b, do not merely copy the hint, but work out the details and give independent definitions if necessary); also prove (as was mentioned in class) that C[G_2,G_4,G_6] is closed under differentiation (here C denotes the complex numbers and you don't need the last two lectures for this; just to be clear, there is no star, so this is NOT a hand-in exercise)
- March 18
Exercises: 3.8.8(a), 5.2.1; these notes: 1*, 2
- March 25
- April 1
Covered: 5.3 (plus more motivation) and 5.4
Exercises: 5.3.2; 5.4: 1, 2*, 3* (for 5.4.1(a) prove the "integral"-equivalent as was stated in class)
- April 8
- April 15
Covered: 5.7 (lecture by Abhijit)
Exercises: do your favorite exercises from 5.7 (no hand-in exercises)
- April 22
Covered: 5.6 and 5.8
Exercises: 5.6.3; 5.8: 3*, 4 (for 3*: you don't have to do ex. 3.5.4, you just may assume that f(z)=eta(t)^2 eta(11t)^2 lies in the given (one-dimensional) space)
- April 29
- May 6
- May 13
Covered: 5.9 and 5.10 + modularity and the BSD-conjecture
Exercises: 5.10.1*, all of the previous exercises that you skipped :-)
- May 20
Plan: Fermat's Last Theorem and more, here are a few very basic Sage commands related to modular forms
- May 27
Oral exam (time to be discussed)