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 Sage.

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 Theorem.

- 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.

- W.A. Stein: Modular Forms, a Computational Approach; as well as his Modular forms database
- J.S. Milne's Course Notes
- T. Apostol: Modular Functions and Dirichlet Series in Number Theory
- J.H. Bruinier, G. van der Geer, G. Harder and D. Zagier: The 1-2-3 of modular forms

Exercises with a star (*) have to be handed in before/at the start of the next lecture.

**February 4**

Covered: 1.1

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**

Covered: 5.2

Exercises: 3.8.8(a), 5.2.1; these notes: 1*, 2**March 25**

No lecture**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**

Covered: 5.5

Exercise: 5.5.1***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**

No lecture**May 6**

No lecture**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)