Modular Forms, Spring 2012

Course material

• We will cover parts of the book "A first course in modular forms" by Fred Diamond and Jerry Shurman (abbreviated below as D-S), Springer Graduate Texts in Mathematics Volume 228.
  This book can be found online at SpringerLink. Access is provided by the Utrecht University Library.
• Some additional handouts will be provided later in the course.
  Here are parts of the notes from a previous course. During the first 3 or 4 lectures, we will have a look at chapters 1,2, and 3.
• The free open-source mathematics software system Sage will be used to perform explicit modular forms computations.
  Introduction notes can be found here.
  There is an online Sage notebook available. Before using it the first time, it is necessary to register first. Don't forget your username and password!
  Alternatively, it is also possible to download Sage and install it under Windows, Mac OSX, Linux, or Solaris.

For background reading and more we can also recommend

• 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

Program

• February 6
  Motivation, introductory examples, SL(2,R)-action on the complex upper half plane, definitions and first properties of modular forms and Eisenstein series.
• February 13
  Read: D-S 1.1
  Exercises: D-S 1.1.3; 1.1.4; 1.1.7
• February 20
  Read: Chapter 1 of these notes; D-S 1.2 up to p.18 line 9
  Exercises: D-S 1.2.2; 1.2.3; 1.2.5; 1.2.6; Let p be a prime number. Prove that every cusp is Gamma_0(p)-equivalent to 0 or infinity, but not both.
• February 27 Read: Sections 2.1, 2.2, and 3.1 of these notes; D-S rest of 1.2
  Exercises: Click here; To be handed in on February 27!
• March 5
  Read: Section 3.3 of these notes; Handout 2
  Exercise: Click here
• March 12
  Read: D-S 4.3
  Exercises: D-S 1.2.11; Create a Sage account (see above) and work through this intro
• March 19
  Read: D-S 5.1 up to p. 166 line 10 from below; Formula (3.16) op p.104 and its proof on p.105
  Exercises: D-S 5.1.2; 5.1.3; 5.1.4
• March 26
  Read: D-S 5.2 up to p. 172 line 8 from below and Proposition 5.2.4 + proof
  Exercises: Click here; To be handed in on April 2!
• April 2
  Read: D-S 5.3
  Exercises: D-S 5.3.1 + see last week
• April 16
  Read: D-S 5.4 and 5.5
  Exercises: D-S 5.4.1; 5.4.2; 5.4.3
• April 23
  Read: D-S 5.6 and 5.7 up to Theorem 5.7.1 (rest is optional);
  Exercises: D-S 5.5.1; 5.6.3
• May 7
  Read: D-S 5.8 up to p. 198; Click here for creating newforms with Sage
  Exercises: Click here; To be handed in on May 7!
• May 14
  Read: D-S 5.9 up to proof of Theorem 5.9.2 (restrict to cusp forms); 5.10 (see 4.9 for some background on the Mellin transform, this is optional)
  Exercises: Your own choice :-)
• May 21
  Read: This handout
  Exercises: None
• June 4
  Read: T.B.A.
  Exercises: Click here; To be handed in on June 4!
• June 11 (or possibly 12 or 13)
  Oral exam