Wintersemester 2007/2008 - Computational Arithmetic of Modular Forms (Modulformen II)

• Dates: Monday and Wednesday 10-12 am at the Institut fuer Experimentelle Mathematik.
• This course will be taught in English.
• It will be accompanied by theoretical and practical exercises.
• Here is the current version of the Lecture Notes.

Some Files

• MagmaIntro
• ModularSymbols

Course Description

This lecture is about computing modular forms and some of their arithmetic properties.

We set the following challenging objectives:

• We explain and completely prove the modular symbols algorithm in as elementary and as explicit terms as possible. The chosen approach is based on group cohomology.
• The devoted student shall be enabled to implement the (group cohomological) modular symbols algorithm over any ring (such that a sufficient linear algebra theory is available in the chosen computer algebra system).
• We introduce the theory of Galois representations attached to modular forms in as explicit terms as possible. We explain some of its number theoretic significance and some computational approaches.
• The devoted student shall be enabled to compute important properties of Galois representations attached to modular forms explicitly.

According to these objectives the lecture consists of two main parts:

• I. Computing Modular Forms
• II. Computational Galois Representations

Due to the diversity of the audience, ranging from students up to PhD students intending to generalise the presented algorithms in different directions, and due to the dual aims, theoretic and algorithmic, the lecture is conceived in parallel layers. Not all layers need be followed by all students and all layers can be reduced individually. The layers are the following:

• Theory. Roughly in 3 of the 4h per week the lecture will introduce theoretical results. All students are expected to attend the lectures. The lectures will be accompanied by exercises concerning the theory presented. Exercises can be handed in and will be corrected. Some time will be devoted to discussing possible solutions.
• Algorithms and implementations. In a lecture in the beginning, programming in some standard computer algebra systems is introduced. In some lectures during the term, algorithms and possibly concrete implementations are presented. Much emphasis is laid on practical issues and students will also be asked to find and implement algorithms. Possible solutions will be discussed.
• Self-learn modules. For the devoted student to gain a more complete picture of the theory than can be presented during the lecture, complementary reading is suggested.

The parallel layers will not necessarily be on a single subject all the time, as it is often necessary to introduce theory first. The lecture is divided up into stages, instead of chapters, in order to emphasize the possible variety of subjects in each stage.

The conception of this lecture is different from every treatment I know, in particular, from William Stein's excellent book ``Modular Forms: A Computational Approach''. Parts will, however, be similar to notes of a series of 4 lectures that I gave at the MSRI Graduate Workshop in Computational Number Theory ``Computing With Modular Forms''. We emphasize the central role of Hecke algebras and focus on the use of group cohomology, since on the one hand it can be described in very explicit and elementary terms and on the other hand already allows the application of the strong machinery of homological algebra. We shall mention geometric approaches only in passing.

Organisational issues will be discussed with all participants and decided together in order to suit everybody.

Prerequisites:

Algebra, complex analysis and basic facts on modular forms (for instance, lecture from previous term; I can indicate good sources to learn from.). We will be using cohomological techniques which we will introduce (partly without proofs); hence, knowledge of homological algebra, algebraic topology etc. is useful, but not essential.

Leistungsnachweis/certificate

A certificate (Leistungsnachweis) can be obtained by regularly and successfully solving the exercises and by passing an oral exam at the end of the term.

Perspectives

If you intend to write your Diplomarbeit, Master's thesis or Staatsarbeit in number theory, this is the lecture to attend. At the end of the lecture, subjects can be obtained.

Last modification: 13 October 2007.