On the arithmetic of modular forms

In this overview talk, we will illustrate the arithmetic significance of the coefficients of modular forms, from classical examples to the modern point of view of Galois representations. From that on, one is naturally lead to questions on the distribution of the coefficients in various senses. We will briefly touch on some of them, report on recent theorems and state some open questions.

When investigating arithmetical properties of coefficients through Galois representations, their ramification properties are an important object to study. One notices that forms of -so called- weight one play a special role that will be explained.

The importance of Galois representations coming from modular forms is that one expects all Galois representations satisfying reasonable "geometric" assumption to arise in such a way. As a special case, we will explain the theorem of Khare and Wintenberger, formerly called Serre's modularity conjecture, and conjectural generalisations.

Whereas Galois representations are usually very hard to calculate directly on a computer, modular forms are pretty simple to compute. We will briefly touch on this and what kind of information one can get. If time allows, we will also touch on an application to the so-called inverse Galois problem.