Der Zusammenhang von Modulformen und Zahlkörpern : Zahlentheorie und Geometrie vereint in der Serre-Vermutung
Der Beweis der Serre-Vermutung, die einen Zusammenhang zwischen Zahlentheorie und Geometrie herstellt, ist erst vor kurzem gelungen. Gabor Wiese gibt einen Überblick über diese Vermutung, die für die Forschung am Institut für Experimentelle Mathematik eine nicht unerhebliche Bedeutung hat.
Recently one of the most important structural conjectures in pure mathematics, Serre’s modularity conjecture, has become a theorem, proved mainly by Khare and Wintenberger. Serre’s conjecture establishes a link between seemingly different areas: number theory and geometry. This link is made through modular forms, which are functions dating back to the 19th century. The main aim of the article is to describe the content of Serre’s conjecture in a non-technical language. Moreover, links to past and ongoing research in Essen are mentioned, as well as some consequences of Serre’s conjecture. The article first surveys the objects involved in Serre’s conjecture: modular forms and Galois representations. According to a theorem by Deligne and Shimura, any Hecke eigenform gives an odd 2-dimensional Galois representation. This is illustrated by means of a simple example. Serre’s conjecture postulates that the converse is also true: any odd 2-dimensional irreducible Galois representation comes from a Hecke eigenform. The remainder of the article is devoted to explaining these objects in more detail. Different types of ’numbers’ and ’fields’ are introduced, discussed and compared: algebraic numbers, real numbers, complex numbers, number fields and finite fields. Subsequently, complex geometry, in particular Riemann surfaces, are touched upon, and geometry over number fields (arithmetic geometry) is mentioned. The article emphasizes the role played by symmetries. It takes the point of view that modular forms link geometry and number theory via symmetries: Möbius transforms, Hecke operators and Galois and Frobenius automorphisms are united as different kinds of symmetries. These objects are explained. Modular forms are presented as objects coming from and being rooted in geometry. The link between modular forms and Galois representations provided by Serre’s modularity conjecture is finally explained in more detail and one small glimpse on its proof is provided. A final section puts Serre’s conjecture into the context of Langlands’ philosophy.