Otmar Venjakob (Universität Heidelberg): Explicit Reciprocity Laws in Number Theory

The quadratic Reciprocity Law for the Legendre or Jacobi-Symbol forms the starting point of all Reciprocity Laws as well as of class field theory. It is closely related to the product formula of the quadratic Hilbert-Symbol over local fields. Various mathematicians have established higher explicit formulae to compute higher Hilbert-Symbols. Analoga were found for formal (Lubin-Tate) groups. Eventually Perrin-Riou has formulated a Resciprocity Law, which allows the explicit computation of local cup product pairings by means of Iwasawa- and $p$-adic Hodge Theory. In this talk I shall try to give an overview of these topics, at the end I will  explain recent developments in this regard.