Vorträge in der Woche 22.09.2008 bis 28.09.2008

Montag, 22.09.2008: Stability and Arithmetic

Lin Weng

Stability plays key roles in arithmetic as well as geometry. In this talk, we will touch two aspects. Globally, we will use stability to introduce a genuine non-abelian zeta functions for number fields. Basic properties and their relations to abelian zetas associated to (reductive groups, maximal parabolics) will be explained. This will reveal the hidden role played by symmetry in the Riemann Hypothesis. Locally, we will outline a conjectural Micro Reciprocity Law connecting de Rham representations with what we call filtered (phi,N;omega) modules. This is an analog of (Narasimhan-)Seshadri's fundamental result on stable parabolic bundles over punctured Riemann surfaces, as a continuation of the works of (Colmez-)Fontine for semi-stable representations. Based on this, we will establish a general non-abelian Class Field Theory for p-adic number fields.