Riigid Geomerty of Curves and Their Jacobians

• Applicant: Professor Dr. Werner Lütkebohmert
• Subject Area: Mathematics
• Term: from 2015 to 2016
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 282330905

In this research program we consider analogs of compact Riemann surfaces, which are defined over non-archimedean fields K. As in the complex case they are smooth projective algebraic curves. One has also the notion of closed paths and, hence, of a homotopy group of closed paths, but this group does not contain so much information as in the complex case. The rank of this group can range from 0 to g, where g is the genus of the curve, whereas in the complex case the rank is 2g. If the rank is g in the non-archimedean case, such curves were introduced by Mumford. In this case one can construct a polarization starting out from the homotopy group of paths as invented by Drinfeld and Manin. One can show that this analytically defined polarization coincides with the canonical theta polarization which is algebraically defined. This fact is widely studied in §2.9 of my forthcoming book `Rigid Geometry of Curves and their Jacobians´. The challenge of this new research program is the study of polarizations in the case where the rank is less than g.

• DFG Programme: Research Grants