Research Projects of Prof. Derenthal
One of the oldest problems in number theory is: what are the rational solutions of diophantine equations? In the language of arithmetic geometry, this can be reformulated as follows: given an algebraic variety defined by a diophantine equation, what are its rational points?
The number theory group is concerned with fundamental questions regarding the existence and distribution of rational points on algebraic varieties: When do rational points exist (Hasse principle)? What is their local distribution (weak and strong approximation)? What is their global distribution (Manin's conjecture)?
We investigate these questions with a combination of algebraic, geometric and analytic methods. Here, universal torsors are a central geometric tool. They can be described explicitly using Cox rings. Quantitative statements regarding rational points are usually obtained via analytic methods. If the Hasse principle or weak/strong approximation fails, this is often explained algebraically via Brauer-Manin obstructions.
These questions are particularly interesting for cubic surfaces, for example. While the geometry of smooth cubic surfaces is well understood, the questions above regarding their arithmetic are open. For some singular cubic surfaces, we have proved Manin's conjecture.
More details on Ulrich Derenthal's work can be found on his personal webpage.