The explicit Bombieri-Lang conjecture for surfaces

点击翻译按钮获取中文摘要

The main focus of this project is the study of the set X(K) of K-rational points on a variety X of general type over a number field K. If X is a curve, then X is of general type if and only if X is a curve of genus at least 2. For these curves, Mordell conjectured and Faltings proved that X(K) is always finite. This statement is the one-dimensional case of the weak form of the (Bombieri-)Lang Conjecture:If X is a variety of general type defined over a number field K, then X(K) is not Zariski dense in X.In general this conjectures is wide open. By results of Faltings, they hold for curves and more generally for subvarieties of abelian varieties.However, even in these cases it is far from clear that a suitable explicit description of X(K) can be determined. In the case of curves, there has been considerable progress in recent years, but for surfaces the situation is quite open. For a surface of general type, the conjecture asserts that outside a finite number of curves of geometric genus at most 1, there are only finitely many rational points.This project aims at proving the conjecture for certain types of surfaces on the one hand and at the development of explicit methods to determine X(K) on the other hand.Bauer is a well-known expert on surfaces of general type and their moduli, whereas Stoll is a leading expert on the arithmetic of curves and algorithmic methods. The proposed project aims at combining this expertise to provide an ideal environment for carrying out PhD and postdoctoral research projects related to the study of rational points on surfaces of general type under the joint supervision of the PIs.