The explicit Bombieri-Lang conjecture for surfaces

曲面的显式 Bombieri-Lang 猜想

• 批准号: 327638107
• 负责人: Professorin Dr. Ingrid Bauer
• 金额: --
• 依托单位: Mathematisches Institut
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2016
• 资助国家: 德国
• 起止时间: 2015-12-31 至 2022-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/327638107?language=en
• 关键词: explicit; Bombieri; Lang; conjecture; surfaces

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.

本项目的主要研究内容是数域K上一般类型簇X上的K-有理点集X（K）。如果X是一条曲线，则X是一般型的当且仅当X是亏格至少为2的曲线。对于这些曲线，Mordell和Faltings证明了X（K）总是有限的。这个陈述是（Besieri-）Lang猜想的弱形式的一维情形：如果X是定义在数域K上的一般类型的簇，则X（K）在X中不是Zesierki稠密的。根据Faltings的结果，它们对曲线和更一般的阿贝尔簇的子簇都成立，然而，即使在这些情况下，也远不能清楚地确定X（K）的适当的显式描述。在曲线的情况下，近年来已经有了相当大的进展，但对于曲面的情况是相当开放的。对于一般型曲面，猜想是指在有限个几何亏格至多为1的曲线之外，只有1/2个有理点。本项目的目的一方面是证明某些类型曲面的猜想，另一方面是发展确定X（K）的显式方法。Bauer是一般型曲面及其模的著名专家，而Stoll是曲线算法和算法方法的领先专家。拟议的项目旨在结合这些专业知识，为在PI的联合监督下开展与一般类型表面上的合理点研究相关的博士和博士后研究项目提供理想的环境。