Algorithmic methods for arithmetic surfaces and regular, minimal models

数域上代数曲线的正则模型和极小模型是在算术几何中起重要作用的算术曲面。本研究项目旨在开发这类算术曲面的算法以及正则模型和极小模型的计算。主要的主题是遵循利普曼的去单元化程序，交叉点配对的功能，特殊的除数，吹气和吹气。在这些算法的基础上，研究了它们在Birch和Swinnerton-Dyer猜想等相关领域中的应用。

Regular and minimal models of algebraic curves over number fields are arithmetic surfaces that play an important role in arithmetic geometry. This research project aims at developing algorithms for such arithmetic surfaces and for the computation of regular and minimal models. The main topics are a desingularisation procedure following Lipman, functionality for the intersection pairing, exceptional divisors, blow ups and blow downs. On the basis of these algorithms applications to the Birch and Swinnerton-Dyer conjecture and other related areas are finally investigated.