Inverse Galois Problem and torsion points on abelian varieties

逆伽罗瓦问题和阿贝尔簇上的扭点

• 批准号: 1940106
• 金额: --
• 依托单位: University of Bristol
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2017
• 资助国家: 英国
• 起止时间: 2017 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-1940106
• 关键词: Inverse; Galois; Problem; torsion; points

One of the biggest unsolved questions in number theory is the Inverse Galois Problem. It is to show that every finite group can occur as a Galois group over the rationals. Abelian groups are well-understood through class field theory, and it is known (though it is very hard) that soluble groups can be realised as well. So a lot of work has been dedicated to simple and almost simple groups, and they can sometimes be realised from Galois groups of torsion points on curves and abelian varieties. This requires being able to construct curves that have some specific behaviour - automorphisms, endomorphisms, reduction types etc. Pip Goodman's research topic is to work in this direction, using recent developments in the theory of curves, their models, and their Jacobians.

数论中最大的未解决问题之一是伽罗瓦逆问题。这是为了证明每个有限群都可以作为有理数上的伽罗瓦群出现。阿贝尔群通过类域理论得到了很好的理解，并且已知（尽管很难）可解群也可以实现。因此，大量的工作一直致力于简单和几乎简单的群体，他们有时可以实现从伽罗瓦群的扭点的曲线和阿贝尔品种。这需要能够构建曲线，有一些特定的行为-自同构，自同态，减少类型等皮普古德曼的研究课题是在这个方向上工作，利用最近的发展理论的曲线，其模型，和他们的雅可比。