Inverse Galois Problem and torsion points on abelian varieties

数论中最大的未解决问题之一是反伽罗瓦问题。目的是证明每一个有限群都可以作为有理数上的伽罗瓦群出现。通过类场理论可以很好地理解阿贝尔群，并且已知（尽管很难）可溶群也可以实现。所以很多工作都致力于简单和几乎简单的群，它们有时可以通过曲线和阿贝尔变体上的扭转点的伽罗瓦群来实现。这需要能够构造具有某些特定行为的曲线——自同构、自同态、约简类型等。皮普·古德曼（Pip Goodman）的研究课题就是在这个方向上工作，利用曲线理论、曲线模型和雅可比矩阵的最新发展。

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.