The Homomorphism Form of Birational Anabelian Geometry

本计画旨在探讨双有理阿那贝耳几何研究领域的新课题与新方向。双有理Anabel几何的主要研究课题是研究Galois群之间的同构，以及在何种程度上可以从Galois群之间的同构开始重建域之间的同构。最近，我和我在京都的合作者玉川昭夫（Akio Tamagawa）一起，在整体域的情况下证明了这样一个理论的一个新的改进版本：数域，或有限域上曲线的函数域。更确切地说，我们证明了两个整体域的三步可解伽罗瓦群之间的同构的存在意味着两个整体域之间的同构的存在。本研究课题的目的是探讨以下问题，这自然产生后，上述结果：从一个开放的同态之间的三步可解的伽罗瓦群的两个整体领域是有可能重建的域之间的嵌入问题？

This project aims to explore new topics and directions in the research area of birational anabelian geometry. The main research topic in birational anabelian geometry is the study of isomorphismsbetween Galois groups of finitely generated fields and the extent to which one can reconstruct isomorphisms between the fields themselves starting from an isomorphism between Galoisgroups. Recently, with my collaborator Akio Tamagawa in Kyoto, we proved a new refined version of such a theory in the case of global fields: number fields, or function fields of curves over finitefields. More precisely, we proved that the existence of an isomorphism between the three step solvable Galois groups of two global fields implies the existence of an isomorphism between the two global fields in question. This research topic aims to explore the following question, which arises naturally after the above result:Starting from an open homomorphism between the three step solvable Galois groups of two global fields is it possible to reconstruct an embedding between the fields in question?