The Homomorphism Form of Birational Anabelian Geometry

The University of Exeter EPSRC DTP (Engineering and Physical Sciences Research Council Doctoral Training Partnership) is offering up to 4 fully funded doctoral studentships for 2019/20 entry. Students will be given sector-leading training and development with outstanding facilities and resources. Studentships will be awarded to outstanding applicants, the distribution will be overseen by the University’s EPSRC Strategy Group in partnership with the Doctoral College.

Supervisors:

Professor Mohamed Saidi, Department of Mathematics, College of Engineering, Mathematics and Physical Sciences
Dr Julio Andrade, Department of Mathematics, College of Engineering, Mathematics and Physical Sciences

Project description:

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 isomorphisms between Galois groups of finitely generated fields and the extent to which one can reconstruct isomorphisms between the fields themselves starting from an isomorphism between Galois groups. 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 finite fields. 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?

A successful applicant will have to study in details the methodology used by myself and Tamagawa in order to prove the above mentioned result, and then start working out ways to adapt this methodology (even partially)to the above question. About a year of study is required to become familiar with this methodology, which includes techniques from algebraic number theory, Galois cohomology, and class field theory. Regular meetings will be held with the first supervisor in order to monitor progress regarding the methodology and discuss new technical difficulties arising and related to the above question.

A potential candidate should be familiar with Galois theory and number theory. Especially the theory of number fields and their extensions. A successful candidate will have to become familiar with the techniques and topics of Galois cohomology and class field theory as outlined for example in the Book by Neuckirch-Shmidt-Winberg : Cohomology of number fields.