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Student
Title
Supervisor
Faculty
Educational Programme
Final Grade
Year of Graduation
Stepan Kazanin
Reconstruction of Fields and Projective Geometry
Mathematics
(Bachelor’s programme)
7
2019
Anabelian geometry is, to put it simply, a field of diophantine geometry concerned with reconstructing varieties over number fields from its etale fundamental group. One result that predates anabelian geometry is a theorem of Neukirch and Uchida (\cite{Neukirch1}, \cite{Neukirch2}, \cite{Neukirch3}), which claims that a number field is determined by its absolute Galois group. This can be restated in the geometric spirit as reconstruction of a $0$-dimensional variety from its etale fundamental group. Substantial work on the $1$-dimensional case was carried out by Shinichi Mochizuki in \cite{Mochizuki1}, \cite{Mochizuki2}. Modern studies retrieve variety from little parts of the fundamental group, such that its pro-$p$ completion or even some structures on Galois cohomology groups. This paper adresses a different but similar problem: reconstructing variety from structures on the multiplicative group of its function field. Our main result, theorem \ref{main}, was never fully stated in the literature, although it can be obtained without much trouble with known methods. In particular, we include full proof of theorem \ref{fundamental} as we were unable to find the most general statement in the literature.

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