Arakelov and Minkowski Theory for S-Integers

Veltman, Niek (2024) Arakelov and Minkowski Theory for S-Integers. Master's Thesis / Essay, Mathematics.

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Abstract

The infrastructure is a group-like structure inside the equivalence class of the unit element of the class group of a real quadratic number field. It induces an algorithm that can compute the regulator of the real quadratic number field. Among other things, this thesis explores the potential of giving an Arakelov theoretical description of the infrastructure for so-called fake real quadratic orders; a specific type of S-integers in an imaginary quadratic number field. To see the utility of Arakelov theory, this thesis describes the Arakelov theoretical description of the original infrastructure. Moreover, it extends Arakelov theory to S-integers. This includes the study of the Arakelov S-class group. Two isomorphic groups are constructed, and the topology is examined. Furthermore, two definitions for reduced Arakelov S-divisors are suggested. To support Arakelov theory for S-integers, also Minkowski theory has been studied for these rings. Namely, this thesis shows how non-zero fractional ideals of S-integers can be viewed as lattices in the S-Minkowski space. Furthermore, it includes an analogue of Minkowski's Convex Body Theorem for the S-Minkowski space. To talk about lattices in this space, the structure of lattices in locally compact groups is examined. This includes the full description of fundamental regions and covolumes of lattices.

Item Type:	Thesis (Master's Thesis / Essay)
Supervisor name:	Muller, J.S. and Top, J.
Degree programme:	Mathematics
Thesis type:	Master's Thesis / Essay
Language:	English
Date Deposited:	20 Nov 2024 09:36
Last Modified:	20 Nov 2024 09:36
URI:	https://fse.studenttheses.ub.rug.nl/id/eprint/34430

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