Universal adelic groups for imaginary quadratic number fields and elliptic curves

Leiden Repository

Universal adelic groups for imaginary quadratic number fields and elliptic curves

Type: Doctoral Thesis
Title: Universal adelic groups for imaginary quadratic number fields and elliptic curves
Author: Angelakis, Athanasios
Publisher: Mathematical Institute , Faculty of Science, Leiden University & Ecole Doctorale de Mathématiques et Informatique, Université Bordeaux I
Issue Date: 2015-09-02
Keywords: Absolute Galois group
Class field theory
Galois representations
Elliptic curves
Abstract: The 1st chapter is of an introductory nature. It discusses the basic invariants of algebraic number fields and asks whether or to which extent such invariants characterize the number field. It surveys some of the older results in the area before focusing on the case of absolute abelian Galois groups that occurs center stage in the next two chapters, and on a question for elliptic curves that can be attacked with the techniques from those two chapters. Chapters 2 & 3 include also the non-trivial proof of the fact that the key criterion to find imaginary quadratic fields with `minimal’ absolute abelian Galois groups can also be used to find Galois groups that are "provably" non-minimal. Chapter 4 moves in a different direction. It explicitly computes adelic point groups of elliptic curves over the field of rational numbers, and shows that the outcome can be made as explicit as in the case of the minimal absolute abelian Galois groups, and, in an even stronger sense than in that case, barely depends on the particular elliptic curve. The results obtained do generalize to arbitrary number fields, and it is this generalization that we plan to deal with in a forthcoming paper.
Description: Promotores: P. Stevenhagen, K. Belabas
With Summary in Dutch, French and Greek
Faculty: Faculteit der Wiskunde en Natuurwetenschappen
Citation: Angelakis, A., 2015, Doctoral Thesis, Leiden University & Université Bordeaux I
Handle: http://hdl.handle.net/1887/34990

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application/pdf Chapter 1 402.5Kb View/Open
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application/pdf Bibliography 198.2Kb View/Open
application/pdf Abstract 260.8Kb View/Open
application/pdf Summary in Dutch 261.2Kb View/Open
application/pdf Summary in French 269.6Kb View/Open
application/pdf Summary in Greek 309.4Kb View/Open
application/pdf Acknowledgements_Curriculum Vitae 163.3Kb View/Open
application/pdf Propositions 189.7Kb View/Open

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