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... Show moreThe 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. Show less
