Classical Conjectures in Iwasawa Theory for the split prime Z_p-extension and the cyclotomic Z_p-extension

by Katharina Müller
Doctoral thesis
Date of Examination:2021-03-26
Date of issue:2021-04-15
Advisor:Prof. Dr. Preda Mihailescu
Referee:Prof. Dr. Preda Mihailescu
Referee:Prof. Dr. Jörg Brüdern
Referee:Prof. Dr. Werner, Bley
Persistent Address: http://dx.doi.org/10.53846/goediss-8553

 

 

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Abstract

English

This thesis constist of three parts. The first one considers the so called split prime Z_p extension over an imaginary quadratic field in which the rational prime p splits. In this setup we discuss the mu=0 conjecture as well as the main conjecture for finite abelian extensions of K. The second part of the thesis concentrates on the cyclotomic Z_p extension of a CM number field. We give a Galois theoretic interpretation of the Gross and the Gross-Kuzmin conjecture under mild assumptions on K. In the third part we specialize to the case of p=2. He we look at the capitulation problem along the cyclotomic Z_2 extension in CM fields and determine the class groups of the finite layers of the cyclotomic Z_2 extension for a certain family of biquadratic base fields.
Keywords: Iwasawa Theory; Main Conjecture; Gross Conjecture; Capitulation; Class Groups; Z_p extensions
 
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