Universiteit Leiden

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Dissertation

Global Fields and Their L-functions

Artin L-functions associated to continuous representations of the absolute Galois group G_K of a global field K capture a lot of information about G_K as well as arithmetic properties of K.

Author
Solomatin, P.
Date
02 March 2021

Artin L-functions associated to continuous representations of the absolute Galois group G_K of a global field K capture a lot of information about G_K as well as arithmetic properties of K. In the first part of the present thesis we develop basic aspects of this framework, starting from the well-known theory of arithmetically equivalent number fields which corresponds to the case of permutation representations of G. Then, based on work of Bart de Smit, we show how to completely recover the isomorphism class of K using Artin L-functions of monomial representations, i.e. representations induced from abelian characters. This allows us to provide an alternative approach to the famous Neukirch-Uchida theorem, which is a central result in anabelian geometry. In the second part of the thesis we shift our attention towards the case of global function fields and show two different approaches to possible generalizations of the results from the first part. Finally in the last part of the dissertation we study invariants of the maximal abelian quotient of G. In particular, we provide more examples of non-isomorphic imaginary quadratic number fields K whose abelianizations of the absolute Galois groups share the same isomorphism class and also prove that infinitely many non-isomorphic pro-finite groups occur as abelianization of G_K for some K. We finish the section with a complete classification of G_K^{ab} in the case of global function fields.

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