From Classical L-Functions to Modern Reciprocity Laws by Sorensen, Claus;

Long description:

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The distinctive compilation of topics in this text provides readers with a smooth and leisurely transition from basic number theory to graduate topics courses on the Langlands program. It is a unique and self-contained resource for number theorists, instructors teaching basic analytic number theory, and the target readership of first and second year graduate students interested in number theory. Portions of the content are also accessible to mathematically mature advanced undergraduates. The copious number of exercises and examples throughout the text, aptly guide the reader. The prerequisite for using the book is a grounded understanding of number fields and local fields.

The book is well designed in its aims to build a bridge from beginning number theory topics to modern, advanced ones. Starting from scratch with the classical theory (Riemann's zeta function, Dirichlet L-functions, Dedekind zeta functions) it merges into a detailed account of Artin L-functions, Tate's thesis, and culminates in a discussion of the Deligne-Serre theorem and related results. These topics have not appeared together in book form.

Table of Contents:

Preface.- 1 Classical zeta and L-functions.- 2 Tate's thesis.- 3 Hecke characters versus Galois characters.- 4 Artin L-functions.- A streamlined introduction to modular forms.- 6. Galois representations attached to weight one forms.- Appendix A Some background material.- References.