From Classical L-Functions to Modern Reciprocity Laws

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.

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.

Claus Sorensen received his Ph.D. in Mathematics from the California Institute of Technology in 2006. Following graduation Sorensen was employed at Princeton University, first as an Instructor from 2006 to 2009, then as an Assistant Professor from 2009-2013. Professor Sorensen's main area of research is Number Theory and Representation Theory centering on the interaction between automorphic forms and Galois representations, commonly referred to as the Langlands program.