I've attempted going past basic number theory several times, and always got lost in its vastness. Do any of you, perhaps, know a good review that pieces together the many concepts involved (Hecke algebras, SL2(ℤ), Fuchsian groups, L-functions, Tate's thesis, Ray class groups, Langlands program, Fourier analysis on number fields, cohomological versions of CFT, Iwasawa theory, modular forms, ...)?


Thanks.



Map of Number Theory
The book you are looking for exists!! And indeed it contains ALL the buzzwords in your question!


It is Manin/Panchishkin's "Introduction to Modern Number Theory". This is a survey book that starts with no prerequisites, contains very few proofs, but nicely explains the statements of central theorems and the notions occurring therein and gives motivations for the questions that are being pursued. You should take a look, at least it can help you decide what you want to study in more detail.


