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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.

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This should be community wiki? –  Anweshi Jan 27 '10 at 12:52

6 Answers 6

up vote 29 down vote accepted

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.

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O Peter, you did it! You mean the Encyclopedia of Mathematics volume, right? –  Anweshi Jan 27 '10 at 16:58
    
Well, Manin and Panshishkin did it, praised may they be! And yes, it is Encyclopaedia of Math. Sciences Nr. 49, Springer, look out for the 2nd edition –  Peter Arndt Jan 27 '10 at 17:08

I don't know of any document that gives an accessible overview of all of the topics that you list. Number theory as a whole may be too big for such a thing to be possible. The number of people who have both deep knowledge of all listed topics and the expository skills to make them accessible may be very small.

However, I have seen some good sources for small clusters of the topics that you list. One book that I would recommend is Anthony Knapp's Elliptic Curves book which, despite its title also covers the basics of the theory of modular forms (Fuchsian groups too) and explains how to deduce that a Hecke newform of weight 2 with integral coefficients corresponds to an elliptic curve (the "easy" converse of the modularity theorem). When I read it, a number of things made sense that hadn't previously. But the said book is kind of long, at 450 pages so it doesn't qualify as a review - I just mention it as a relatively accessible and interesting place to get started.

As I've mentioned elsewhere on this site, I like Andre Weil's "Two lectures on number theory. past and present." L'Enseignement Mathematique. Revue Internationale. fie Serie. 20: 87-110. 1974

available here. It focuses on the early history of number theory and touches on little of what you say but may be useful to you in placing modern developments in context.

You might also get something out of browsing Henri Darmon's "Rational Points on Modular Elliptic Curves" which is at a more advanced level than Knapp's book. My experience is that it goes too fast for a beginner to understand in detail, but it gives some idea of different topics in (the more structural parts of) number theory interact with one another.

I look forward to other responses to this question.

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Darmon's book is great, but who proofreads these things? :) –  Pete L. Clark Jan 27 '10 at 8:34

Your question about one book for number theory is like a non-mathematician asking about one book for all mathematics. It is simply not possible. It is a growing subject in various directions. The best I can attempt is to give a book each for each direction, approximating your question. It is impossible to give anything better than this.

For Analytic Number Theory, what you ask can be achieved by:

Iwaniec And Kowalski, Analytic Number Theory.

This is THE book. It is quite comprehensive. Includes L-functions, modular forms, random matrices, whatever.

For algebraic number theory, the book:

Cassels and Frohlich, Algebraic Number Theory

would tell you all about developments upto Classfield Theory and Tate's thesis. Includes the cohomological version. This is a MUST for algebraic number theorists.

For Langlands' program, use the reference that Pete gives.

For Iwasawa theory, there are two books by Coates and Sujatha.

You might want to know a bit more about the applications of algebraic geometry into number theory. The way to go is through Silverman on elliptic curves, Q. Liu's book, Serre's books, etc..

A historic overview up to the time of Legendre can be found in Weil's book, "Number theory through history: From Hammurapi to Legendre".

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This is a comment to Anweshi. I posted it as an answer because of its length.

@Anweshi - There's a funny story about Weil's Basic Number Theory which Shimura describes on page 139 of his autobiography My Life. Apparently, when the book came out, another member of the IAS protested vehemently about the title; class field theory should not be called basic he reasoned (the member took offense to the title, thinking that Weil had implied he did not possess even a basic knowledge of number theory). The reason that Weil titled the book as he did, was of course because the book did not contain any new results - only results that were by that time well known. Although I look at the book occasionally, my personal feeling is that any book that brings up the Haar measure on a locally compact group before the end of page 3 is perhaps not basic enough.

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Yes, strange guy, he does the whole theory of local fields based on this Haar measure invention of him. Maybe it is natural in some perspective, and more modern, as he claims. On the other hand he refuses to deal with cohomology and restricts himself to central simple algebras. I scratched my head all the while and moreover his writing style is extremely hard to decipher. –  Anweshi Jan 27 '10 at 15:09
    
I think Shimura indeed supports Weil's statement that it was basic number theory from some perspective? –  Anweshi Jan 27 '10 at 15:11
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I can't imagine that Weil didn't notice, intend, or appreciate that other meaning of the title Basic Number Theory. It's a nasty trick to recommend it to someone without explanation. –  Douglas Zare Jan 28 '10 at 11:50

There are quite a few good books on class field theory. I think that Janusz' book Algebraic Number Fields stands out among them as one of the friendliest (Milne's notes are also quite good). As far as local class field theory is concerned, I think that one would be hard pressed to find a better resource than Serre's book Local Fields. Finally, Artin and Tate's book Class Field Theory was recently reprinted by the AMS. It is a fantastic book, but definitely falls under the 'you should only attempt to read this book after already having gained an understanding of the field' heading.

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Cassels&Frohlich too is good. Weil's book Basic Number Theory offers a rather strange perspective, and is quite hard to read -- but instructive. –  Anweshi Jan 27 '10 at 14:41

Ramakrishnan and Valenza, "Fourier analysis in number fields" is quite nice. Serre's "Cours d'arithmétique" is pretty good too -- and starts basic ("basic" as in "simple" not in "base" -- it has nothing to do with Weil's joking title).

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