Liu's book, Serre's books, etc.Ī historic overview up to the time of Legendre can be found in Weil's book, "Number theory through history: From Hammurapi to Legendre". The way to go is through Silverman on elliptic curves, Q. Shafarevich confirmed me the 1st variation and stated that this e-book could be any further the publication approximately classification box idea. You might want to know a bit more about the applications of algebraic geometry into number theory. This is a MUST for algebraic number theorists.įor Langlands' program, use the reference that Pete gives.įor Iwasawa theory, there are two books by Coates and Sujatha. A great deal of his work was directed towards.
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Would tell you all about developments upto Classfield Theory and Tate's thesis. Andr Weils work laid the foundation for abstract algebraic geometry and the modern theory of abelian varieties. Includes L-functions, modular forms, random matrices, whatever.Ĭassels and Frohlich, Algebraic Number Theory Iwaniec And Kowalski, Analytic Number Theory. It is impossible to give anything better than this.įor Analytic Number Theory, what you ask can be achieved by: The best I can attempt is to give a book each for each direction, approximating your question. It is a growing subject in various directions. Your question about one book for number theory is like a non-mathematician asking about one book for all mathematics. I look forward to other responses to this question. 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. 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. 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. past and present." L'Enseignement Mathematique. 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.Īs I've mentioned elsewhere on this site, I like Andre Weil's "Two lectures on number theory.
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When I read it, a number of things made sense that hadn't previously.
#WEIL BASIC NUMBER THEORY HOW TO#
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).
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However, I have seen some good sources for small clusters of the topics that you list. 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. Number theory as a whole may be too big for such a thing to be possible. I don't know of any document that gives an accessible overview of all of the topics that you list.