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Anweshi
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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".

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..

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".

Source Link
Anweshi
  • 7.4k
  • 11
  • 75
  • 99

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..