3
$\begingroup$

I am studying class field theory. I need good reference books, notes, or other materials which explain the following topics: ideles and ideals, Haar measure and integration on local fields, Fourier analysis on local fields, Fourier transform, multiplicative characters, and local zeta functions.

I know the book "Algebraic Number Theory" by Cassels and Fröhlich. I studied its chapter on global fields, and I know the definition and first few properties of adeles and ideles. Then I tried to read the chapter on Tate's thesis but I find it very difficult. Actually, I am not so good in analysis and I don't have a clear conception of the Haar measure.

It will be very helpful if someone suggests textbooks appropriate as an introduction to the subject of class field theory and where the above topics are well explained with all the details.

Improve this question
$\endgroup$
1

3 Answers 3

Reset to default
3
$\begingroup$

Ramakrishnan and Valenza's Fourier Analysis on Number Fields has all the topics you name, but doesn't prove the main theorems of class field theory. If you want a book which has all of this and also the class field theory proofs (but moves a lot faster) try Weil's Basic Number Theory.

Improve this answer
$\endgroup$
2
$\begingroup$

Ramakrishnan and Valenza, Fourier Analysis on Number Fields gives a gentle treatment of Tate's thesis.

Improve this answer
$\endgroup$
0
$\begingroup$

Try Deitmar and Echterhoff: Principles of Haromic Analysis. The second edition contains a chapter on adeles and ideles.

Improve this answer
$\endgroup$

Not the answer you're looking for? Browse other questions tagged .