I am studing Class field theory. I need a good reference books, notes etc. which explains the following topics : Ideles and ideals, haar volume measure and integration on local fields, Fourier analysis on local fields, Fourier transform, ultiplicative charaters and local zeta functions.

I kow the book by Cassels and Fröhlich and I learnt the 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 thesis but I find it very difficult. Actually I am not so good in Analysis, and I don't have a clear idea of Haar measure.

It will be very helpful if someone suggests me some books where the above topics are well explained with all details and accessable for first reading.

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    $\begingroup$ - Fourier analysis on local fields, by M. H. Taibleson, 1975. $\endgroup$ Oct 26 '14 at 16:36

As already commented, the book Fourier Analysis on local fields by Taibleson is a reference for Fourier analysis on number fields. This book contains all the basic facts of Fourier analysis on local fields and has a spirit similar to books covering Fourier analysis on Euclidian spaces.

Another good reference is the book Fourier Analysis on number fields by Ramakrishnan & Valenza. The content of this book overlaps with Taibleson's book at several places, but distinguishes itself from Taibleson's in that it contains a thorough treatment of commutative harmonic analysis and number theory including topics as the existence and uniqueness of the Haar measure, the Pontryagin duality theorem, adeles & ideles, class field theory et cetera. In fact, the book covers all the background of harmonic analysis and number theory that is necessary in order to understand and appreciate the content of Tate's thesis.

Although both aforementioned books are devoted to Fourier analysis on local fields, the book by Ramakrishnan & Valenza focuses more on the number theoretical aspects whereas the book by Taibleson focuses more on the analysis on local fields.


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