3
$\begingroup$

My long term goal for this "reading/study project" is to understand roughly what the Langlands conjectures are about. A more modest short term goal though, and more realistic one, is to understand:

1) the basics of Algebraic Number Theory, with a lot of examples worked out,

2) the basics of class field theory, but explained from a modern point of view, using adeles and ideles, eventually. However, I would like the abstraction to be gradual, so to speak, and motivated by a few worked out examples.

My background includes of course the usual Graduate Algebra courses, but I did read on my own quite a bit of commutative algebra and algebraic geometry. So I know what is a Noetherian ring, localization, the ring of integers in a number field, a UFD, a PID, the Galois group etc. I would like to know more about Dedekind domains and "onwards".

I would possibly like references that make analogies with algebraic geometry say, via schemes, for instance (I know the basics of schemes but I prefer an approach with many examples, kind of like, say, Eisenbud and Harris's book "The Geometry of Schemes", but with more examples worked out on the Algebraic Number Theory side).

I realize that my requirements may not all be met at once, but I will take whatever I can get, so to speak, in terms of advice and recommendations. Thank you!

$\endgroup$
  • 1
    $\begingroup$ From what you're describing, Neukirch's book could be a good start. Or Milne's notes $\endgroup$ – Mathmo123 May 16 '18 at 15:54
  • 1
    $\begingroup$ See math.stackexchange.com/questions/90972/… $\endgroup$ – lhf May 16 '18 at 15:54
  • 1
    $\begingroup$ J.S. Milne's Algebraic Number Theory and Fields and Galois Theory $\endgroup$ – emma May 16 '18 at 16:00
  • 1
    $\begingroup$ And, when you get to it, Benedict Gross's (youtube) Eilenberg lectures are a great intro to the Langlands program.... $\endgroup$ – peter a g May 16 '18 at 16:02
  • $\begingroup$ @Mathmo123, yes, I have stumbled on Milne's notes, and love them. Thank you for Neukirch's book. I will check it out. lhf, thank you, yes I just checked 2 posts which ask for references for class field theory. It seems each one would like to learn things in a certain way, but only a limited number of books/approaches are available. $\endgroup$ – Malkoun May 16 '18 at 16:03
0
$\begingroup$

I thank everybody for their comments. I have decided to use Milne's lecture notes http://jmilne.org/math/CourseNotes/index.html. I am starting with his algebraic number theory notes for now. It would take a while before getting to the Langland's program, but anyway, the process is fun. I appreciate more now the definition of Dedekind domains, which, from what I understand, took a while to formulate as the right setting for the rings of integers in number fields for instance (talking as a non-expert of course).

Edit: Neukirch's book is really nice by the way. I am now reading parts of it. Thank you @Mathmo123 for suggesting it, and I thank Prof. Khuri-Makdissi for also suggesting it (among others).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.