I am currently interested in algebraic number theory and I wanted to learn more about it. The problem is that the course algebraic number theory at my university is for people without any background in algebraic geometry. But for me this seems not that interesting because as far as I know algebraic number theory can be done using algebraic geometry and since I know a bit about schemes, cohomology and varietes I wanted to ask if there is any reference for learning algebraic number theory which uses algebraic geometry.

Since I did not go into detail of what I know about algebraic geometry I would be glad if you could also mention prerequisites for the given reference.

  • $\begingroup$ If it helps, arithmetic geometry is a field that is concerned with the number theory-side of algebraic geometry, so to speak $\endgroup$
    – Krijn
    Jan 24 '18 at 13:40
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    $\begingroup$ What is your background in algebraic number theory? Are you familiar with the content of a first course (global/local fields) and looking for more? Or are you looking for a geometric approach to a first course? $\endgroup$
    – Mathmo123
    Jan 24 '18 at 14:04
  • $\begingroup$ I don't know of a reference that covers a first course in ANT purely geometrically. But Neukirch's book does give a geometric outlook on top of the more algebraic one (e.g. there is a chapter on one-dimensional schemes). $\endgroup$
    – Mathmo123
    Jan 24 '18 at 14:07

You'll find that Milne has a collection of (in my opinion) excellent notes here.

He indicates which of his documents depend on one another by means of a 'required' and 'useful' column. His Abelian varieties notes require AG and ANT, and suggest the CFT notes would be useful. I believe it would be feasible to gain a deep appreciation for the ANT you care about by using the Abelian varieties notes to give yourself direction, and the CFT and ANT notes to understand the content.


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