# Where's the best place for an algebraic geometer to learn some algebraic number theory?

There are lots of introductions to number theory out there, but typically they are streamlined to assume as little prerequisite knowledge as possible. I'm looking for a text which does the opposite -- assumes you are fluent in algebraic geometry, and builds on that knowledge to introduce number theory. This is kind of a converse to this question. I'd like to believe, for example, that if you know enough algebraic geometry, then you could start number theory from scratch and get through class field theory in a semester.

(Not that I'm actually fluent in algebraic geometry, but I do know more algebraic geometry than I know number theory, and as a matter of fact, I'd like to use learning number theory as a springboard to learn more algebraic geometry, so I'm more than happy to look up unfamiliar algebraic geometry concepts as they arise. Possibly the real answer is that for anybody really fluent in algebraic geometry, the translations are so painfully obvious that a book about it is unneccessary...)

After all, I know lots of algebraic geometry was designed by the Grothendieck school to generalize stuff from number theory, but somehow this gets lost in translation when you learn from a text like Hartshorne which emphasizes the case where everything is over $$\mathbb C$$.

Questions:

1. Is there a text out there written as an introduction to algebraic number theory for people who know a lot of algebraic geometry?

2. Alternatively, has somebody written a "dictionary" which translates the distinctive number theory terminology (things like conductors, differents, discriminants,... come to mind) into algebraic geometry?

• It may not have a lot on the number-theoretic side, but did you have a look at the examples of arithmetic schemes in "The Geometry of Schemes", by Eisenbud and Harris? This may help a bit with the "dictionary" part of your question (or it may be too basic for you perhaps). – Malkoun Dec 13 '18 at 22:20
• Thanks, that was edifying. But again it's less systematic than I was hoping for. – Tim Campion Dec 13 '18 at 22:33
• If i remember correctly, Serre's local fields uses quite a bit of algebraic geometry language (or at least commutative algebra). This is a comment and not an answer because I don't remember exactly. – Asvin Dec 14 '18 at 3:51
• I have a vague memory that there was a book by Shafarevich and coauthors, or maybe with Shafarevich only as an (co)editor, that treated ANT from more algebro-geometric perspective. I will have to check later. – M.G. Dec 14 '18 at 8:23
• I must have missed something when I read [Hartshorne]...Where does it say that it restricts to working over $\mathbb C$? – Sándor Kovács Dec 14 '18 at 23:52