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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?

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    $\begingroup$ 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). $\endgroup$
    – Malkoun
    Dec 13, 2018 at 22:20
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    $\begingroup$ 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. $\endgroup$
    – Asvin
    Dec 14, 2018 at 3:51
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    $\begingroup$ 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. $\endgroup$
    – M.G.
    Dec 14, 2018 at 8:23
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    $\begingroup$ I must have missed something when I read [Hartshorne]...Where does it say that it restricts to working over $\mathbb C$? $\endgroup$
    – Sándor Kovács
    Dec 14, 2018 at 23:52
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    $\begingroup$ @SándorKovács True, Hartshorne develops most things in greater generality. But I seem to recall for example that most of the exercises involving specific schemes were over an algebraically closed field of characteristic 0 -- it seemed like you'd have to read between the lines a bit to apply things to number theory. Was I mistaken? If I go back to Hartshorne with fresh eyes, will I find more number theory than I remember? A quick search reveals only a handful of places in the book where the word "number field" appears, for example... $\endgroup$
    – Tim Campion ♦
    Dec 15, 2018 at 0:31

3 Answers 3

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Algebraic Number Theory by Jürgen Neukirch, states this as its aim:

The present book has as its aim to resolve a discrepancy in the textbook literature and to provide a comprehensive introduction to algebraic number theory which is largely based on the modern, unifying conception of (one-dimensional) arithmetic algebraic geometry.

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    $\begingroup$ I do like this book, and think it would be a good place to start too, given the OP's background! I also do not know ANT, and chose this book too. I also ordered ANT by Serge Lang, as well as "Number Fields" by Marcus. But they have not arrived yet, so I cannot comment yet on their contents. They have been recommended to me though by an expert. $\endgroup$
    – Malkoun
    Dec 13, 2018 at 22:17
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    $\begingroup$ Funny, Neukirch is actually what I've been reading from, and although I've learned a lot from it, it's not quite what I'm looking for. For instance, in Chapter I there is no algebraic geometry except for an impressionistic overview in section 13. I was really hoping for something where the algebraic geometry is more tightly integrated. I'm currently reading Chapter IV, section 3 on abstract Kummer theory, and I was a bit dismayed at some of the roundaboutness introduced because Neukirch apparently doesn't want to assume the reader is familiar with homological algebra. $\endgroup$
    – Tim Campion ♦
    Dec 13, 2018 at 22:20
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T. Szamuely had written two chapters (about Dedekind schemes and finite étale covers thereof) for his book Galois Groups and Fundamental Groups which weren't included in the final version of the book. Nevertheless, they are available here and they explain much of the basic theory underlying algebraic number theory using the language of schemes.

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(I'm certainly not the right person to answer this question.) Anyway, I think the "divisor theoretic" approach (to valuations, ramification, etc.) that Borevich and Shafarevich take in chapter 3 of "Number Theory" is based in Algebraic Geometry, and perhaps easy to follow to someone who knows Algebraic Geometry. This also appears in Koch's "Algebraic number theory" (edited by Parshin and Shafarevich), which is the book I think @M.G. is referring in his comment.

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