27

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?

YCor
  • 60,149
Tim Campion
  • 60,951
  • 1
    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
  • 2
    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 03:51
  • 2
    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 08:23
  • 1
    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
  • 1
    @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... – Tim Campion Dec 15 '18 at 00:31
  • @TimCampion: I didn't say that Hartshorne works with number fields, but there is a big difference between that and only working over $\mathbb C$. There is a lot there about general schemes. For instance pretty much everything in Chapter II and a lot in Chapter III applies to schemes over number fields. For instance he does cohomology of affine schemes and projective spaces over an arbitrary noetherian ring, so for example over a number field. – Sándor Kovács Dec 15 '18 at 01:11
  • @TimCampion (cont'd): Of course,[Hartshorne] is not a number theory book, but you are saying you want to learn more algebraic geometry. Then why not read Hartshorne? You seem to be saying it is too specialized. I never heard that one before. – Sándor Kovács Dec 15 '18 at 01:12
  • @TimCampion: In any case, just so I would not be simply contrarian, here is a recommendation: Michael Rosen has a book entitled "Number Theory in Function Fields". One could argue that that's truly a cross between number theory and algebraic geometry as the "function field" case is usually the "geometric" version of a number theory question originally posed over number fields. – Sándor Kovács Dec 15 '18 at 01:16
  • @TimCampion (cont'd): A good example of this is Mordell's Conjecture. The number field case of that was done by Faltings, using something called "Parshin's trick". Parshin came up with that trick as he gave a new proof of the function field case which had been proved earlier by Manin in a different way. – Sándor Kovács Dec 15 '18 at 01:16
  • @TimCampion (ps) Having looked at your comment below, I'm afraid Rosen's book will also not be what you are looking for. Perhaps you will have to write that book. :) – Sándor Kovács Dec 15 '18 at 01:19
  • @SándorKovács Thanks, this is helpful! I'm starting to suspect that I'm most likely to find what I'm looking for in some text written with the aim of detailing the number field / function field analogy. – Tim Campion Dec 15 '18 at 21:33
  • 1
    By the way, Qing Liu's textbook on algebraic geometry is better than Hartshorne's in this aspect. – Z. M Dec 09 '22 at 17:32

3 Answers3

8

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.

Carlo Beenakker
  • 177,695
  • 1
    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. – Malkoun Dec 13 '18 at 22:17
  • 8
    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. – Tim Campion Dec 13 '18 at 22:20
5

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.

Gabriel
  • 943
3

(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.

efs
  • 3,099