Timeline for Where's the best place for an algebraic geometer to learn some algebraic number theory?
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
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| Dec 9, 2022 at 17:32 | comment | added | Z. M | By the way, Qing Liu's textbook on algebraic geometry is better than Hartshorne's in this aspect. | |
| Mar 24, 2021 at 10:01 | history | edited | YCor |
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| Mar 24, 2021 at 9:58 | answer | added | Gabriel | timeline score: 6 | |
| Dec 19, 2018 at 16:08 | answer | added | efs | timeline score: 3 | |
| Dec 15, 2018 at 21:33 | comment | added | Tim Campion | @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. | |
| Dec 15, 2018 at 1:19 | comment | added | Sándor Kovács | @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. :) | |
| Dec 15, 2018 at 1:16 | comment | added | Sándor Kovács | @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. | |
| Dec 15, 2018 at 1:16 | comment | added | Sándor Kovács | @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. | |
| Dec 15, 2018 at 1:12 | comment | added | Sándor Kovács | @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. | |
| Dec 15, 2018 at 1:11 | comment | added | Sándor Kovács | @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. | |
| Dec 15, 2018 at 0:31 | comment | added | Tim Campion | @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... | |
| Dec 14, 2018 at 23:52 | comment | added | Sándor Kovács | I must have missed something when I read [Hartshorne]...Where does it say that it restricts to working over $\mathbb C$? | |
| Dec 14, 2018 at 8:23 | comment | added | M.G. | 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. | |
| Dec 14, 2018 at 3:51 | comment | added | Asvin | 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. | |
| Dec 13, 2018 at 22:33 | comment | added | Tim Campion | Thanks, that was edifying. But again it's less systematic than I was hoping for. | |
| Dec 13, 2018 at 22:20 | comment | added | Malkoun | 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). | |
| Dec 13, 2018 at 22:04 | answer | added | Carlo Beenakker | timeline score: 9 | |
| Dec 13, 2018 at 19:49 | history | asked | Tim Campion | CC BY-SA 4.0 |