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hi everybody, I have already finished the Hartshorne's algebraic geometry from chapter 1 to chapter 4, so I'd like to find some suggestions about the next step to study arithmetic geometry. I want to know that how to use the scheme theory and their cohomology to solove the arithmeic problem.Would you like to recommend me some of these kind of books and papers?

Thank you very much!

PS: I also want to learn some materials about moduli theory, if you like, could you recommend me some books or papers ?

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From my point of view, it is very important to have a sound basis in algebraic number theory. –  Regenbogen Apr 16 '10 at 20:57

6 Answers 6

up vote 14 down vote accepted

My suggestion, if you have really worked through most of Hartshorne, is to begin reading papers, referring to other books as you need them.

One place to start is Mazur's "Eisenstein Ideal" paper. The suggestion of Cornell--Silverman is also good. (This gives essentially the complete proof, due to Faltings, of the Tate conjecture for abelian varieties over number fields, and of the Mordell conjecture.) You might also want to look at Tate's original paper on the Tate conjecture for abelian varieties over finite fields, which is a masterpiece.

Another possibility is to learn etale cohomology (which you will have to learn in some form or other if you want to do research in arithemtic geometry). For this, my suggestion is to try to work through Deligne's first Weil conjectures paper (in which he proves the Riemann hypothesis), referring to textbooks on etale cohomology as you need them.

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Do you happen to know where those Deligne papers were published? –  Joel Dodge Apr 16 '10 at 17:18
    
IHES, I think. It is easy to find on mathscinet, or probably just typing "Weil I Deligne" into Google will find it. –  Emerton Apr 16 '10 at 20:35
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Matt, not that it matters, but due to the multi-year gap between the two papers probably it wasn't called Weil I, for the same reason World War I wasn't called that at the time of its "creation" (who expected WWII?). –  BCnrd Apr 17 '10 at 3:12
    
I wondered about this, but nevertheless, typing in "Weil I Deligne" gives the wikipedia entry as the first link, which in turn gives the reference. (This is what I thought would happen. As it turns out, it is titled "La conjecture de Weil: I" (!), and did appear in Publications of the IHES, vol. 43.) –  Emerton Apr 17 '10 at 3:38
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And the introduction states straight away that Deligne intends to publish a second article on the subject. The delay seems to be part of the never ending turmoil concerninh SGA 4,5 and 5. –  Olivier Apr 20 '10 at 9:29

If you can find a (say, library) copy of Cornell and Silverman's Arithmetic Geometry I would highly recommend it. It is a comprehensive treatment of the arithmetic theory of abelian varieties using the modern scheme-theoretic language. Lamentably it's basically impossible to buy a copy these days (there's usually one available on-line from some obscure seller for something like $950). I also agree with the above recommendations of Liu's Algebraic Geometry and Arithmetic Curves. It builds scheme theory from scratch (even developing the necessary commutative algebra in first chapter) and has an eye towards arithmetic applications throughout. In particular, the end of the book has a great chapter on reduction of curves. If you want a treatment of elliptic curves in extreme generality (using scheme language) then you might be interested in Katz' and Mazur's Arithmetic Moduli of Elliptic Curves. I emphasize however, that this particular book is very difficult (at least for me it is).

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+1: Katz-Mazur is especially excellent. –  stankewicz Apr 16 '10 at 13:47
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Mumford's "Abelian Varieties" (note the appendices), Ch. 6 of his GIT book, sga3 on quotients, "Neron Models" for good techniques & much more, books on etale cohomology (Freitah-Kiehl, Milne), SGA1 (skip boring expose on fibered categories), Tate's papers on p-divisible groups & Honda-Tate theory, Katz' paper on p-adic properties of modular forms, FGA Explained (for Hilbert & Picard schemes, needed for relative Jacobians and much more). Illuminating to read EGA I (e.g., good intro to formal schemes, needed for serious deformation theory and Tate curve and beyond). –  BCnrd Apr 16 '10 at 14:21
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Because it uses Drinfeld's notion of a "Drinfeld basis" to define p-power level structures in char. p. This gives an important tool that is not in Deligne--Rapoport. (I should add, I don't know if this makes it "preferable", but it is the main technical innovation of Katz--Mazur.) –  Emerton Apr 16 '10 at 16:12
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To augment Emerton's comment, KM works nicely over $\mathbf{Z}$ but gives no conceptual technique at cusps whereas DR provides good technique at the cusps but inverts the level. Funny part is that when KM deal with cusps, they list axioms concerning Tate curve and one needs DR techniques to justify everything in their axioms. So the KM approach to handling cusps requires theory of generalized elliptic curves even though KM never mentions that concept, so they sidestep if their proper $\mathbf{Z}$-curves are moduli spaces for generalized elliptic curves with Drinfeld structure (they are). –  BCnrd Apr 16 '10 at 19:24
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OK, I'm not endorsing anything, but there's this website called gigapedia.org. It may or may not help in situations where a book is out of print and unattainable for less than US$1000. –  Pietro KC Apr 17 '10 at 5:37

An apology first: This is more a supplement to Charles' answer than an answer itself. This was originally a set of comments, but I was not able to format the comments so as to be readable.

"Arithmetic of Elliptic curves" is particularly recommended for those who want a first look at arithmetic applications of cohomology. Chapter 8 proves the Mordell-Weil theorem using Galois cohomology. Pretty much everything in this book is good though and the only overlap with Hartshorne is in the first two chapters. It's the canonical book for elliptic curves for a reason!

"Rational Points on Elliptic curves" would probably not be so exciting for someone who's already gone through Hartshorne.

"Advanced Topics" is exactly that, but maybe a little more friendly than most topics books. The chapters are essentially free standing. Of particular interest might be the chapter on Elliptic surfaces which give a peek at ℤ schemes in (almost) all their glory.

I've only glanced through Hindry-Silverman, so I couldn't say much either way.

"An Invitation to Arithmetic Geometry" for this reader would primarily serve to highlight how Algebraic Number Theory intersects Arithmetic Geometry, I think.

"Algebraic Geometry and Arithmetic Curves" is a fantastic reference for Arithmetic Geometry, and there's quite a lot of overlap with Hartshorne.

edit: For moduli of elliptic curves, Chapter 1 (Modular forms) of "Advanced topics" is a good place to start, and Katz-Mazur is a good eventual target. Between those two, there are lots of books on modular forms and moduli spaces to fill the gap. I'm partial to Diamond and Shurman, but the original works of Shimura deserve recognition here. Your mileage may vary.

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"Algebraic Geometry and Arithmetic Curves" by Liu might be good, it covers a lot of the same material, but does it more arithmetically.

There's also "An Invitation to Arithmetic Geometry" by Lorenzini

Also, don't discount the "series" by Silverman: "Rational Points on Elliptic Curves" (with Tate), "Arithmetic of Elliptic Curves", "Advanced Topics in the Arithmetic of Elliptic Curves" and "Diophantine Geometry" with Hindry.

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thank you so much for answer my question. actually, I think silverman's books did not use scheme theory. I hope can find some materials which can show me the power of scheme and sheaf cohomology. I know Lorenzini's book, but I think I don't like his writting style. –  kiseki Apr 16 '10 at 12:52
    
Some chapters in Advanced Topics in the Arithmetic of Elliptic Curves by Silverman do use scheme theory. –  David Corwin Jul 15 '10 at 16:11

In addition to the mentioned Cornell--Silverman book there is another Cornell--Silverman (+Stevens) collection named "Modular forms and Fermat's last theorem" (http://www.springer.com/mathematics/numbers/book/978-0-387-98998-3), which I can warmly recommend. It's available in paperback.

The purpose of the volume is to cover the material used in the proof of Fermat's last theorem. Therefore a lot of arithmetic geometry is covered at a reasonable graduate-level (maybe a few more demanding surveys, though). Brian Conrad from previous comments is responsible for one nice paper in the volume.

I especially like Tate's paper on Finite group schemes and Mazur's on deformation theory of Galois representations.

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That book contained my first published mention in mathematics so you get +1 ;-) –  Kevin Buzzard Apr 17 '10 at 12:15
    
Thanks! :) Now I have to go find out where that citation/mention is. –  Daniel Larsson Apr 17 '10 at 12:28

Considering it is now two full years since the OP asked this question this reply is (probably) purely for archival purposes if someone (like me) happens to stumble upon this question and finds it useful.

Professor Emerton's detailed comment on Professor Tao's blog is incredibly useful as a roadmap found here.

Also, Professor Ellenberg has a webpage for prospective students who wish to be advised by him. On it he has recommended books to read in pursuit of this path.

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