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I was thinking to start reading the proof of Mordell-Weil Theorem for abelian varieties over number fields, after getting done with the proof in the case of elliptic curves over number field and I wasn't sure how to go on about it.

As of now, I am familiar with the basic Algebraic Geometry (classical, affine and projective varieties' theory included in the first introduction to the course, although I'll have to brush up on Riemann- Roch theorem quite a bit) and also the proof in case of Elliptic Curves over number fields using cohomological approach.

I've heard that the more generalized result in case of abelian varieties uses quite a bit of AG, so this post is to ask what kind of AG is needed to be able to read/understand the proof? Can someone recommend some references for it with possibly some reviews about them or maybe suggest a path that one should follow if they're interested in the proof?

Thank you in advance.

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    $\begingroup$ Did suggestions given in mathoverflow.net/questions/334366/… worked for you to read about Mordell-Weil Theorem for elliptic curves ?? $\endgroup$ – Praphulla Koushik Jul 25 '19 at 13:47
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    $\begingroup$ @abx The question/answer that you cite is definitely not an answer to this question. The proof of the Mordell-Weil theorem for elliptic curves can be done much more explicitly, and in a more elementary fashion, than for higher dimensional abelian varieties. There are a number of places to read the higher dimensional proof (Lang's Diophantine Geometry, Mumford's Abelian Varieites, ...), and a discussion of their respective merits would be good answers to this question, as would a discussion of the requisite background. $\endgroup$ – Joe Silverman Jul 25 '19 at 14:04
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    $\begingroup$ OK, sorry, I was too hasty. Voting to reopen. $\endgroup$ – abx Jul 25 '19 at 14:40
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    $\begingroup$ @Joe Silverman: How about your book 'Diophantine geometry'? I just found it, although it has all the necessary AG in an appendix, do you think someone with as little AG background as what I mentioned above, should be okay with reading it? (Of course,I'll also check out other references you mentioned as well, thanks for them btw) $\endgroup$ – Shreya Jul 25 '19 at 16:35
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    $\begingroup$ The AG chapter in my book with Hindry is more of a "pre-pendix". In any case, you can try reading the proof of MW, referring back to the AG chapter as needed. $\endgroup$ – Joe Silverman Jul 25 '19 at 18:37
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In the Abelian Variety case, as in the Elliptic Curve case, there are two parts to the proof - the weak mordell weil theorem and then proving the full version using heights. The weak Mordell-Weil theorem can be proven for abelian varieties quite easily using that multiplication by n is generically etale for AV's and standard arguments using the Kummer sequence. I sketch it here (with quite a few typos and only implicitly working with Abelian Varieties but hopefully it helps): https://asving.com/2017/11/19/weak-mordell-weil-as-a-consequence-of-hermite-minkowski/.

As for the part using heights, it's again quite formal given the definition of an appropriate height functions, it requires the theorem of the cube and the fact that abelian varieties are projective. For example, see here: http://www.msri.org/attachments/workshops/301/HtSurveyMSRIJan06.pdf. In short, I would suggest learning about Abelian varieties by themselves (enough to understand why multiplication by $n$ is generically etale, why they are projective and on the way, the theorem of the cube). Then, you can figure out how to extend the classical Mordell-Weil to AVs using these tools.

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  • $\begingroup$ I have a question- can we prove this theorem for abelian varieties without using Kummer pairing approach as you said in your first para and instead using an approach similar to Galois cohomology one? $\endgroup$ – Shreya Aug 6 '19 at 16:48
  • $\begingroup$ Also, as you said in your last paragraph, learning so and so about abelian varieties is sufficient, could you recommend some reference for it? I tried looking into Mumford's text, but got lost. $\endgroup$ – Shreya Aug 6 '19 at 16:50
  • $\begingroup$ I think I indicate the proof using the Kummer sequence which is the galois cohomology approach. The Kummer sequence is different from the Kummer pairing but they are closely related. You might try Moonen's notes on Abelian Varieties. Milne is decent too. $\endgroup$ – Asvin Aug 6 '19 at 18:42

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