Can someone help in how to approach reading Mordell-Weil Theorem for abelian varieties? 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.  
 A: 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.
