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Does anyone know a reference for the proof of the finite generation of the Mordell-Weil group over finitely generated fields?

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It is in Lang's book "Fundamentals of diophantine geometry", chapter 6: google book preview

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Neron did the finitely generated ground field (his thesis), and Lang-Neron together did a generalization. I needed to use Lang-Neron (really just Neron's part) in a paper, but personally found the Lang book unsuitable as a reference because everything hinges in the theory of the Chow trace whose development Lang gives only in the Weil Foundations style which is really hard to understand (in contrast with the modern spirit of the rest of that book). So I wrote an expository article on "modern" approach to Chow trace and deduction of Lang-Neron from it. It's on my webpage. –  BCnrd Apr 10 '10 at 17:07
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Pete, I've decided to stop posting "answers" on MO, only making comments. It keeps things short & sweet, and I don't care about the "point" stuff. –  BCnrd Apr 10 '10 at 18:08
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Dear Brian, Leaving aside "points", there are some technical reasons to prefer that sufficiently significant comments should appear as answers: essentially, we want to use the system in such a way so that a later reader gets pointed quickly to the most useful responses. As a fellow longwinded answerer, I appreciate that the length restriction for comments can be useful. Probably what will happen is that people will convert some of your comments to answers, with attribution to you. I suppose you'll be okay with that? –  Pete L. Clark Apr 11 '10 at 4:54
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Pete, if someone is inspired by a comment to write their own answer in their own words then that's fine, but I do not have in mind that such a thing should actually happen much. There can also be a virtue in things not being too easy to find (for non-reference questions): will lead some to think more on their own. –  BCnrd Apr 11 '10 at 12:38
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If you don't care for points, you could for example make all of your (future) answers community wiki. –  Kevin H. Lin Apr 12 '10 at 8:08
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