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Sep 11, 2014 at 17:34 comment added user9072 @JoseCapco it was not "all" in the 50's. The early papers from Weil, especially the Mordell--Weil mentioned in the answer, are from around 1927 to 1929, and there were some contributions in between.
Sep 11, 2014 at 17:25 comment added Jose Capco Thanks Joe. I like your detailed answer, especially about what your wrote about the observation of Northcott. I marked this as the answer. I hope I havent discredited Keith, I also upvoted his comment. So to put it mildly, it was in the 50's when all these things about height was developed. It is interesting to note that it came about because of a problem in elliptic curve then made its way into algebraic number theory and older maths.
Sep 11, 2014 at 17:10 vote accept Jose Capco
Sep 11, 2014 at 14:22 history edited Joe Silverman CC BY-SA 3.0
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Sep 11, 2014 at 14:11 history answered Joe Silverman CC BY-SA 3.0