I've read in some references (some of them important, like page $328$ from Heights in Diophantine Geometry, by Bombieri and Gubler) that André Weil proved in his PhD thesis that the rank of an abelian variety (over a number field) is finite.
I've read Weil's thesis and what I found was this (I'm paraphrasing):
If $C$ is a plane algebraic curve over a number field $K$, with arbitrary genus, then its jacobian variety $J(C)$ has a structure of a finitelly generated abelian group.
I don't know much about abelian varieties, but I have the feeling that what Weil proved was not in that level of generality mentioned in the book (i.e., for any abelian variety).
Is it true that the case of abelian varieties can be reduced to that of the Jacobian of a plane curve?
If it is, how so?