TITLE: Over which fields does the Mordell-Weil theorem hold?
QUESTION [31 upvotes]: According to a well-known theorem of Mordell, the group of rational points $E(\mathbf{Q})$ of an elliptic curve $E/\mathbf{Q}$ is finitely generated. Weil generalized this theorem to abelian varieties over number fields.
Less well-known is the following generalization, due I believe to Néron : if $K$ is a field of finite type (that is, finitely generated over its prime field) and $A$ is an abelian variety over $K$, then $A(K)$ is finitely generated. There is an even more general statement, the Lang-Néron theorem, for relative field extensions which are finitely generated (see Brian Conrad's article for the precise statement and a proof of this theorem).
Q1. Are there other fields $K$ for which the group of $K$-rational points of an abelian variety over $K$ is always finitely generated?
In the other direction, there exist fields $K$ for which $A(K)$ is clearly never finitely generated whenever $\operatorname{dim}(A) \geq 1$. For example $K=\mathbf{C}$, in which case it follows from the description af abelian varieties as complex tori. If $K$ is a finite extension of $\mathbf{Q}_p$, then $A(K)$ contains a finite-index subgroup isomorphic to $\mathcal{O}_K^{\operatorname{dim} A}$, so $A(K)$ is again never finitely generated. Other examples I can think of are complete discretely valued fields and algebraically closed fields. Note that we often have the stronger result that $A(K) \otimes \mathbf{Q}$ is infinite-dimensional (except when $K=\overline{\mathbf{F}}_p$, in which case $A(K)$ is a torsion group).
Q2. Are there other fields $K$ for which the group of $K$-rational points of a non-trivial abelian variety over $K$ is never finitely generated?

REPLY [5 votes]: This is a partial answer to Q2, in particular a full answer to the question asked by François Brunault in the comments:

By the way, if K is ferile, will A(K)⊗Q be necessarily be infinite-dimensional? 

Here, A is a non-trivial abelian variety over K, and fertile = large = ample = for every geometrically integral smooth K-variety the set of K-rational points is either empty or Zariski-dense.
If K is algebraic over a finite field, the answer is NO, as already pointed out by Laurent Moret-Bailly. In all other cases, the answer is YES: 
For fields of infinite transcendence degree, this was proven in the paper On the rank of abelian varieties over ample fields with Sebastian Petersen using elementary methods.
For fields of characteristic zero, in the same paper we deduce the claim using the Mordell-Lang conjecture as proven by Faltings, Raynaud and possibly others.
For fields of positive characteristic and transcendence degree at least one, in Ranks of abelian varieties and the full Mordell-Lang conjecture in dimension one we deduce the claim from a recent result of Rössler by a similar approach.