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This question is closely related the question Over which fields does the Mordell-Weil theorem hold?

I consider the following question:

(1) Let $K$ be a finitely generated field extension of $\mathbb F_p$, $K^\text{per}$ its perfect closure, and $A$ be an abelian variety over $K$. Is $A(K^{\text per})$ finitely generated?

I know that the answer is yes if $A$ is a non-isotrivial elliptic curve (this is Theorem 3.3 of this 2010 paper by Ghioca). But because of this very article (and because there is also a 2017 paper by Rössler proving a somewhat similar statement for an abelian variety satisfying some conditions), I am pretty sure that there is no proof of a positive answer in full generality to my question (1) at this time.

So my real questions are: is the answer to question (1) conjectured to be "Yes"? If so, is there a reference which states this conjecture? Or is it a "folklore conjecture"? Or on the contrary, is there a known counter-example?

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  • $\begingroup$ If I understand correctly, can't you always reduce the problem to a minimal field over which the Abelian variety is not isotrivial? And if this field happens to be a finite field, then the group contains $A(\overline{\mathbb F_p})$ which is not finitely generated. On the other hand, if the Abelian variety is not isotrivial over the function field of a curve, we can extend it a neron model over the curve and then the Mordell Weil group would just be the set of sections of the Neron Model. Any map from a curve to an Abelian variety would factor through the Jacobian of the curve (ctd...) $\endgroup$
    – Asvin
    Commented Nov 5, 2019 at 18:10
  • $\begingroup$ (ctd...) and therefore the set of sections would be a subgroup of group of homomorphisms which is essentially described by the endomorphism of the Jacobian which is finite generated. So shouldn't it be true in general that over a curve, any non isotrivial Abelian variety has finitely generated Mordell Weil group? $\endgroup$
    – Asvin
    Commented Nov 5, 2019 at 18:11
  • $\begingroup$ In my previous comment, you don't need to consider the Neron model, you can simply say that any map from $Spec K^{per}$ to the abelian variety extends to a map from the curve to the Abelian variety and proceed from there. I don't know what happens in higher transcendence degree. $\endgroup$
    – Asvin
    Commented Nov 5, 2019 at 18:48
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    $\begingroup$ Let me just add that if question (1) had a positive answer, then the torsion in $A(K^{\text{per}})$ would be finite. This has been proved recently by Ambrosi and D'Addezio (arxiv.org/abs/1811.08423, Theorem 1.2.2). $\endgroup$
    – Riccardo Pengo
    Commented Dec 6, 2019 at 8:22

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