This is a follow-up to this earlier question on elliptic curves: [Largest rank assumed by infinitely many elliptic curves](https://mathoverflow.net/questions/302461/largest-rank-assumed-by-infinitely-many-elliptic-curves)


Let $g \geq 1$ be an integer. For each $g$, what is known about the largest positive integer $r(g)$ for which we know there exist infinitely many abelian varieties $A$ of dimension $g$ which are not products of lower dimensional abelian varieties (for example, not a product of elliptic curves) such that the Mordell-Weil rank of $A$ is at least $r(g)$? In the first question, it was essentially answered by Michael Stoll that $r(1) \geq 19$, and in another comment it was mentioned that recent work by Bjorn Poonen and various coauthors, as well as earlier heuristics due to Andrew Granville, indicate that $r(1) = 21$ exactly (that is, there ought to be only finitely many elliptic curves with Mordell-Weil rank exceeding 21). 


Moreover, if there are any known constructions for such $A$'s for fixed genus $g \geq 2$, do they arise as Jacobians of curves of genus $g$, or is there some other construction?



Largest ranks achieved by abelian varieties of fixed dimension
Here is a construction (as far as I know, originally due to Mestre)
that gives a lower bound $r(g) \ge 4g + 5$.


Write
$$ \prod\_{i=1}^{4g+6} (x - t\_i) = h(x)^2 - f(x)\,, $$
where the $t\_i$ are independent indeterminates, $h$ is monic of degree
$2g+3$ and $f$ has degree $2g+2$. Then there are $4g+6$ points
$P\_i = (t\_i, h(t\_i))$ on the hyperelliptic curve $C$ given by
$y^2 = f(x)$. Generically, the only relation between them in the Jacobian
is that
$$ \sum\_{i=1}^{4g+6} P\_i - (2g+3) D\_\infty = \operatorname{div}(y - h(x)) \,,$$
where $D\_\infty$ is the sum of the two points at infinity.
So most specializations of the $t\_i$ will produce a curve whose
Jacobian (is simple and) has Mordell-Weil rank at least $4g+5$.


It is certainly possible to tweak the construction to get somewhat
higher rank. (I had used it in some early papers of mine to get
high-rank examples of genus 2 Jacobians over $\mathbb Q$.) But I doubt
that one can get superlinear growth in this way. Anyway, the question
is whether we should *expect* $r(g)$ to grow faster than linearly.


In view of Joe Silverman's comment below, it should be added that
Néron already had a construction that produces infinitely many
(again hyperelliptic) Jacobians of curves of genus $g$ with rank
$\ge 3g + 6$. This is the Corollaire on page 163 of Néron's paper.


