TITLE: Are ranks of Jacobians over number fields unbounded?
QUESTION [7 upvotes]: Fix a number field $K$.

Is the rank of $J(K)$ unbounded, where $J$ ranges over the Jacobians of all smooth, projective, geometrically connected curves over $K$?
Does there exist an integer $g$ such that the rank of $J(K)$ is unbounded, where $J$ now ranges over the Jacobians of all smooth, projective, geometrically connected curves of genus $g$ over $K$?

I expect (perhaps naively) that the answer to 1. is "yes". Maybe one can write down an explicit family of superelliptic curves and use descent to show that their ranks are not bounded. Or else there may be a construction that, given an elliptic curve $E$ and integer $r$, produces a curve $C$ such that $\mathrm{Jac}(C)$ contains a factor $E^r$ up to isogeny. But I can't make either approach work.
On the other hand, I would be surprised if 2. were known, since it is so famously open in the case $g=1$.

REPLY [3 votes]: The answer to Question 1 is already contained in the following paper:
ANDRÉ NÉRON,
Problèmes arithmétique et géométriques rattachés à la notion de rang d’une courbe algébrique dans un corps, Bulletin de la S. M. F., tome 80 (1952), p. 101-166.
http://archive.numdam.org/ARCHIVE/BSMF/BSMF_1952__80_/BSMF_1952__80__101_0/BSMF_1952__80__101_0.pdf
The corollary to Theorem 7 (see page 155 of the article) says that for any given number field $K$, there are infinitely many curves $C$ of genus $g$ such that $\text{Jac}(C)(K)$ has rank at least $3g+5$. (With a bit more work, one can get $3g+6$. And with even more work, for $g=1$, the author gets up to rank at least $11$.)