Fix a number field $K$.


1. Is the rank of $J(K)$ unbounded, where $J$ ranges over the Jacobians of all smooth, projective, geometrically connected curves over $K$?
2. 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 $g=1$.



Are ranks of Jacobians over number fields unbounded?
The answer to 1. is yes. Take $J\_0(N)$ the Jacobian of the modular curve $X\_0(N)$ over the rationals. Since all elliptic curves of conductor dividing $N$ are factors of $J\_0(N)$ and there are infinitely many isogeny classes of elliptic curves over the rationals with positive rank. My guess, just like yours, is that 2. is open.


