3 problems

Silverman's density conjecture. For all $b\in\mathbb{P}^1(\mathbb{Q})$ outside a set of density $0$ in $\mathbb{P}^1(\mathbb{Q})$,

Rank-zero fibre conjecture. There are infinitely many $b\in\mathbb{P}^1(\mathbb{Q})$ such that

Non-isotrivial elliptic-surface rank conjecture. If $\pi$ is non-isotrivial, then both $\mathscr{J}(X,\pi)$ and $\mathscr{N}(X,\pi)$ are infinite.