Dear Felipe, as long as we restrict ourselves to elliptic curves over $\mathbb{Q}$, the only results known in this direction are that the 2, 3, 5, 7 and 13 primary torsion of Tate-Shafarevich groups can be arbitrarily large. These are due to Kramer, Cassels, Fisher, Matsuno and Matsuno, respectively. It is still unknown , whether it is true that for any $p$, the $p$ primary torsion of sha can be arbitrarily largefor any $p$. . It is not even known whether the $p$-torsion of sha can be non-trivial for arbitrary $p$.
As far as I know, almost everybody nowadays believes that ranks of elliptic curves over $\mathbb{Q}$ can be arbitrarily large, but when pressed for evidence, most people point to the function field case. In fact, it is not even known, whether the rank can get arbitrarily large over number fields of uniformly bounded degree.
Dear Felipe, as long as we restrict ourselves to elliptic curves over $\mathbb{Q}$, the only results known in this direction are that the 2, 3, 5, 7 and 13 primary torsion of Tate-Shafarevich groups can be arbitrarily large. These are due to Kramer, Cassels, Fisher, Matsuno and Matsuno, respectively. It is still unknown, whether the $p$ primary torsion of sha can be arbitrarily large for any $p$. It is not even known whether the $p$-torsion of sha can be non-trivial for arbitrary $p$.
As far as I know, almost everybody nowadays believes that ranks of elliptic curves over $\mathbb{Q}$ can be arbitrarily large, but when pressed for evidence, most people point to the function field case. In fact, it is not even known, whether the rank can get arbitrarily large over number fields of uniformly bounded degree.