# Is the Tate-Shafarevich group of a rational elliptic curve finite?

It seems that Lan Nguyen proved in a preprint on arxiv of 2013 that the Tate-Shafarevich group of a rational elliptic curve is finite. However, I couldn't find any published version thereof. So is it now known that the Tate-Shafarevich group of a rational elliptic curve is finite?

• the paper only deals with plane cubics locally isomorphic to $E$, so it's if anything a proof that the 3-torsion in Sha is finite (the error is p12, six lines up: there are all sorts of crazy homogeneous spaces that can contribute to Sha and it's certainly not true that they're all plane cubics). It also claims to prove that if Sha has an element of order 3 then $E$ has CM by a cube root of 1, which is surely not true. – znt Nov 24 '16 at 19:58
MO is not the place to discuss the validity of preprints, but I think it is safe to say that the finitiness of the Tate-Shafarevich group for elliptic curves over $\mathbb{Q}$ is considered an open problem for rank $\geq 2$.
• In fact, for every single elliptic curve over $\mathbb{Q}$ of rank at least $2$, the finiteness of its Tate-Shafarevich group is an open problem. – Alex B. Nov 24 '16 at 20:31