It seems that Lan Nguyen proved [in a preprint on arxiv](https://arxiv.org/abs/1309.7675) 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?


Many thanks in advance.



Is the Tate-Shafarevich group of a rational elliptic curve finite?
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 $\geq 2$.


As far as I can tell the paper was never published.


