The fine Shafarevich-Tate (ST) group appears to not have been studied in much depth except for one paper by Wuthrich (2007). However, it only talks about fine ST group for elliptic curves, which makes me wonder if the results are in fact specific to elliptic curves or whether they can be generalised to Abelian varieties (specially something like Theorem 3.5 where it is proven that for elliptic curves over $\mathbb{Q}$ of rank 1, the $p$-primary part of the ST group and the fine ST group are equal.)

It is mentioned that it is believed for number fields in general, the fine ST group is expected to be small. But other than some examples, I did not see a moral reason to expect it.

Murty-Lim (2015) showed that the if one varies over $\mathbb{Z}/p$ extensions of number fields then the $p$-primary part of fine Selmer group of elliptic curves (their results are for Abelian variety) has unbounded growth. This combined with the results of Clark-Sharif from 2010, (where they proved that the $p$-torsion of the ST group of elliptic curves over number fields have unbounded growth in degree $p$ extensions) might suggest that $p$-primary part of the fine ST group can have unbounded growth.

Is there any recent work done on this?


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