Let $A$ be an abelian variety over a number field $K$, $\text{Sha}(A/K)$ its Tate-Shafarevich group, $\ell$ a prime.


Is it known that the $\ell$-primary torsion subgroup $\text{Sha}(A/K)\{\ell\}$ is trivial for almost all primes $\ell$?



Tate-Shafarevich group over number fields
No. 


It is always difficult to "prove" that something is "not known", but this may do: I claim it is not even known when $K=\mathbb Q$, $A$ is an *elliptic curve* $E$. In fact in this case, the result you ask for is not even known when $E$ *has CM*. In fact, even in this very special case, it is not known that the $l$-primary torsion subgroup of the Sha is *finite* for almost all primes $l$, while you ask for *trivial* instead of finite. Actually, it is not even known that this $l$-primary torsion subgroup is finite for *infinitely many primes $l$*. See for example Theorem 1.1 of Coates-Liang-Sujatha here in <https://link.springer.com/article/10.1007/s00032-010-0127-2>


