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 $\ell$?



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

  • 2
    $\begingroup$ Good job Joël! I had a hard time coming up with anything different than No as an answer. $\endgroup$
    – Olivier
    Feb 20 '18 at 13:01

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