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For abelian variety $A/K$, divisibility problem (i.e. $\forall n≧1$, $Ш(A/K)⊂p^nH^1(G_K,A)$ holds for fixed prime $p$?) was asked by Cassels in 1962 and even now discussed.

On the other hand, once group structure of $Ш(A/K)$ is known, the question of Cassels holds for enough large $p$. But in my understanding, even the group structure of $Ш(A/K)$ is known, divisibility problem is not finished for small $p$.

My question is, which is more difficult, Cassels divisibility problem or group structure of $Ш(A/K)$ ?

I believe group structure is much more harder question, but does Cassels problem help understanding of group structure of $Ш(A/K)$?

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    $\begingroup$ "group structure" is not a well-posed problem. The divisibility for large $p$ follows from the (conjectured) finiteness of Sha, simply because every finite abelian group is divisible by all primes not dividing its order. For primes dividing the order, it's hard to tell how the two are related, since Cassels' problem then asks about existence of certain classes outside Sha itself. $\endgroup$
    – Wojowu
    Commented Mar 3, 2023 at 6:17
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    $\begingroup$ I also think that Cassels' question of the divisibility in the Weil-Châtelet group is not really about the group structure of Sha as an abelian group (which I consider a well-posed problem). The opposite would: If one could prove that elements in Sha are not divisible in the ambient group, then the $p$-primary part of Sha would be finite. Therefore, the divisibility Cassels conjectures sort of says it is hard to prove that Sha is finite. $\endgroup$
    – Chris Wuthrich
    Commented Mar 3, 2023 at 11:07

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