Let $E/\mathbb{Q}$ be an elliptic curve defined over $\mathbb{Q}$. For a given prime $p$, the $p$-Selmer group $\operatorname{Sel}_p(E)$ of $E$ and the $p$-part of the Tate-Shafarevich $Ш_E[p]$ group are related by the following exact sequence:

$$\displaystyle 0 \rightarrow E(\mathbb{Q})/pE(\mathbb{Q}) \rightarrow \operatorname{Sel}_p(E) \rightarrow Ш_E[p] \rightarrow 0,$$

where $E(\mathbb{Q})$ denotes the Mordell-Weil group of the elliptic curve $E$.

In a series of papers (see below for some references), Bhargava and Shankar showed that the average size of the $p$-Selmer group for elliptic curves over any `large' family of elliptic curves over $\mathbb{Q}$ is equal to $3,4, 6$ respectively for $p = 2,3,5$. Are there any analogous results for the $p$-part of $Ш_E$ for the same primes, perhaps upper or lower bounds?

Both lower bounds and upper bounds for this average have interesting consequences. If one can prove a non-trivial lower bound, for example suppose the average $2$-rank of $Ш_E$ is at least $0.8$, then from the equality

$$\displaystyle r_2(\operatorname{Sel}_2(E)) = r(E) + r_2(E(\mathbb{Q})[2]) + r_2(Ш_E[2])$$

and the fact that the average $2$-Selmer rank is at most $1.5$ and that the average of $r_2(E(\mathbb{Q})[2])$ is 0, we see that the average of the Mordell-Weil rank $r(E)$ is at most $1.5 - 0.8 = 0.7$, which beats the current best bound (also due to Bhargava and Shankar, as a consequence of their proof that the average $5$-Selmer size is 6). If one can prove a non-trivial upper bound for $Ш_E[p]$, then one can use it to produce more curves with positive rank.

Are such averages known, whether over a `large' family or not? I am particularly interested in the case $p = 2$.






What's known is very little, but there is a very compelling conjectural picture of the distribution of Mordell-Weil, Selmer, and Sha in the paper of Bhargava, Kane, Lenstra, Poonen, and Rains:



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