A cup product in Galois cohomology of Elliptic curve

Question

Let $E$ be an elliptic curve over a field $K$. I am mostly interested in the case $K$ is a number field or a local field but the following question is valid for all $K$.

Let $p$ be any prime power which is comprime to the characteristics of $K$. Galois cohomology gives a cup product map

$$H^{1}(K,E[p]) \times H^{1}(K,E[p]) \to H^{2}(K,E[p] \otimes E[p])$$

Composing this map with the map induced by the Weil pairing gives a map

$$ \phi: H^{1}(K,E[p]) \times H^{1}(K,E[p]) \to H^2(K,\mu_{p}). $$

Are there any known examples where the above map $\phi$ is non-trivial?

In theory, it is possible to compute both sides explicitly when p is small. However, computing the cup product is quite hard (for me.)

Answer 1

One can use the exact sequence $$ 0 \to E(K)/mE(K) \to H^1(K,E[m]) \to H^1(K,E)[m] \to 0 $$ to define a pairing $$ E(K)/mE(K) \times H^1(K,E)[m] \to H^2(K,\mu_m) $$ by taking $(Q,\xi)$ to $\phi(Q'\cup \xi')$, where $Q'$ is the image of $Q$ in $H^1(K,E[m])$ and $\xi'$ is any pullback of $\xi$ to $H^1(K,E[m])$.

For $K/\mathbb Q_p$, Tate proved that this is a perfect pairing (Tate local duality). Since it's easy to find examples with $E(K)/mE(K)\ne0$, this gives examples where your map $\phi$ is non-trivial. And indeed, if you take $m=p^k$, then you can force $E(K)/p^kE(K)$ to be quite large. (In this case, we have $H^2(K,\mu_m)\cong \frac1m\mathbb Z/\mathbb Z$, so this piece of the Brauer group makes a natural target for a pairing.)

For global fields, one does something similar to construct the Cassels-Tate pairing on the Tate-Shafarevich group, but the target space is instead $H^2(K,\mathcal{C}[m])$, where $\mathcal{C}$ is the idele class group. This is preferable to $H^2(K,\mu_m)$, since the Brauer group $H^2(K,\mathbb G_m)$ is huge.