Let $E/K$ be an elliptic curve over a number field. Let $M=K(E[p])$.
I want to know $H^1(M/K,E[p])$: for $p=2$, it is $0$, but what about the case $p>2$?
Is it always zero?
In fact, I want to know if the restriction map $$ \operatorname{Sel}_p(E/K) \rightarrow \operatorname{Sel}_p(E/M) $$ is injective.
The vanishing of $H^1(M / K, E[2])$ seems to be something of an exception; for all $p > 2$, there are elliptic curves over number fields such that $H^1(M / K, E[p]) \simeq \mathbb{F}_p$. For brevity, let $G = \mathrm{Gal}(M / K)$.
By the transfer map, if the order of $G$ is prime to $p$, $H^1(G, E[p]) = 0$. Similarly, the transfer map gives for any subgroup $H$ containing a $p$-Sylow subgroup, $P$, the map on cohomology $H^1(G, E[p]) \to H^1(H, E[p])$ is injective. Since $G \leq \mathrm{GL}_2(\mathbb{F}_p)$, any $p$-Sylow subgroup, $P$, is cyclic, and for $p > 2$, a direct calculation shows $H^1(P, \mathbb{F}_p^2) \neq 0$. It is generated by a cocycle sending a generator of $P$ to an element of $\mathbb{F}_p^2$ not fixed by $P$.
Let $N$ be a normaliser of $P$. By the inflation-restriction sequence, $H^1(N, \mathbb{F}_p^2) \simeq H^1(P, \mathbb{F}_p^2)^{N / P}$. Writing $P$ as the subgroup $\begin{pmatrix}1 & * \\ 0 & 1\end{pmatrix}$, and computing the action of $N$ on the cohomology of $P$ shows the cohomology of $N$ is non-trivial if and only if it consists of matrices of the form $\begin{pmatrix}a^2 & * \\ 0 & a\end{pmatrix}, a \in \mathbb{F}_p^{\times}$. Combining this with the fact that any subgroup of $\mathrm{GL}_2(\mathbb{F}_p)$ with $p$ dividing the order and more than 1 $p$-Sylow subgroup contains $\mathrm{SL}_2(\mathbb{F}_p)$, shows that $N$ must be the entire group if the cohomology is non-trivial.
Over $\mathbb{Q}$ there is a further restriction, namely, the determinant of a mod $p$ representation must be surjective. Despite this, there are infinitely many examples, for $p = 3, 5$ parameterised by suitable modular curves. Two example curves are LMFDB label 19a3 and LMFDB label 50a2 where the Galois cohomology for the 3 and 5 torsion respectively is non-trivial.
For both of these examples, the rational $p$-Selmer group is trivial, and so must inject into the $p$-Selmer group over $M$. It seems plausible that there are elliptic curves where $\mathrm{Sel}_p(E / K) \to \mathrm{Sel}_p(E / M)$ fails to be injective. For example, if there is a modular curve parameterising elliptic curves such that there is a $K$-rational 3-torsion point, and over the field extension generated by the other 3-torsion points, the $K$-rational 3-torsion point becomes divisible by 3 (these conditions should be able to be imposed by a constraint on the mod 9 Galois representation, and from that congruence subgroup you can get the modular curve). For a generic point on this modular curve, the corresponding elliptic curve would have $E(K) / 3E(K)$ not injecting into $E(M) / 3E(M)$, and so by the snake lemma, $\mathrm{Sel}_p(E / K)$ would not inject into $\mathrm{Sel}_p(E / M)$.
