nt.number theory – Galois cohomology of symmetric powers of the p adic Tate module of an elliptic curve


Let $E$ be an elliptic curve over a number field $K$. Let us fix an odd prime $p$ and $S$ a finite set of primes of $K$ containing both bad primes for $E$ and primes lying over $p$. Let $mathcal{O}_S$ be the ring of $S$-integers in $K$. Let $V_p$ be the $p$-adic Tate module as usual, i.e. $V_p=T_pEotimes_{mathbb{Z}_p}mathbb{Q}_p$. Time to time I found literature which seems to treat two (etale) cohomology groups $H^1(mathcal{O}_S,mathrm{Sym}^{2k+1}V_p(1))$ and $H^1(mathcal{O}_S,V_p(k+1))$ for integers $kgeq0$ as the same group — here, $(-)(1)$ and $(-)(k+1)$ means the Tate twists as it should be. I have a several question in this context:

  1. Is there a natural map between them?
  2. Is it an isomorphism? If not, does it become an isomorphism under additional assumptions? — for example, $E$ is an elliptic curve over an imaginary quadratic field $K$ with CM by $mathcal{O}_K$.
  3. If there is no natural map, can we construct a map between them? Again, is the constructed map an isomorphism (under some conditions)?

I am not sure, but probably this is a question which might be well-known to the experts. Any reference would be helpful. Thank you very much in advance.