# 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.