Reference request for the isomorphism $H^1(G_{K_v},E)[n]\cong (E(K_v)/nE(K_v))^*$ in the context of Tate-duality

Question

Let $E/K$ be an elliptic curve over a number field $K$. Let $M_K$ be the set of all places of $K$. Let $K_v$ be a completion of $K$ at $v$.

I'm searching for a reference for the statement of the following proposition:

For a positive integer $n\ge 2$, $H^1(G_{K_v},E)[n] \cong (E(K_v)/nE(K_v))^*$.

When $n$ is a prime, Theorem 1.4 of https://kskedlaya.org/kolyvagin-seminar/duality.pdf and Proposition 7.5 of B. Gross, "Kolyvagin's work on modular elliptic curves," London Math. Soc. Lecture Note Ser. 153 (1991), 235-256 cover this. But for an arbitrary positive integer $n \ge 2$, I don't see the proposition.

In the proof of Lemma 4.2 of the paper(https://www.researchgate.net/publication/254983039_Elliptic_curves_with_large_Tate-Shafarevich_groups_over_a_number_field), it is stated that Corollary I.3.4 in Milne's "Arithmetic duality theorems" corresponds to the claim for general $n$. However, unfortunately, this does not seem to be the corresponding proposition.

Could you please tell me refference for this ?

If anyone is aware of any literature regarding this proposition, I would greatly appreciate your guidance. Thank you very much in advance.

Answer 1

The original proof is by Tate in WC groups over $\mathfrak{p}$-adic fields. Nowadays, it is often derived from local Tate duality $H^1\bigl(K_v,E[n]\bigr)\times H^1\bigl(K_v,E[n]\bigr)\to \mathbb{Z}/n\mathbb{Z}$ and the additional fact that the image of the Kummer map $\kappa\colon E(K_v)/nE(K_v) \to H^1\bigl(K_v,E[n]\bigr)$ is its own orthogonal complement.