A question on the cohomology of elliptic curves over local fields

Question

Let $K$ be a number field,$\nu$ a nonarchimedian prime of $K$, $K_{\nu} $ the completion of $K $ at $\nu $ with maximal unramified extension $K_{\nu}^{unr} $. Let $E $ be an elliptic curve defined over $K $, and let $E_{p^n} $ denote its $p^n $-divisors, where $p $ is a (rational) prime that is not a multiple of $\nu $.

The authors of the paper that I'm studying claim that $$H^1 (K_{\nu}^{unr}/K_{\nu},E_{p^n})\subseteq ker (H^1 (K_{\nu}, E_{p^n})\rightarrow H^1 (K_{\nu},E)) $$. It totally escapes me why that is the case. I might not be seeing the wood for all the trees here. Can someone help me out?

Answer 1

By the Kummer sequence, your kernel is isomorphic to $E(K_\nu)/p^nE(K_\nu)$ via the connecting homomorphism. If you also assume that your elliptic curve has good reduction ath $\nu$, then the cocycle you get via the connecting homomorphism will be unramified, which I think is what you want (plus use inflation-restriction). This all comes down to the fact that the $p^n$-torsion generates an unramified extension of $K_\nu$ provided that (1) $E$ has good reduction at $\nu$ and (2) $\nu$ does not divide $p$. (However, you didn't say that $E$ has good reduction at $\nu$, in which case I don't immediately see why your inclusion should be true.)