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Let $K$ be a number field, $E$ an elliptic curve over $K$, and $p$ an odd prime. If $v$ is a place of $K$, we know by the Kummer injection that $E(K_v)/pE(K_v) \hookrightarrow H^1 (K_v, E[p])$ for every $v$. By Tate local duality, we know that the image is self-dual (i.e. it is its own orthogonal complement under the Tate pairing).

Let $q \ne p$, $q > 3$ be a prime with $F_q$ the finite field of $q$ elements. Let $C$ be a smooth, geometrically connected curve $F_q$, $K = F_q^s(C)$ ($K$ is a geometric function field) , where $F_q^s$ is the separable closure of $F_q$. Following Ellenberg Prop 2.1 (http://www.math.wisc.edu/~ellenber/CMECTFF.pdf), we know that the Kummer injection factors through the tame fundamental group. We also have the local descent map

$E(K_v)/pE(K_v) \hookrightarrow H^1 (\pi_v, E[p])$, where $\pi_v$ is the local tame fundamental group ($\cong Gal(K_v^{tame}/K_v)$).

Questions:

Is there a version of Tate local duality for geometric function fields?

If there is a Local duality version as in the number field case, is the image self-dual?

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