Will someone kindly explain Kato's dual exponential map? I am reading this article by Rubin. Will somebody how to derive the formula given in equation 2 of section 5? It states thus:

$z$ corresponds to the map
$$x\mapsto Tr_{\mathbb{Q}_{n,p}/{\mathbb{Q}}}\lambda_E(x)exp^*(z)$$

Will somebody explain to me how this can be seen? Or at least how to begin? This result is used in several papers but didn't find an easy proof anywhere. Thank you!!
 A: This formula is actually more or less tautological. The point is that there are isomorphisms of finite-dimensional $\mathbf{Q}_p$-vector spaces
$$ E(\mathbf{Q}_{n, p}) \otimes \mathbf{Q}_p \cong \Big(H^1_{/S}(\mathbf{Q}_{n, p}, V)\Big)^\vee $$
(induced by local Tate duality as explained a few lines up in Rubin's paper), and
$$\operatorname{tan}(E / \mathbf{Q}_{n, p}) \cong \Big( \operatorname{cotan}(E / \mathbf{Q}_{n, p}) \Big)^\vee. $$
The second formula is not quite as tautological as it looks, because $\operatorname{cotan}(E / \mathbf{Q}_{n, p})$ is the dual of $\operatorname{tan}(E / \mathbf{Q}_{n, p})$ as a vector space over $\mathbf{Q}_{n, p}$, not over $\mathbf{Q}_{p}$; but composing the obvious $\mathbf{Q}_{n, p}$-linear duality pairing $\operatorname{tan} \times \operatorname{cotan} \to \mathbf{Q}_{n, p}$ with the trace map $\operatorname{Tr}_{\mathbf{Q}_{n, p} / \mathbf{Q}_{p}}$.
With these identifications, the dual-exponential map 
$$ \exp^*_E: H^1_{/S}(\mathbf{Q}_{n, p}, V) \to \operatorname{cotan}(E / \mathbf{Q}_{n, p})$$
is by definition the dual of the exponential map
$$\exp_E: E(\mathbf{Q}_{n, p}) \to \operatorname{tan}(E / \mathbf{Q}_{n, p})$$
and that is all that Rubin's formula (2) is saying -- the somewhat mysterious appearance of the trace map is just there to convert $\mathbf{Q}_{n, p}$-linear duality between the tangent and cotangent spaces to $\mathbf{Q}_{p}$-linear duality, as above.
You might like to look at Kato's own account of this theory, in Asterisque 295 (2004), for an authoritative version. 
PS. Of course this is not the whole story; in order to actually prove anything about the dual exponential map, one needs to first give a new, much more direct definition of the dual-exp, and then prove that the map given by the new definition coincides with the dual of the exponential, so suddenly the duality that was true by definition becomes a serious theorem. This deep, but purely local, theorem is due to Kato and is in "Lectures on Hasse-Weil L-functions via $\mathbf{B}_{\mathrm{dR}}$, I" (generally known as the Trento Lectures).
