Thus, we have a (completelycompletely different!) example of a situation where the non-vanishing of the derivative of a zeta function implies the existence of an explicit non-trivial cohomology class which accounts"accounts" for the non-vanishing (in the first case, viewing a rational point on $E$ as a degree $0$ Galois cohomology class).
As René points out, $D$ is always nontrivial, for trivial reasons. What is interesting to me is not so much the nontriviality of $D$, but rather that one can deduce this nontriviality from $\zeta'(\Delta, 0)\neq 0$.
Another important difference that I should point out betweeen the two situations is that $\zeta'(\Delta, 0)\neq 0$ does not imply $\zeta(\Delta, 0)= 0$, because there is no analogue of BSD. In fact, $\zeta(\Delta, 0)$ is essentially $\mathrm{vol}(M)$.
Nevertheless, I am still curious.
