Let $C$ be a curve over $\mathbb Q$ with a point $P$ on $Pic^1$. For each $\mathbb Q$-rational point $Q$, $Q-P$ is a point on the Jacobian $J$. We can use the map $H^0(\mathbb Q, J) \to H^1(\mathbb Q,J[n])$ to obtain a class in $H^1(\mathbb Q,J[n])$ associated to each rational point.

Rather than trying to find a local obstruction to existence of rational points on $C$, we could try to find a local obstruction to existence of rational points in each $H^1(\mathbb Q, J[n])$ class. A local obstruction at $\mathbb Q_p$ would occur when the class, projected down to $H^1(\mathbb Q_p, J[n])$ does not arise from any $\mathbb Q_p$-rational point of $C$.

If $C$ does not have rational points, must there be some $n$ such that each class in $H^1(\mathbb Q, J[n])$ has a local obstruction at some $p$ or at $\infty$?