Weil-Chatelet Groups of Rational Elliptic Surfaces

We classify pairs $(S, \gamma)$, consisting of a rational elliptic surface $S$ and a Galois cover $\gamma$ of the base, which satisfy a condition we call $\mathcal{L}$-stability. We explain how to use the theory of Mordell-Weil lattices to compute the kernel of the restriction maps of Weil-Chatelet groups for $\mathcal{L}$-stable pairs. We also prove results about the injectivity of restriction maps of Weil-Chatelet groups for some pairs which are not $\mathcal{L}$-stable.