Chevalley-Weil formula for hypersurfaces in ${\mathbf{P}}^n$-bundles over curves and Mordell-Weil ranks in function field towers

Let $X$ be a complex hypersurface in a ${\mathbf{P}}^n$-bundle over a curve $C$. Let $C^{\mathrm{\prime }}\to C$ be a Galois cover with group $G$. In this paper we describe the $\mathbf{C}[G]$-structure of $H^{p,q}(X{\times }_C{C}^{\mathrm{\prime }})$ provided that $X{\times }_C{C}^{\mathrm{\prime }}$ is either smooth or $n=3$ and $X{\times }_C{C}^{\mathrm{\prime }}$ has at most ADE singularities.% and the $\mathbf{C}[G]$-structure of the cohomology of its resolution of singularities. As an application we obtain a geometric proof for an upper bound by Pal for the Mordell-Weil rank of an elliptic surface obtained by a Galois base change of another elliptic surface. If the Galois group of the base field acts trivially on the Galois group of the cover $C^{\mathrm{\prime }}\to C$ then we show that the bound of Pal is weaker than the bound coming from the Shioda-Tate formula.