  Let $X$ be a complex hypersurface in a $\mathbf{P}^n$-bundle over a curve
$C$. Let $C'\to C$ be a Galois cover with group $G$. In this paper we describe
the $\mathbf{C}[G]$-structure of $H^{p,q}(X\times_C C')$ provided that
$X\times_C C'$ is either smooth or $n=3$ and $X\times_C C'$ 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 $C'\to C$ then we show that the
bound of Pal is weaker than the bound coming from the Shioda-Tate formula.
