Chevalley-Weil formula for hypersurfaces in Pⁿ -bundles over curves and Mordell—Weil ranks in function field towers

Abstract

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.