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

TL;DR

This paper investigates the Galois module structure of cohomology groups of hypersurfaces in projective bundles over curves, providing a geometric proof for bounds on Mordell-Weil ranks in elliptic surface base changes.

Contribution

It describes the Galois representation on cohomology of hypersurfaces in P^n-bundles over curves, extending to singular cases, and applies this to Mordell-Weil rank bounds in elliptic surfaces.

Findings

Galois module structure of cohomology groups determined for certain hypersurfaces.

Provides a geometric proof of an upper bound for Mordell-Weil ranks.

Shows the Pal bound is weaker than the Shioda-Tate bound under specific conditions.

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.