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 Pⁿ-bundle over a curve C. Let C'→C be a Galois cover with group G. In this paper we describe the C[G]-structure of $H^p,q$(X x$_{c}$ C C') provided that X x$_{c}$ C' is either smooth or n = 3 and X x$_{c}$ C' has at most ADE singularities. As an application we obtain a geometric proof for an upper bound by Páal for the Mordell—Weil rank of an elliptic surface obtained by a Galois base change of another elliptic surface.