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

Remke Kloosterman

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.

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

Abstract

Let be a complex hypersurface in a -bundle over a curve . Let be a Galois cover with group . In this paper we describe the -structure of provided that is either smooth or and has at most ADE singularities.% and the -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 then we show that the bound of Pal is weaker than the bound coming from the Shioda-Tate formula.

Paper Structure

This paper contains 4 sections, 29 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Weierstrass models and Projective bundles
  3. The $\mathbf{C}[G]$-structure of $H^{p,q}(X')$
  4. The $\mathbf{C}[G]$-structure of the cohomology of Weierstrass models

Key Result

Proposition 1.1

Let $\pi: X\to C$ be an elliptic surface and set $\mathcal{L}=(\pi_*N_{S/X})^*$. Let $f:C'\to C$ be a ramified Galois cover with group $G$ and let $X'\to C'$ be the smooth minimal elliptic surface birational to $X\times_C C'$. Suppose that over each branch point of $f$ the fiber of $\pi$ is smooth o

Theorems & Definitions (62)

  • Proposition 1.1
  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • ...and 52 more