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Calculating the Mordell-Weil rank of elliptic threefolds and the cohomology of singular hypersurfaces

Klaus Hulek, Remke Kloosterman

Abstract

In this paper we give a method for calculating the rank of a general elliptic curve over the field of rational functions in two variables. We reduce this problem to calculating the cohomology of a singular hypersurface in a weighted projective 4-space. We then give a method for calculating the cohomology of a certain class of singular hypersurfaces, extending work of Dimca for the isolated singularity case.

Calculating the Mordell-Weil rank of elliptic threefolds and the cohomology of singular hypersurfaces

Abstract

In this paper we give a method for calculating the rank of a general elliptic curve over the field of rational functions in two variables. We reduce this problem to calculating the cohomology of a singular hypersurface in a weighted projective 4-space. We then give a method for calculating the cohomology of a certain class of singular hypersurfaces, extending work of Dimca for the isolated singularity case.

Paper Structure

This paper contains 14 sections, 38 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Relation between the Mordell-Weil group and cohomology of singular hypersurfaces
  3. Set-up
  4. Miranda's construction
  5. Comparing Mordell-Weil ranks
  6. Cohomology of hypersurfaces in ${\mathbf{P}}$
  7. Cohomology of hypersurfaces in ${\mathbf{P}}$: general results
  8. Cohomology of a surface with isolated ADE-singularities
  9. Calculation of $H^4_{\Sigma}(Y,{\mathbf{C}})$, local information
  10. Glueing local information
  11. Method for calculating $MW(\pi)$
  12. Examples
  13. Examples
  14. An application

Key Result

Theorem 1.1

Let $\pi: X \to S$ be an elliptic threefold $X$ over a rational surface $S$ and let $Y$ be a minimal model of $X/S$ in ${\mathbf{P}}(2n,3n,1,1,1)$. Assume that $H^4(Y,{\mathbf{Q}})$ has a pure weight 4 Hodge structure. Then

Theorems & Definitions (88)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Shioda-Tate-Wazir, Waz
  • Remark 3.1
  • ...and 78 more