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Rank jumps for Jacobians of Hyperelliptic curves on K3 surfaces

Ander Arriola Corpion, Cecília Salgado

Abstract

We study Mordell-Weil rank jumps on families of jacobians of a pencil of genus-2 curves on a K3 surface defined over a number field k. We exhibit a finite extension l/k over which the subset of fibers for which the rank jumps is infinite. Moreover, we describe further geometric conditions on the K3 surface under which the rank jumps on a non-thin set of fibers.

Rank jumps for Jacobians of Hyperelliptic curves on K3 surfaces

Abstract

We study Mordell-Weil rank jumps on families of jacobians of a pencil of genus-2 curves on a K3 surface defined over a number field k. We exhibit a finite extension l/k over which the subset of fibers for which the rank jumps is infinite. Moreover, we describe further geometric conditions on the K3 surface under which the rank jumps on a non-thin set of fibers.

Paper Structure

This paper contains 15 sections, 20 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Methodology and relation to the literature
  3. Organization of the text
  4. Background
  5. Fibered surfaces and Jacobians
  6. Abelian fibrations and specialization
  7. Multisections of fibrations and rank jumps
  8. Thin subsets
  9. Pencils of hyperelliptic curves of genus two on K3 surfaces
  10. The pull-back of lines
  11. Rank jumps for pencils of hyperelliptic curves of genus 2
  12. Rank jumps in the complete linear system
  13. Examples
  14. The Shioda-Tate formula for higher genus fibrations.
  15. Examples

Key Result

Theorem 1.1

Assume that $B$ contains a geometrically irreducible non-linear component defined over $k$. Then, there is a field extension $l/k$ of degree at most 12 such that $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Definition 1
  • Definition 2
  • Theorem 2.1
  • Theorem 2.2: Djamanenko-Manin
  • proof : Proof of Theorem \ref{['specialization']}
  • ...and 39 more