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Rank jumps and Multisections of elliptic fibrations on K3 surfaces

Alice Garbagnati, Cecília Salgado

Abstract

We consider the countably many families $\mathcal{L}_d$, $d\in\mathbb{N}_{\geq 2}$, of K3 surfaces admitting an elliptic fibration with positive Mordell--Weil rank. We prove that the elliptic fibrations on the very general member of these families have the potential Mordell--Weil rank jump property for $d\neq 2,3$ and moreover the Mordell--Weil rank jump property for $d\equiv 3\mod 4$, $d\neq 3$. We provide explicit examples and discuss some extensions to subfamilies. The result is based on the geometric interaction between the (potential) Mordell--Weil rank jump property and the presence of special multisections of the fibration.

Rank jumps and Multisections of elliptic fibrations on K3 surfaces

Abstract

We consider the countably many families , , of K3 surfaces admitting an elliptic fibration with positive Mordell--Weil rank. We prove that the elliptic fibrations on the very general member of these families have the potential Mordell--Weil rank jump property for and moreover the Mordell--Weil rank jump property for , . We provide explicit examples and discuss some extensions to subfamilies. The result is based on the geometric interaction between the (potential) Mordell--Weil rank jump property and the presence of special multisections of the fibration.

Paper Structure

This paper contains 20 sections, 23 theorems, 31 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Genus 1 fibrations and Elliptic surfaces
  4. Multisections of genus 1 fibrations
  5. Rank jump and base change
  6. Lattice polarized K3 surfaces and genus 1 fibrations
  7. K3 surfaces with more than one genus 1 fibration.
  8. K3 surfaces with infinite automorphisms group and one genus 1 fibration.
  9. K3 surfaces with an elliptic fibration and low Picard number
  10. Bisections of genus 1 and elliptic fibrations
  11. Bisections of genus 1 fibrations
  12. Elliptic K3 surfaces of Picard number 3
  13. Automorphisms of K3 surfaces in $\mathcal{L}_d$
  14. Smooth rational bisections on elliptic K3 surfaces
  15. Smooth genus 1 bisections on elliptic K3 surfaces
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $X$ be a K3 surface defined over a number field such that $\mathop{\mathrm{Pic}}\nolimits(X)=U \oplus \langle -2d\rangle$, for $d\in\mathbb{N}$, or equivalently a K3 surface defined over $k$ admitting an elliptic fibration (possibly over a field extension of $k$) and with Picard number 3. Assume

Theorems & Definitions (54)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.8
  • Lemma 2.9
  • proof
  • Corollary 2.10
  • ...and 44 more