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Elliptic Surfaces

Matthias Schuett, Tetsuji Shioda

TL;DR

This survey provides a comprehensive, lattice-theoretic framework for elliptic surfaces with section, focusing on rational elliptic surfaces and elliptic K3 surfaces. It develops the interplay between the geometry of singular fibres, the Mordell-Weil group and lattice realisations (Néron–Severi, MWL, essential lattice), and base-change phenomena, with Tate’s algorithm and Kodaira–Néron models at the core. It delivers detailed classifications (e.g., extremal and semi-stable cases), explicit constructions via cubic pencils and universal elliptic curves, and finiteness results akin to Shafarevich’s theorem in the function-field setting. The work bridges geometry, arithmetic, and lattice theory to illuminate how fibre types constrain sections, automorphisms, and moduli, with powerful implications for the arithmetic of rational surfaces and elliptic K3 surfaces.

Abstract

This survey paper concerns elliptic surfaces with section. We give a detailed overview of the theory including many examples. Emphasis is placed on rational elliptic surfaces and elliptic K3 surfaces. To this end, we particularly review the theory of Mordell-Weil lattices and address arithmetic questions.

Elliptic Surfaces

TL;DR

This survey provides a comprehensive, lattice-theoretic framework for elliptic surfaces with section, focusing on rational elliptic surfaces and elliptic K3 surfaces. It develops the interplay between the geometry of singular fibres, the Mordell-Weil group and lattice realisations (Néron–Severi, MWL, essential lattice), and base-change phenomena, with Tate’s algorithm and Kodaira–Néron models at the core. It delivers detailed classifications (e.g., extremal and semi-stable cases), explicit constructions via cubic pencils and universal elliptic curves, and finiteness results akin to Shafarevich’s theorem in the function-field setting. The work bridges geometry, arithmetic, and lattice theory to illuminate how fibre types constrain sections, automorphisms, and moduli, with powerful implications for the arithmetic of rational surfaces and elliptic K3 surfaces.

Abstract

This survey paper concerns elliptic surfaces with section. We give a detailed overview of the theory including many examples. Emphasis is placed on rational elliptic surfaces and elliptic K3 surfaces. To this end, we particularly review the theory of Mordell-Weil lattices and address arithmetic questions.

Paper Structure

This paper contains 149 sections, 70 theorems, 244 equations, 10 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Elliptic curves
  3. Analytic description
  4. Algebraic description
  5. Weierstrass equation
  6. Group operations
  7. Discriminant
  8. j-invariant
  9. Normal form for given j-invariant
  10. Group structure over number fields
  11. Integral points
  12. Bad places
  13. Elliptic surfaces
  14. Example: Cubic pencil
  15. Sections
  16. ...and 134 more sections

Key Result

Lemma 2.3

A cubic curve $E$ is smooth if and only if $\Delta\neq 0$.

Figures (10)

  • Figure 1: Elliptic surface with section and singular fibres
  • Figure 2: Irreducible singular fibres
  • Figure 3: Singular fibre of type ${\mathop{\rm I}}_n\; (n>1)$
  • Figure 4: Reducible additive singular fibres
  • Figure 5: Horizontal and vertical divisors
  • ...and 5 more figures

Theorems & Definitions (92)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Theorem 2.4
  • Definition 2.5: Normal form for j
  • Theorem 2.6: Mordell-Weil
  • Theorem 2.7: Siegel
  • Theorem 2.8: Shafarevich
  • Definition 3.1
  • Theorem 5.1
  • ...and 82 more