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On elliptic K3 surfaces

Ichiro Shimada

TL;DR

This work addresses the problem of classifying all possible configurations of singular fibers and Mordell–Weil torsion for complex elliptic $K3$ surfaces. It develops a lattice-theoretic framework based on discriminant forms, Jordan decompositions, and root lattices, and then uses Nikulin’s and Nishiyama’s results to translate geometric questions into explicit algebraic criteria. The main contributions are (i) a complete, computable catalog of realizable pairs $(oldsymbol{\,oldsymbol{\, ext{Sigma}}},G)$, (ii) a precise existence criterion linking ADE-type data to the torsion part of the Mordell–Weil group via overlattice and discriminant-form conditions, and (iii) a Maple-based algorithm that assembles the full dataset $oldsymbol{\mathcal{P}}$ from seed data through elementary transformations. This lattice-theoretic classification provides a concrete, verifiable map from singular fiber types to possible Mordell–Weil structures, enabling both geometric and arithmetic applications in the study of elliptic $K3$ surfaces and their moduli.

Abstract

We classify all the possible configurations of singular fibers and the torsion parts of Mordell-Weil groups of complex elliptic K3 surfaces. The complete list of 3279 configurations is attached.

On elliptic K3 surfaces

TL;DR

This work addresses the problem of classifying all possible configurations of singular fibers and Mordell–Weil torsion for complex elliptic surfaces. It develops a lattice-theoretic framework based on discriminant forms, Jordan decompositions, and root lattices, and then uses Nikulin’s and Nishiyama’s results to translate geometric questions into explicit algebraic criteria. The main contributions are (i) a complete, computable catalog of realizable pairs , (ii) a precise existence criterion linking ADE-type data to the torsion part of the Mordell–Weil group via overlattice and discriminant-form conditions, and (iii) a Maple-based algorithm that assembles the full dataset from seed data through elementary transformations. This lattice-theoretic classification provides a concrete, verifiable map from singular fiber types to possible Mordell–Weil structures, enabling both geometric and arithmetic applications in the study of elliptic surfaces and their moduli.

Abstract

We classify all the possible configurations of singular fibers and the torsion parts of Mordell-Weil groups of complex elliptic K3 surfaces. The complete list of 3279 configurations is attached.

Paper Structure

This paper contains 21 sections, 17 theorems, 47 equations, 1 figure, 11 tables.

Table of Contents

  1. Introduction
  2. Main results
  3. Tortion parts of Mordell-Weil groups
  4. $ADE$-types of singular fibers
  5. From $ADE$-types to configurations of singular fibers
  6. Miscellaneous facts
  7. Local invariants of lattices
  8. Local invariants
  9. Existence of lattices over $\mathord{\mathbb Z}$ with given local data
  10. Theory of discriminant forms
  11. Definitions
  12. Discriminant forms and overlattices
  13. Localization and discriminant form
  14. Existence of even lattices with a given discriminant form
  15. Over $\mathord{\mathbb Z}\sb p$
  16. ...and 6 more sections

Key Result

Theorem 2.1

The torsion part of the Mordell-Weil group of an elliptic $K3$ surface is isomorphic to one of the following :

Figures (1)

  • Figure 6.1: Dynkin diagram

Theorems & Definitions (18)

  • Theorem 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Theorem 2.10
  • Theorem 2.12
  • ...and 8 more