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Mordell–Weil Lattices

  • Matthias Schütt
  • Tetsuji Shioda
Book

Table of contents

  1. Front Matter
    Pages i-xvi
  2. Matthias Schütt, Tetsuji Shioda
    Pages 1-7
  3. Matthias Schütt, Tetsuji Shioda
    Pages 9-37
  4. Matthias Schütt, Tetsuji Shioda
    Pages 39-56
  5. Matthias Schütt, Tetsuji Shioda
    Pages 57-77
  6. Matthias Schütt, Tetsuji Shioda
    Pages 79-114
  7. Matthias Schütt, Tetsuji Shioda
    Pages 115-143
  8. Matthias Schütt, Tetsuji Shioda
    Pages 145-159
  9. Matthias Schütt, Tetsuji Shioda
    Pages 161-190
  10. Matthias Schütt, Tetsuji Shioda
    Pages 191-228
  11. Matthias Schütt, Tetsuji Shioda
    Pages 229-286
  12. Matthias Schütt, Tetsuji Shioda
    Pages 287-315
  13. Matthias Schütt, Tetsuji Shioda
    Pages 317-353
  14. Matthias Schütt, Tetsuji Shioda
    Pages 355-406
  15. Back Matter
    Pages 407-431

About this book

Introduction

This book lays out the theory of Mordell–Weil lattices, a very powerful and influential tool at the crossroads of algebraic geometry and number theory, which offers many fruitful connections to other areas of mathematics.

The book presents all the ingredients entering into the theory of Mordell–Weil lattices in detail, notably, relevant portions of lattice theory, elliptic curves, and algebraic surfaces. After defining Mordell–Weil lattices, the authors provide several applications in depth. They start with the classification of rational elliptic surfaces. Then a useful connection with Galois representations is discussed. By developing the notion of excellent families, the authors are able to design many Galois representations with given Galois groups such as the Weyl groups of E6, E7 and E8. They also explain a connection to the classical topic of the 27 lines on a cubic surface.

Two chapters deal with elliptic K3 surfaces, a pulsating area of recent research activity which highlights many central properties of Mordell–Weil lattices. Finally, the book turns to the rank problem—one of the key motivations for the introduction of Mordell–Weil lattices. The authors present the state of the art of the rank problem for elliptic curves both over Q and over C(t) and work out applications to the sphere packing problem.

Throughout, the book includes many instructive examples illustrating the theory.

Keywords

Mordell--Weil lattice lattices and sphere packings elliptic curves and surfaces K3 surface Galois representations and algebraic equations

Authors and affiliations

  • Matthias Schütt
    • 1
  • Tetsuji Shioda
    • 2
  1. 1.Institut für Algebraische GeometrieLeibniz Universität HannoverHannoverGermany
  2. 2.Department of MathematicsRikkyo UniversityTokyoJapan

Bibliographic information