Authors:
- Is the first comprehensive introduction of Mordell–Weil lattices that does not assume extensive prerequisites
- Shows that the theory of Mordell–Weil lattices itself is very powerful yet relatively easy to master and apply
- Demonstrates with many examples and applications how Mordell–Weil lattices connect with several areas of mathematics
Part of the book series: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics (MATHE3, volume 70)
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Table of contents (13 chapters)
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Front Matter
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Back Matter
About this book
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 withelliptic 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.Reviews
“The monograph is very well-written … and the structure is well-planned. … it is highly recommended to those readers that want to learn about the history and the recent developments on this subject.” (Piotr Pokora, zbMATH 1433.14002, 2020)
Authors and Affiliations
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Institut für Algebraische Geometrie, Leibniz Universität Hannover, Hannover, Germany
Matthias Schütt
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Department of Mathematics, Rikkyo University, Tokyo, Japan
Tetsuji Shioda
Bibliographic Information
Book Title: Mordell–Weil Lattices
Authors: Matthias Schütt, Tetsuji Shioda
Series Title: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics
DOI: https://doi.org/10.1007/978-981-32-9301-4
Publisher: Springer Singapore
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Singapore Pte Ltd. 2019
Hardcover ISBN: 978-981-32-9300-7Published: 29 October 2019
Softcover ISBN: 978-981-32-9303-8Published: 29 October 2020
eBook ISBN: 978-981-32-9301-4Published: 17 October 2019
Series ISSN: 0071-1136
Series E-ISSN: 2197-5655
Edition Number: 1
Number of Pages: XVI, 431
Number of Illustrations: 23 b/w illustrations, 9 illustrations in colour
Topics: Algebraic Geometry, Commutative Rings and Algebras, Field Theory and Polynomials, Category Theory, Homological Algebra, Non-associative Rings and Algebras