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＜電子ブック＞
Mordell–Weil Lattices / by Matthias Schütt, Tetsuji Shioda
(Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics. ISSN:21975655 ; 70)

版	1st ed. 2019.
出版者	(Singapore : Springer Nature Singapore : Imprint: Springer)
出版年	2019
本文言語	英語
大きさ	XVI, 431 p. 32 illus., 9 illus. in color : online resource
著者標目	*Schütt, Matthias author Shioda, Tetsuji author SpringerLink (Online service)
件　名	LCSH:Algebraic geometry LCSH:Commutative algebra LCSH:Commutative rings LCSH:Algebraic fields LCSH:Polynomials LCSH:Algebra, Homological LCSH:Nonassociative rings FREE:Algebraic Geometry FREE:Commutative Rings and Algebras FREE:Field Theory and Polynomials FREE:Category Theory, Homological Algebra FREE:Non-associative Rings and Algebras
一般注記	Introduction -- Lattices -- Elliptic Curves -- Algebraic surfaces -- Elliptic surfaces -- Mordell--Weil Lattices -- Rational Elliptic Surfaces -- Rational elliptic surfaces and E8-hierarchy -- Galois Representations and Algebraic Equations -- Elliptic K3 surfaces 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 HTTP:URL=https://doi.org/10.1007/978-981-32-9301-4

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