Details
Mordell-Weil Lattices
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, Band 70
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139,09 € |
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| Verlag: | Springer |
| Format: | |
| Veröffentl.: | 17.10.2019 |
| ISBN/EAN: | 9789813293014 |
| Sprache: | englisch |
Dieses eBook enthält ein Wasserzeichen.
Beschreibungen
<p></p><p></p><p>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.</p>
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 <i>E</i><sub>6</sub>, <i>E</i><sub>7</sub> and <i>E</i><sub>8</sub>. They also explain a connection to the classical topic of the 27 lines on a cubic surface.<div><p>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 <b>Q</b> and over <b>C</b>(<i>t</i>) and work out applications to the sphere packing problem.</p> Throughout, the book includes many instructive examples illustrating the theory.<p></p><p></p><p></p></div>
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 <i>E</i><sub>6</sub>, <i>E</i><sub>7</sub> and <i>E</i><sub>8</sub>. They also explain a connection to the classical topic of the 27 lines on a cubic surface.<div><p>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 <b>Q</b> and over <b>C</b>(<i>t</i>) and work out applications to the sphere packing problem.</p> Throughout, the book includes many instructive examples illustrating the theory.<p></p><p></p><p></p></div>
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.
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
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