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Mordell-Weil Lattices (Hardcover, 1st ed. 2019)
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Mordell-Weil Lattices (Hardcover, 1st ed. 2019)
Series: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 70
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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.
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