TY - BOOK ID - 48304819 TI - Mordell–Weil Lattices AU - Schütt, Matthias. AU - Shioda, Tetsuji. PY - 2019 SN - 9813293012 9813293004 PB - Singapore : Springer Singapore : Imprint: Springer, DB - UniCat KW - Lattice theory. KW - Lattices (Mathematics) KW - Space lattice (Mathematics) KW - Structural analysis (Mathematics) KW - Algebra, Abstract KW - Algebra, Boolean KW - Group theory KW - Set theory KW - Topology KW - Transformations (Mathematics) KW - Crystallography, Mathematical KW - Algebraic geometry. KW - Commutative algebra. KW - Commutative rings. KW - Algebra. KW - Field theory (Physics). KW - Category theory (Mathematics). KW - Homological algebra. KW - Nonassociative rings. KW - Rings (Algebra). KW - Algebraic Geometry. KW - Commutative Rings and Algebras. KW - Field Theory and Polynomials. KW - Category Theory, Homological Algebra. KW - Non-associative Rings and Algebras. KW - Rings (Algebra) KW - Homological algebra KW - Homology theory KW - Category theory (Mathematics) KW - Algebra, Homological KW - Algebra, Universal KW - Logic, Symbolic and mathematical KW - Functor theory KW - Algebra KW - Classical field theory KW - Continuum physics KW - Physics KW - Continuum mechanics KW - Mathematics KW - Mathematical analysis KW - Algebraic geometry KW - Geometry KW - Algebraic rings KW - Ring theory KW - Algebraic fields KW - Geometry, Algebraic. KW - Field theory (Physics) KW - Categories (Mathematics) KW - Algebra, Homological. UR - https://www.unicat.be/uniCat?func=search&query=sysid:48304819 AB - 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. ER -