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  3. LMSST: 24 Lectures on Elliptic Curves
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    • Publisher:
      Cambridge University Press
      Publication date:
      05 June 2012
      21 November 1991
      ISBN:
      9781139172530
      9780521425308
      DOI:
      https://doi.org/10.1017/CBO9781139172530
      Dimensions:
      Weight & Pages:
      Dimensions:
      (228 x 152 mm)
      Weight & Pages:
      0.191kg, 144 Pages
      • Subjects:
      Mathematics (general), Mathematics, Number Theory, Geometry and Topology
      Series:
      London Mathematical Society Student Texts (24)
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    Subjects:
    Mathematics (general), Mathematics, Number Theory, Geometry and Topology
    Series:
    London Mathematical Society Student Texts (24)
    Information

    Book description

    The study of (special cases of) elliptic curves goes back to Diophantos and Fermat, and today it is still one of the liveliest centres of research in number theory. This book, which is addressed to beginning graduate students, introduces basic theory from a contemporary viewpoint but with an eye to the historical background. The central portion deals with curves over the rationals: the Mordell-Weil finite basis theorem, points of finite order (Nagell-Lutz) etc. The treatment is structured by the local-global standpoint and culminates in the description of the Tate-Shafarevich group as the obstruction to a Hasse principle. In an introductory section the Hasse principle for conics is discussed. The book closes with sections on the theory over finite fields (the 'Riemann hypothesis for function fields') and recently developed uses of elliptic curves for factoring large integers. Prerequisites are kept to a minimum; an acquaintance with the fundamentals of Galois theory is assumed, but no knowledge either of algebraic number theory or algebraic geometry is needed. The p-adic numbers are introduced from scratch, as is the little that is needed on Galois cohomology. Many examples and exercises are included for the reader. For those new to elliptic curves, whether they are graduate students or specialists from other fields, this will be a fine introductory text.

    Reviews

    ‘… an excellent introduction … written with humour.’

    Source: Monatshefte für Mathematik

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