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  3. Elliptic Curves and Big Galois Representations
  • Cited by 7
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    • Publisher:
      Cambridge University Press
      Publication date:
      04 May 2010
      31 July 2008
      ISBN:
      9780511721281
      9780521728669
      DOI:
      https://doi.org/10.1017/CBO9780511721281
      Dimensions:
      Weight & Pages:
      Dimensions:
      (228 x 152 mm)
      Weight & Pages:
      0.41kg, 288 Pages
      • Subjects:
      Mathematics (general), Mathematics, Number Theory, Geometry and Topology
      Series:
      London Mathematical Society Lecture Note Series (356)
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    Subjects:
    Mathematics (general), Mathematics, Number Theory, Geometry and Topology
    Series:
    London Mathematical Society Lecture Note Series (356)
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    Book description

    The arithmetic properties of modular forms and elliptic curves lie at the heart of modern number theory. This book develops a generalisation of the method of Euler systems to a two-variable deformation ring. The resulting theory is then used to study the arithmetic of elliptic curves, in particular the Birch and Swinnerton-Dyer (BSD) formula. Three main steps are outlined: the first is to parametrise 'big' cohomology groups using (deformations of) modular symbols. Finiteness results for big Selmer groups are then established. Finally, at weight two, the arithmetic invariants of these Selmer groups allow the control of data from the BSD conjecture. As the first book on the subject, the material is introduced from scratch; both graduate students and professional number theorists will find this an ideal introduction. Material at the very forefront of current research is included, and numerical examples encourage the reader to interpret abstract theorems in concrete cases.

    Reviews

    "This research monograph contains much that has not been published elsewhere, and will be useful for specialists in the field who want to catch up on the author's work."
    Neil P. Dummigan, Mathematical Reviews

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