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MA426 Content

Content: We hope to cover the following topics in varying levels of detail:

  1. Non-singular cubics and the group law; Weierstrass equations.
  2. Elliptic curves over the rationals; descent, bounding E(\Q)/2E(\Q) , heights and the Mordell-Weil theorem, torsion groups; the Nagell-Lutz theorem.
  3. Elliptic curves over complex numbers, elliptic functions.
  4. Elliptic curves over finite fields; Hasse estimate, application to public key cryptography.
  5. Application to diophantin equations: elliptic diophantine problems, Fermat's Last Theorem.
  6. Application to integer factorisation: Pollard's $ p-1 $ method and the elliptic curve method.

Leads to: Ph.D. studies in number theory or algebraic geometry.

Books:

Our main text will be Washington; the others may also be helpful:

  • Lawrence C. Washington, Elliptic Curves: Number Theory and Cryptography, Discrete Mathematics and its applications, Chapman & Hall / CRC (either 1st edition (2003) or 2nd edition (2008)
  • Joseph H. Silverman and John Tate, Rational Points on Elliptic Curves, Undergraduate Texts in Mathematics, Springer-Verlag, 1992.
  • Anthony W. Knapp, Elliptic Curves, Mathematical Notes 40, Princeton 1992.
  • J. W. S. Cassels, Lectures on Elliptic Curves, LMS Student Texts 24, Cambridge University Press, 1991.