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MA426 Elliptic Curves

Lecturer: Sam Chow

Term(s): Term 1

Status for Mathematics students: List C

Commitment: 30 lectures

Assessment: 85% by 3-hour examination, 15% coursework

Formal registration prerequisites: None

Assumed knowledge:

MA249 Algebra II: Groups and Rings:

  • Basic theory of groups, rings, fields

MA257 Introduction to Number Theory:

  • Primes
  • Divisibility
  • Congruences

Useful background:

MA4L7 Algebraic Curves:

  • Affine and projective plane curves
  • Bezout's theorem
  • Singular points
  • Change of coordinates

MA3B8 Complex Analysis:

  • Holomorphic and meromorphic functions
  • Laurent series
  • Poles
  • Residues

MA4A5 Algebraic Geometry:

  • Algebraic varieties
  • Rational maps
  • Morphisms

Synergies: The following modules go well together with Elliptic Curves:

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

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

Additional Resources