Lecturer: Harry Schmidt
Term(s): Term 2
Commitment: 30 lectures
Assessment: 85% 3 hour Oral exam, 15% coursework
Formal registration prerequisites: None
- Affine and projective plane curves
- Bezout's theorem
- Singular points
- Change of coordinates
- Algebraic varieties
- Rational maps
Synergies: The following modules go well together with Elliptic Curves:
MA3B8 Complex Analysis:
- Holomorphic and meromorphic functions
- Laurent series
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 , 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 diophantine 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.
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.