Lecturer: Harry Schmidt

Term(s): Term 2

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

MA3B8 Complex Analysis:

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

Useful background:

MA4L7 Algebraic Curves:

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

MA4A5 Algebraic Geometry:

• Algebraic varieties
• Rational maps
• Morphisms

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

• MA3A6 Algebraic Number Theory
• MA4H9 Modular Forms
• MA3D5 Galois Theory
• MA4M3 Local Fields

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: Mordell's equation, Siegel's theorem

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