# MA426 Elliptic Curves

**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

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

**Useful background:**

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

- 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 , 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.

**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.