# MA626 Elliptic Curves

**Lecturer: Prof. David Loeffler**

**Term(s):** Term 2

**Commitment:** 30 lectures

**Assessment:** 85% Oral exam, 15% coursework

**Prerequisites:** This is a sophisticated module making use of a wide palette of tools in pure mathematics. In addition to a general grasp of first and second year algebra and analysis modules, the module involves results from MA246 Number Theory (especially factorisation, modular arithmetic). Parts of MA3B8 Complex Analysis, MA3D5 Galois Theory, MA3A6 Algebraic Number Theory or MA4A5 Algebraic Geometry may be helpful but are not essential.

**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 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.