# MA426 Content

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

1. Non-singular cubics and the group law; Weierstrass equations.
2. Elliptic curves over the rationals; descent, bounding $E(\Q)/2E(\Q)$ , heights and the Mordell-Weil theorem, torsion groups; the Nagell-Lutz theorem.
3. Elliptic curves over complex numbers, elliptic functions.
4. Elliptic curves over finite fields; Hasse estimate, application to public key cryptography.
5. Application to diophantin equations: elliptic diophantine problems, Fermat's Last Theorem.
6. 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.