MA426 Elliptic Curves

Introductory description

See principle module aims.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

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()/2E(), 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.

Learning outcomes

By the end of the module, students should be able to:

• The students are expected to become familiar with the group law on elliptic curves, the degree of an isogeny, the parallelogram identity, elliptic curves over finite fields, the Reduction Theorem and the Lutz-Nagell Theorem, explicit 2-descents in the presence of a 2-torsion point and computations of the Mordell-Weil group.

Subject specific skills

Elliptic curves are at the heart of modern number theory, drawing tools from commutative algebra, algebraic geometry, complex analysis, abstract algebra. The students taking this module develop and strengthen the knowledge acquired in their mathematical studies. Moreover, elliptic curves find applications in cryptography and are used in encoding and decoding of messages sent over the internet.

Transferable skills

Elliptic curves are at the heart of modern number theory, drawing tools from commutative algebra, algebraic geometry, complex analysis, abstract algebra. The students taking this module develop and strengthen the knowledge acquired in their mathematical studies. Moreover, elliptic curves find applications in cryptography and are used in encoding and decoding of messages sent over the internet.