Subsections

## MTH416: Arithmetic of elliptic curves

[Cr:4, Lc:3, Tt:1, Lb:0]

#### Course Outline

The course aims to introduce elliptic curves and their moduli with an emphasis on curves over finite and number fields. Statements of theorems will be explained in detail and some relevant proofs will be given. Some examples of classical problems that can be studied using elliptic curves will be taken up and the use of the SAGE system to make calculations on elliptic curves will be introduced.

• Analytic theory: Doubly periodic functions and the Weierstrass form.
• Modular theory: Lattices in complex numbers and their classification.
• Algebraic theory: Tate-Weierstrass eqation and group law.
• Conversions between different forms of elliptic curves: Recognising ellptic curves hidden in various problems.
• Elliptic curves over finite-fields: Endomorphisms and Frobenius.
• Elliptic Curves over number-fileds: Mordell-Weil theorem.
• Calculations: Calculating points on elliptic curves, calculating rank of an ellptic curve, calculating modular forms.