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

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.

The following advanced topics could also be addressed:

- L-functions of elliptic curves: The terms of the L-function. The statement of the Birch and Swinnerton-Dyer conjecture and its similarity with the Dirichlet unit theorem.
- Modularity of Elliptic curves: Shimura-Taniyama-Weil conjecture and the statement of the Theorem of Wiles-Taylor.

- R. V. Gurjar et al, Elliptic Curves Narosa/NBHM (2006).
- J. H. Silverman, The Arithmetic of Elliptic Curves Springer GTM 106 (1986).