Proofs of many famous problems in Number Theory (including Fermat’s Last Theorem) rely on understanding some properties about Elliptic Curves, which makes this topic inevitable and very interesting . One of the seven ‘Millennium Prize Problems’ stated by the Clay Mathematics Institute is the Birch Swinnerton-Dyer Conjecture, which is a statement regarding Elliptic Curves.

In this talk I will briefly describe the idea of a proof of the Mordell-Weil Theorem and introduce the n-Selmer group and Tate-Shafarevitch group associated to Elliptic Curves. I will define the L-function and some other arithmetic invariants attached to the Elliptic Curves and state the celebrated Birch-Swinnerton-Dyer Conjecture.