Functional form and theoretic definition of an elliptic curve.

Relation to Abelian groups called E(K).

Abelian varieties.

Mordell-Weil group.

Rank of E(K) through the Birch-Swinnerton-Dyer conjectural formula.

Given a prime p and a subgroup of elements, P, such that pP = 0 determines a subgroup called E(K)[p]. Using the algebraic closure of K, say K-bar, we can get the subgroup E(K-bar)[p]. Under suitable conditions this group is isomorphic to (Z/pZ) squared. The Galois group, Gal(K-bar/K), permutes this group. Which gives rise to Galois representations. These ideas were used by Andrew Wiles to prove Fermat’s Last Theorem. Related to Langlands program, automorphic forms and modular forms.

If E is an elliptic curve over C, the complex field, then we have a complex manifold. Induces transformations which are simply translations. This gives rise to Hodge theory. Elliptic functions with periods.

Moduli space gives rise to an object called an orbifold and modular curves.