School of Mathematics and Statistics
Number Theory Seminar 

In this talk I will introduce the basic definitions and concepts concerning abelian varieties. An abelian variety is a variety which admits an abelian group law, e.g. elliptic curves. After a small number of general remarks, I will talk exclusively about abelian varieties over the complex numbers. While it is well known that every lattice in C corresponds to an elliptic curve (and vice versa), the corresponding statement is false for general abelian varities. The main aim of my talk is to introduce the concepts of a Riemann form and a theta function; these give us (respectively) (*) a criterion for deciding when a complex torus can be thought of as an abelian variety, and(*) a means of embedding the variety in projective space. I'll discuss representations of isogenies between abelian varieties if time permits. 