We define an abelian varieties. We define isogenies of abelian varieties and give some properties of isogenies. We prove some basic properties including the following: abelian varieties are commutative, the relative dualizing sheaf is trivial and we prove that all rational maps from a nonsingular variety to an abelian variety extend to a morphism.​

We show that all abelian varieties are projective, not just proper. The proof can be thought of as a generalization of the fact that every elliptic curve embeds into a 2-dimensional projective space via a Weierstrass equation.

In the notes and time permitting, we discuss which n it is possible to embed an abelian variety of dimension g into projective space of dimension n. It turns out that in general, for an abelian variety of dimension g, the smallest such value of n is 2g+1. Note that the case of elliptic curves, all of which can be embedded in 2-dimensional projective space via a Weierstrass equation are an exception.

We define the dual abelian variety of a given abelian variety, explain its universal property and sketch the proof of existence. 

We give an overview of some basic theorems in étale cohomology and explain why the Zariski site sails. We then compute the étale cohomology of abelian varieties and relate it to their Tate module.

We study endomorphism rings in the case of elliptic curves. We show that there are only 3 possible types of endomorphism ring. We will be particulatly inteserted in elliptic curves with complex multiplication. We show by considering class groups of orders of imaginary quadratic number fields that there are precisely 13 elliptic curves defined over Q (up to isomorphism). We also discuss how CM elliptic curves are related to class field theory.

We give a result allowing us to count the number of F_q^m points on an abelian variety over a closure of F_q using the Frobenius endomorphism. We then define the zeta function of an abelian variety and discuss how this gives a Riemann Zeta function.

We will concentrate on elliptic curves over number fields and discuss how one might actually carry out computations. More specifically, I'll discuss how to determine the places of good reduction, the torsion group, and coefficients of the L-series.

We discuss heights on abelian varieties, prove a finiteness result and sketch the proof of the Mordell-Weil theorem.

We introduce Jacobian varieties of curves, first abstractly as representable functors and then constructing them as abelian varieties. We prove some basic properties of the Jacobian variety, such as its dimension, and briefly show some applications such as to the Weil Conjectures. 

Precise topics to be determined.

Precise topics to be determined.

Precise topics to be determined