Lecture 1: The rank of elliptic curves

Abstract:  Cubic equations in two variables, or elliptic curves,

have been in the forefront of number theory since since the time of Fermat.

I will focus on the group of rational points, which Mordell proved was

finitely generated. I will review the conjecture of Birch and Swinnerton-

Dyer, which attempts to predict the rank of this group from the average

number of points (mod p), and will discuss the progress that has been

made on this conjecture to date.

Lecture 2: The arithmetic of hyperelliptic curves

Abstract: Hyperelliptic curves first appeared in work of Abel, who generalized

Euler's addition laws for elliptic integrals. Abel defined their genus g as the number

of integrals of the first kind. Every hyperelliptic curve of genus g has an affine

equation of the form y^2 = F(x), where F(x) is a separable polynomial of degree

2g+2 or 2g+1. Abel, Legendre, Jacobi, and Riemann studied these curves over

the real and complex numbers. In this talk, I will focus on curves defined over the

rational numbers, and will study the set of their rational solutions. Faltings proved

that when the genus g is at least 2, this set is finite. Using ideas of Bhargava,

one can now show that it is usually empty.

Lecture 3: Heegner points on modular curves

Abstract: In this talk, we will briefly review the theory of complex multiplication

and define certain special points, called Heegner points, on the modular curves

X_0(N). Following Birch, we will consider the divisor classes supported on these

points in the Jacobian, and will discuss methods that can be used to show that these classes are non-trivial. We will end with applications to the conjecture of Birch and Swinnerton-Dyer for elliptic curves over the rational numbers.