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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.