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.
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.
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.