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.
Virginia Mathematics Lectures archive