Date & Time: Monday, July 30, 2007, 14:00-15:00.
Venue: Ramanujan Hall
Title: The Arithmetic of Elliptic Surfaces
Speaker: Ritabrata Munshi, Rutgers University
Abstract: A family of elliptic curves (or an elliptic surface) is given by a Weierstrass equation $E_t : y^2 = x^3 + A(t)x + B(t)$, where $A(t)$ and $B(t)$ are polynomials. Little is known about the rational points on such families. However there are interesting predictions. For example, a conjecture of Mazur states that if one fiber $E_t$ has infinite number of rational points, then there is a dense set of fibers with infinite number of rational points. In the special case when deg A <= 4 and deg B <= 6, there is a more precise conjecture of Manin about the number of rational points of a fixed height. In this talk I will describe recent results towards these conjectures. I will also discuss other open problems regarding distribution of Mordell-Weil ranks.