The Elliptic Curves and Modular Forms Seminar

Organized by William Stein

Wednesdays 1-2pm in Science Center 507 at Harvard University

Fall 2004
Archive of Previous Years

K3 surfaces with Picard number one and infinitely many rational points

Ronald van Luijk (of UC Berkeley)

February 9, 2005 at 1pm

Not much is known about the arithmetic of K3 surfaces in general. Once the Picard number, which is the rank of the Neron-Severi group, is high enough, more structure is known and more can be said. But still we don't know of a single K3 surface whose set of rational points has been proved to be neither empty, nor Zariski dense. Also, until recently, not even a single K3 surface was known with Neron-Severi rank 1 and infinitely many rational points. We will give an explicit example of such a surface over Q, where even the Picard number over the algebraic closure is equal to 1. This solves an old problem, that has been attributed to Mumford.