Sections to the 2-nilpotent fundamental group and rational points
Jordan Ellenberg

Let X/K be a variety over a number field and Xbar its base change to the algebraic closure. Then there is a natural map X(K) -> H^1(G_Q,pi_1(Xbar)) which is the subject of a great deal of interest (e.g. the Section Conjecture asserts that under some circumstances this map is more or less a bijection.) It is by now well known that very small quotients of the etale fundamental group still carry lots of arithmetic information about X. In this spirit, let pi^(2) be the quotient of pi_1(Xbar) by the second term of its lower central series, so that pi^(2) is a group of nilpotence class 2. We will discuss the problem of understanding the cohomology set H^1(G_Q,pi^(2)) and what it has to say about rational points on curves.