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.