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4:00 pm Tuesday, March 23, 2010 Algebraic Geometry Seminar: Etale pi_1 obstructions to rational pointsby
Kirsten Wickelgren (Harvard/AIM) in HB 227- Grothendieck's anabelian conjectures say that hyperbolic algebraic curves over number fields should be K(pi,1)'s in algebraic geometry. It follows that conjecturally the rational points on such a curve are the sections of etale pi_1 of the structure map. We will use these sections to approach the problem of distinguishing the rational points of a curve from the rational points of its Jacobian, where the curve is viewed as embedded inside its Jacobian using some fixed rational point. More specifically, we will use cohomological obstructions of Jordan Ellenberg coming from the etale fundamental group to obstruct a rational point of the Jacobian from lying on the curve. We will relate Ellenberg's obstructions to Massey products, and explicitly compute versions of the first and second for P^1- {0,1, infty}. Over R, we show the first obstruction alone determines the connected components of real points of the curve from those of the Jacobian, giving a strengthening of the section conjecture over R.
Submitted by evanmb@gmail.com |