For a variety defined over a global function field K, e.g., Fp(t), the simplest cohomological obstruction is the "elementary obstruction" of Colliot-Thélène and Sansuc. Assuming this vanishes, what additional geometric hypotheses imply existence of a rational point? Combining "rational simple connectedness" in characteristic 0 with work of Esnault on rational points over finite fields, Chenyang Xu and I give new, uniform proofs of three classical results: Lang's proof that K is C2, a theorem of Brauer-Hasse-Noether that twisted forms of Grassmannians over K admit rational points, and the split case of Harder's theorem / Serre's "Conjecture II" over K. I will focus on the geometry of rational curves over C, and how one
"transports" this to results over a field such as K.
Yale 2013 >