Symplectic geometry and Selmer schemes

Title: Symplectic geometry and Selmer schemes

 

Abstract: The rational points of a hyperbolic curve over a number field inject into non-abelian Galois cohomology with coefficients given by the curve's geometric etale fundamental group. This suggests that studying such Galois cohomology sets should yield Diophantine consequences; an instance of this is realized in the Chabauty-Kim method. The central objects in this method are Selmer schemes which are obtained by considering "Selmer intersections" of group cohomological functors. 

 

In this talk, we'll introduce these group cohomological functors and Selmer schemes. We'll show that a wide class of group cohomological functors can be shown to possess symplectic structures and that the related Selmer schemes can be naturally realized as intersections of multiple Lagrangians. Lastly, we'll explain some of the utility of this description, especially towards effective Diophantine questions.