Étale Homotopy Obstructions of Arithmetic Spheres

Abstract

Let $K$ be a field of characteristic $\ne 2$ and let $X$ be the affine variety over $K$ defined by the equation $$X:a_0{x}_0^2+\cdots +a_n{x}_n^2=1$$ where $n\ge 0$ and $a_i\in K$. In this paper we compute the lowest mod 2 étale homological obstruction class to the existence of a $K$-rational point on $X$, and show that it is the cup product of the form $$o_{n+1}=[a_0]\cup \cdots \cup [a_n].$$ Our computation is an étale-homotopy analogue of the topological fact that Stiefel-Whitney classes are the homological obstructions to find a section to the unit sphere bundle of a real vector bundle.

42 pages