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arXiv · 1902.03404

\'{E}tale Homotopy Obstructions of Arithmetic Spheres

Edo Arad·Shachar Carmeli·Tomer M. Schlank
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Abstract

Let $K$ be a field of characteristic $\ne 2$ and let $X$ be the affine variety over $K$ defined by the equation $$ X:\ a_0x_0^2 + \cdots + a_nx_n^2 = 1 $$ where $n\ge 0$ and $a_i\in K$. In this paper we compute the lowest mod 2 \'{e}tale 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 \'{e}tale-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.

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Edo Arad, Shachar Carmeli, Tomer M. Schlank. 2019-02-09. \'{E}tale Homotopy Obstructions of Arithmetic Spheres. https://arxiv.org/abs/1902.03404

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