[Paper Review] Étale Homotopy Obstructions of Arithmetic Spheres

Edo Arad, Shachar Carmeli+1 more|arXiv (Cornell University)|Feb 9, 2019

Homotopy and Cohomology in Algebraic Topology4 citations

TL;DR

This paper establishes a precise link between étale homotopy theory and arithmetic geometry by showing that the lowest mod 2 étale homological obstruction to a rational point on the affine quadric $X: \sum a_i x_i^2 = 1$ over a field $K$ of characteristic $\neq 2$ is the cup product $[a_0] \cup \cdots \cup [a_n]$ in $H^*_{\text{ét}}(K, \mathbb{Z}/2)$, which coincides with the top Hasse-Witt invariant. The result is an étale homotopy analogue of the topological obstruction theory for unit sphere bundles.

ABSTRACT

Let $K$ be a field of characteristic $ e 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 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.

Motivation & Objective

• To connect classical arithmetic invariants—specifically Hasse-Witt classes—for quadratic forms with modern étale homotopy obstruction theory.
• To establish that the lowest mod 2 étale homological obstruction to a rational point on a sphere defined by $\sum a_i x_i^2 = 1$ is the cup product of the coefficients $[a_i]$.
• To provide an étale homotopy-theoretic interpretation of the classical Hilbert symbol and Hasse-Witt invariants as obstruction classes.
• To generalize topological obstruction theory (Stiefel-Whitney classes) to the arithmetic setting using the étale homotopy type of classifying stacks $BO_{n,K}$.

Proposed method

• Utilizes the relative étale homotopy type and pro-étale sheaf cohomology to define obstruction classes in $\operatorname{Pro} \mathrm{Shv}_\infty(\text{Spec } K, \mathbb{Z}/2)$.
• Applies higher obstruction theory in $\infty$-topoi to compute obstructions for global sections of sheaves associated to sphere bundles.
• Constructs a universal sphere bundle $\widetilde{S}_{n+1} \to BO_{n+1,K}$ and computes its étale cohomology, showing $H^*_{\text{ét}}(BO_{n+1,K}, \mathbb{Z}/2)$ is a polynomial algebra generated by Hasse-Witt classes $HW_1, \dots, HW_n$.
• Uses the Beck-Chevalley condition and smooth base change for higher stacks to ensure compatibility of pullbacks under morphisms of schemes.
• Leverages the Whitney product formula for Hasse-Witt classes to decompose the obstruction for a diagonalized quadratic form as a cup product.
• Reduces the general case to the 1-dimensional case via factorization through $Q_1^{n+1} \to Q_{n+1}$, where the first Hasse-Witt class pulls back to $[a_i]$.

Experimental results

Research questions

• RQ1 What is the étale homotopy-theoretic obstruction to the existence of a rational point on the affine quadric $X: \sum a_i x_i^2 = 1$?
• RQ2 How does the mod 2 étale homological obstruction relate to the Hasse-Witt invariants of the quadratic form?
• RQ3 Can the classical topological obstruction (Stiefel-Whitney classes) be lifted to an arithmetic setting using étale homotopy theory?
• RQ4 Is the cup product $[a_0] \cup \cdots \cup [a_n]$ the precise obstruction class in $H^*_{\text{ét}}(K, \mathbb{Z}/2)$ for rational points on the sphere?
• RQ5 Does the obstruction theory for sphere bundles in the étale setting recover the classical Hasse-Witt invariant $HW_n(B)$?

Key findings

• The lowest mod 2 étale homological obstruction to a rational point on the sphere $X: \sum_{i=0}^n a_i x_i^2 = 1$ is the cup product $[a_0] \cup \cdots \cup [a_n]$ in $H^*_{\text{ét}}(K, \mathbb{Z}/2)$.
• This obstruction class coincides with the top Hasse-Witt invariant $HW_{n+1}(B)$ of the quadratic form $B = \sum a_i x_i^2$.
• The obstruction arises as the pullback of the $n$-th Stiefel-Whitney class $HW_n$ from the classifying stack $BO_{n+1,K}$ via the classifying map $f_B: \mathrm{Spec}\,K \to Q_{n+1}$.
• For a 1-dimensional sphere $ax^2 = 1$, the obstruction $f^*HW_1$ is precisely the class $[a] \in H^1_{\text{ét}}(K, \mathbb{Z}/2) \cong K^\times / (K^\times)^2$.
• The obstruction theory for sphere bundles in the étale setting is compatible with pullbacks and respects the Whitney product formula, ensuring consistency with classical topological analogues.
• The computation confirms that the étale homotopy obstruction theory recovers the classical arithmetic obstruction defined by the Hilbert symbol and Hasse-Witt invariants.