Approximation theorems for classifying stacks over number fields

Technical Summary: Approximation Theorems for Classifying Stacks over Number Fields

Problem Statement
The paper addresses a specific approximation problem for algebraic stacks over a number field $k$. Given a connected linear algebraic group $G$ and a finite set of non-archimedean places $v_1, \dots, v_n$ of $k$, the central question is: given a collection of $G$-torsors $P_v$ defined over the completions $k_{v_i}$, does there exist a global $G$-torsor $P$ defined over $k$ that approximates the local data? In the language of stacks, this asks whether the map from the set of $k$-points of the classifying stack, $BG(k)$, to its adelic points, $BG(\mathbb{A}_k)$, has a dense image. The paper investigates this under the constraint of the Brauer-Manin obstruction, seeking conditions under which local torsors can be approximated by a global torsor that is $v$-adically close to the prescribed local torsors.

Methodology and Technical Framework
The paper employs a combination of topological constructions on algebraic stacks, cohomological methods, and reduction techniques via special groups.

1. Topologizing Adelic Points of Stacks:
   A primary technical hurdle addressed is the construction of a topology on the adelic points $X(\mathbb{A}_k)$ of an algebraic stack $X$. The author notes that a previous construction in the literature ([7]) contains an error regarding the triviality of projective modules over adele rings. To correct this, the paper introduces the concept of liftable stacks (Definition 2.12 and 2.14). A stack is $S$-liftable if it admits a presentation $P \to X$ by a separated algebraic space $P$ such that the map on points is surjective and satisfies specific local triviality conditions.

   • Key Lemma: The paper proves that for a finite subset $S$ of places, every $GL(V)$-torsor and $SL(V)$-torsor over the ring of $S$-adeles is trivial (Proposition 2.2 and Corollary 2.3). This allows the construction of a quotient topology on $X(\mathbb{A}_k)$ via a lifting presentation, ensuring the topology is well-defined and independent of the choice of presentation (Proposition 2.17).

2. The Brauer-Manin Pairing for Stacks:
   The author extends the classical Brauer-Manin obstruction to algebraic stacks.

   • Spreading Out: Using simplicial techniques and the big étale site, the paper establishes that cohomology classes (specifically Brauer classes) on a stack over a number field can be "spread out" to a model over an open subscheme of the ring of integers (Theorem 3.7, Corollary 4.3).
   • Pairing Construction: The Brauer-Manin pairing $\langle -, - \rangle: X(\mathbb{A}_k) \times Br(X) \to \mathbb{Q}/\mathbb{Z}$ is constructed by summing local Hasse invariants. The paper proves the continuity of this pairing with respect to the adelic topology (Proposition 4.5, 4.6).
   • Obstruction: The vanishing locus of this pairing, $X(\mathbb{A}_k)^{Br}$, is shown to be a closed subset containing the rational points $X(k)$, providing an obstruction to strong approximation.

3. Reduction to Special Groups:
   The core strategy for proving approximation theorems involves reducing the problem for a general group $G$ to the case of the special linear group $SL(V)$.

   • Isomorphism of Brauer Groups: By fixing a faithful representation $G \hookrightarrow SL(V)$, the classifying stack $BG$ is identified with the quotient stack $[X/SL(V)]$ where $X = G \backslash SL(V)$. The paper proves that the pullback map $Br([X/SL(V)]) \to Br(X)$ is an isomorphism (Proposition 5.3).
   • Equivariant Approximation: This reduction allows the problem to be shifted to the variety $X \times_G SL(V)$. The paper leverages known results on strong approximation for $SL(V)$ (a theorem of Kneser) and equivariant strong approximation for homogeneous spaces (results from [5] and [8]).

Key Contributions and Results

• Correction of Topological Construction: The paper provides a corrected and rigorous framework for defining the topology on the adelic points of algebraic stacks, resolving issues related to non-trivial projective modules over adele rings found in prior work.
• Strong Approximation with Brauer-Manin Obstruction for $BG$: The main result (Theorem 5.5) states that for a connected linear algebraic group $G$ over a number field $k$, and a finite set of places $S$ containing all archimedean places and at least one finite place, strong approximation holds for the classifying stack $BG$ off $S$ with respect to the Brauer-Manin obstruction. Specifically, the image of $BG(k)$ is dense in the Brauer-Manin fixed point locus of $BG(\mathbb{A}_k)$.
• Generalization to Quotient Stacks: Theorem 5.4 establishes a criterion for strong approximation for quotient stacks $[Y/G]$. It shows that if strong approximation holds for the associated space $Y \times_G SL(V)$, then it holds for the quotient stack $[Y/G]$. This applies to quotients of "groupic" varieties, including certain quotients of toric varieties.
• Equivariant Strong Approximation: The paper utilizes and clarifies the relationship between strong approximation for the group $G$ and the homogeneous space $G \backslash SL(V)$, demonstrating how the approximation properties of the group transfer to the classifying stack via the Brauer-Manin pairing.

Significance and Claims
The paper claims to answer a "very concrete question" regarding the approximation of local $G$-torsors by a global one, framing it as a strong approximation property for the classifying stack. The significance lies in:

1. Extending Classical Number Theory: It extends the classical theory of strong approximation and the Brauer-Manin obstruction, traditionally applied to schemes and varieties, to the realm of algebraic stacks, specifically classifying stacks.
2. Concrete Application: It provides a definitive condition (the vanishing of the Brauer-Manin obstruction) under which a collection of local $G$-torsors can be approximated by a global $G$-torsor.
3. Methodological Rigor: By correcting the topological foundations for adelic points of stacks, the paper enables further study of approximation properties in the stacky setting, which is essential for understanding the arithmetic of moduli problems and torsors.

The author notes that while the result for $BG$ relies on the connectedness of $G$ and the properties of $SL(V)$, the framework developed is applicable to a broader class of quotient stacks generated by groupic varieties. The paper does not claim to solve strong approximation for all algebraic stacks but restricts its main theorem to classifying stacks of connected linear algebraic groups and specific quotient stacks derived from them.