Insufficiency of the algebraic Brauer--Manin obstruction for homogeneous spaces

Technical Summary: Insufficiency of the algebraic Brauer–Manin obstruction for homogeneous spaces

Problem Statement
The paper addresses the failure of the Hasse principle (local-global principle) for rational points on algebraic varieties. Specifically, it investigates whether the Brauer–Manin obstruction, a cohomological mechanism introduced by Manin, is sufficient to explain all such failures for homogeneous spaces of connected linear algebraic groups.

While the sufficiency of the Brauer–Manin obstruction is established for varieties with "tame" geometry (e.g., geometrically rationally connected varieties) and for homogeneous spaces with connected or commutative stabilizers, the case of finite non-abelian stabilizers remains largely open. Previous work by Demarche, Lucchini Arteche, Neftin, Demeio, and Lagarde demonstrated that the algebraic part of the Brauer–Manin obstruction is insufficient for weak approximation in the case of finite constant metabelian stabilizers. However, the existence of homogeneous spaces where the algebraic obstruction fails to explain the failure of the Hasse principle itself was unknown. This paper aims to construct such a counterexample.

Methodology
The authors construct a specific counterexample using a homogeneous space $X$ of the special linear group $SL_n$ over a number field $k$ containing a primitive $p$-th root of unity (where $p$ is an odd prime). The construction relies on the machinery developed by Borovoi and Kunyavskii, further refined by the author in previous work [Lin24].

1. Group Construction: The geometric stabilizer $F$ of the homogeneous space is constructed as a central extension of $M \oplus M$ by $Z$, where $M = \mathbb{F}_p^d$ and $Z$ is a quotient of $M \otimes M$. The extension is defined via a surjective, non-degenerate homomorphism $\phi: M \otimes M \to Z$.
2. Numerical Data: The authors choose $d = [k:\mathbb{Q}] + 3$ and define the inclusion $A \otimes A \to M \otimes M$ (where $A=\mathbb{F}_p$) by mapping the generator to the identity matrix $Id$. The group $F$ thus has nilpotency class 2 and order $p^{d^2+2d-1}$.
3. Cohomological Setup: A class $\beta \in H^2(k, Z)$ is constructed using global class field theory. This class vanishes at all places except for a set of distinct finite places $\{v_1, \dots, v_d\}$, where its local invariants correspond to the images of rank-one matrices $e_i \otimes e_i$ under $\phi$.
4. Local vs. Global Analysis:
   • Local Existence: Using Tate's local duality and the fact that the rank of the local cohomology groups $H^1(k_v, \mathbb{F}_p)$ is small (bounded by 2 for places not dividing $p$), the authors show that local solutions $x_v, y_v$ exist for the equation $\beta|_{k_v} = \phi^*(x_v \cup y_v)$ at every place $v$. This ensures $X(k_v) \neq \emptyset$ for all $v$.
   • Global Non-Existence: The authors prove that no global solution exists. They utilize a uniform bound on the rank of the matrix $A_v = \text{Inv}_v(x \cup y)$ derived from the dimension of $H^1(k_v, \mathbb{F}_p)$. For sufficiently large $d$, the local constraints force $A_v$ to be exactly $e_i \otimes e_i$ at the specific places and $0$ elsewhere. Summing these local invariants yields the identity matrix $Id$, which is non-zero, contradicting the global reciprocity law (which requires the sum of local invariants to vanish).

Key Contributions and Results
The paper establishes the following main result (Theorem 1):

• Counterexample Construction: There exists a homogeneous space $X$ of $SL_n$ over $k$ with finite $p$-group stabilizers of nilpotency class 2 such that:

  1. $X$ has points in every completion $k_v$ (local solubility).
  2. $X$ has no rational points ($X(k) = \emptyset$).
  3. The unramified algebraic Brauer group satisfies $Br_{nr,1}(X) = Br_0(X)$.

• Transcendental Obstruction: Since the algebraic part of the Brauer–Manin obstruction is trivial (equal to the constant part), the failure of the Hasse principle cannot be explained by algebraic classes. The authors demonstrate that the full unramified Brauer group $Br_{nr}(X)$ is isomorphic to $\mathbb{Z}/p$ modulo constants. By constructing an explicit transcendental class $A_L \in Br_{nr}(X)$ using a linear form $L: M \otimes M \to \mathbb{F}_p$, they show that this class yields a non-vanishing Brauer–Manin pairing:
  $$\sum_v \text{inv}_v(A_L(P_v)) = 1 \in \mathbb{Z}/p$$
  This confirms that the failure of the Hasse principle is explained by a transcendental Brauer–Manin obstruction.

Significance
The paper provides the first example of a homogeneous space of a connected linear algebraic group where the Hasse principle fails due to a transcendental Brauer–Manin obstruction, while the algebraic obstruction is insufficient. This result highlights the necessity of considering transcendental classes in the arithmetic of homogeneous spaces with finite non-abelian stabilizers. It confirms that for such spaces, the "tame" geometry conjectures (which often rely on abelian duality) do not suffice, and the full power of the Brauer–Manin obstruction, including transcendental elements, is required to control rational points. The method exploits the interplay between the nilpotency class of the stabilizer, the rank of local cohomology groups, and global reciprocity laws.