Cohomological Obstructions for Varieties over $p$-adic Function Fields

Technical Summary: Cohomological Obstructions for Varieties over p-adic Function Fields

Problem Statement
The paper addresses the arithmetic of smooth integral varieties defined over $p$-adic function fields (function fields of smooth projective curves over $p$-adic fields). Specifically, it investigates the obstructions to the local-global principle and weak approximation. While the Brauer–Manin obstruction is well-understood over number fields, the paper seeks to establish a systematic obstruction theory over $p$-adic function fields. A central question is identifying the "finest" obstruction arising from arithmetic dualities and determining whether the unramified cohomological obstruction provides a stronger constraint than the classical Manin obstruction (defined via $H^3_{\text{ét}}$) or descent obstructions.

Methodology
The author employs a combination of étale cohomology, motivic complexes, and arithmetic duality theories.

1. Cohomological Framework: The study utilizes the étale sheaf $\mathbb{Q}/\mathbb{Z}(2)$ and motivic complexes $\mathbb{Z}(2)$. The paper defines obstructions via pairings between adelic points $X(\mathbb{A})$ and cohomology groups, specifically $H^3_{\text{ét}}(X, \mathbb{Q}/\mathbb{Z}(2))$ (Manin obstruction) and the unramified cohomology group $H^3_{\text{nr}}(X, \mathbb{Q}/\mathbb{Z}(2))$.
2. Functorial Approach: Obstructions are constructed using contravariant functors from the category of $K$-schemes to sets, comparing the image of adelic points under evaluation maps against local-global compatibility conditions.
3. Spectral Sequences: The Čech spectral sequence and the Hochschild–Serre spectral sequence are utilized to compute cohomology groups for specific geometric constructions, particularly for quotients of affine spaces by group actions.
4. Cohomological Invariants: The paper leverages the theory of cohomological invariants (specifically the Rost invariant) to identify unramified cohomology groups with groups of invariants for semi-simple simply connected groups.
5. Arithmetic Dualities: The analysis relies on Poitou–Tate sequences and local invariants (isomorphisms between $H^3(K_v, \Lambda)$ and $\mathbb{Q}/\mathbb{Z}$) to establish exact sequences and duality pairings.

Key Contributions and Results

• Comparison of Obstructions (Theorem 1.1):
  The paper establishes a hierarchy among cohomological obstructions. It proves that for a smooth integral $K$-variety $X$ and a commutative linear algebraic $K$-group $M$:

  • The unramified obstruction is finer than the Manin obstruction: $X(\mathbb{A})^{H^3_{\text{nr}}} \subseteq X(\mathbb{A})^{H^3_{\text{ét}}}$.
  • The unramified obstruction is finer than obstructions arising from any element $\alpha \in H^i_{\text{ét}}(X, M)$ for $i \ge 1$.
  • For finite $M$, the unramified obstruction is finer than obstructions from unramified cohomology classes.
  • The paper demonstrates that the unramified obstruction coincides with the obstruction defined by $H^4_{\text{ét}}(X, \mathbb{Z}(2))$ via motivic complexes.

• Existence of a Strictly Stronger Unramified Obstruction (Theorem 1.2):
  The author constructs an explicit example of a variety $X = Y/G$ (where $G$ is a semi-simple simply connected group acting freely on an open subset of affine space) such that:

  • The Manin obstruction is trivial ($X(\mathbb{A})^{H^3_{\text{ét}}} = X(\mathbb{A})$).
  • The unramified obstruction is non-trivial ($X(\mathbb{A})^{H^3_{\text{nr}}} \subsetneq X(\mathbb{A})$).
    This result relies on the properties of the Rost invariant $R$ and the existence of places $v$ where $H^1(K_v, G) \neq 1$ but the Rost kernel is trivial.

• Relation to Descent Obstructions (Theorem 1.3):
  The paper compares the unramified obstruction with the descent obstruction. It proves that for a linear $K$-group $G$ embedded in $SL_n$, the following are equivalent:

  • The unramified obstruction on the quotient $X = SL_n/G$ coincides with the set of adelic points generated by $SL_n(\mathbb{A})$ and rational points $X(K)$.
  • For any smooth integral variety $Z$ and any $G$-torsor $f: Y \to Z$, the unramified obstruction is contained in the descent obstruction ($Z(\mathbb{A})^{H^3_{\text{nr}}} \subseteq Z(\mathbb{A})_f$).
    The proof utilizes contracted products and the isomorphism of unramified cohomology under torsor twists.

Significance and Claims
The paper claims to provide a systematic understanding of obstruction theory over $p$-adic function fields, analogous to developments over number fields.

• Refinement of Obstructions: It identifies the unramified obstruction as the finest obstruction among those arising from arithmetic dualities and Brauer groups in this context.
• Counterexample: It resolves the question of whether the unramified obstruction is strictly stronger than the Manin obstruction by providing a concrete counterexample where the Manin obstruction fails to detect a failure of the local-global principle, while the unramified obstruction succeeds.
• Motivic Connection: It clarifies the relationship between different cohomological degrees ($H^3_{\text{nr}}$ vs. $H^4_{\text{ét}}$) via motivic complexes, showing they define the same obstruction set.
• Limitations: The paper modestly notes that a full comparison with the general descent obstruction is currently limited by the lack of a suitable generalization of the Kottwitz exact sequence for $p$-adic function fields.

The work builds upon and extends previous studies by Harari, Scheiderer, Szamuely, Tian, and others regarding local-global principles for reductive groups, and connects to Colliot-Thélène's conjectures on weak approximation.