Remarks on Brauer-Manin obstruction for Weil restrictions

Abstract: Given a finite extension $K/k$ of number fields and a smooth quasi-projective variety $X$ over $K$. If the abelianized fundamental group of $X$ is trivial, we prove that there is a natural identification between Brauer-Manin sets of $X$ and its Weil restriction $R_{K/k}X$. If $X$ is projective and $Pic(X\times_{K}\overline{k})$ is a torsion-free abelian group, we prove that there is a natural identification between algebraic Brauer-Manin sets of $X$ and $R_{K/k}X$.

• The paper establishes that for varieties with trivial abelianized étale fundamental groups, the Brauer-Manin set is preserved under Weil restriction.
• It proves that for projective varieties with torsion-free Picard groups, a natural isomorphism exists between the algebraic Brauer-Manin sets of the original variety and its Weil restriction.
• Advanced cohomological techniques, including the Hochschild-Serre spectral sequence and torsor theory, underpin the identification of these isomorphisms.

Brauer-Manin Obstruction and Weil Restrictions: Natural Identifications

Introduction

This paper investigates the connection between the Brauer-Manin obstruction for rational points and Weil restriction of scalars for algebraic varieties over number fields. Specifically, it addresses the existence (and preservation) of the Brauer-Manin set under the Weil restriction functor, focusing on varieties with trivial abelianized étale fundamental groups or those with torsion-free geometric Picard groups. These questions trace back to inquiries posed by Colliot-Thélène and Poonen, and the paper combines advanced techniques in cohomology, Galois module theory, and the arithmetic of torsors.

Background and Motivation

For a smooth quasi-projective variety $V$ over a number field $k$, the Brauer-Manin set $V(\mathbf{A}_k)^{\Br}$ defines the adelic points orthogonal to elements of the Brauer group under the Brauer-Manin pairing. The failure of the Hasse principle is often explained via a "Brauer-Manin obstruction," and the study of this phenomenon under various field extensions and constructions is central in arithmetic geometry.

Given a finite extension $K/k$ and a variety $X$ over $K$, the Weil restriction $_{K/k}X$ is an essential tool for relating the geometry and arithmetic of $X$ to that of varieties over $k$. An open question asks if the existence and structure of the Brauer-Manin obstruction for $X$ are equivalent to those for $k$0 via the canonical identification of adelic points

$k$1

and crucially, whether $k$2 preserves the Brauer-Manin sets exactly.

Prior work addressed analogous questions for descent obstructions and the étale Brauer-Manin set, but the algebraic Brauer-Manin case remained elusive except in very particular cases.

Main Results

The principal contributions can be summarized as precisely targeted isomorphisms of Brauer-Manin sets under stringent geometric conditions:

1. Varieties with Trivial $k$3: If $k$4 is a smooth quasi-projective variety over $k$5 with trivial abelianized geometric étale fundamental group, then the Brauer-Manin sets are preserved under $k$6:

$k$7

This result establishes equivalence of the existence of Brauer-Manin obstructions between $k$8 and $k$9 in this setting.

2. Projective Varieties with Torsion-Free Picard Group: For a smooth projective $V(\mathbf{A}_k)^{\Br}$0 such that $V(\mathbf{A}_k)^{\Br}$1 is a torsion-free abelian group (i.e., the geometric Picard group is free), there is a natural identification between the algebraic Brauer-Manin set (where only the algebraic part of the Brauer group acts) of $V(\mathbf{A}_k)^{\Br}$2 and that of its Weil restriction.

These theorems are underpinned by advanced spectral sequence arguments (notably from Hochschild-Serre), Galois module manipulations, and explicit consideration of torsors defined by multiplicative-type group schemes.

Technical Approach

The arguments rely on several key structural results:

• The Hochschild-Serre spectral sequence is employed to show that cohomology classes of torsors for finite abelian group schemes are controlled by Galois cohomology under the hypothesis on the étale fundamental group.
• The authors leverage the compatibility between cohomological invariants and Weil restriction, specifically proving canonical isomorphisms at the level of Galois modules for tori and Picard groups.
• Birational invariance of the Brauer group is invoked in certain cases, and density arguments (Harari's formal lemma) are used to deduce statements about the structure and closure properties of the Brauer-Manin set.
• Explicit commutative diagrams clarify the behavior of torsors and their types under restriction, connecting the abstract isomorphisms to concrete cohomological data.

One notable technical point is the identification, via the Kummer sequence and Pontryagin duality, that a smooth projective variety $V(\mathbf{A}_k)^{\Br}$3 over a number field $V(\mathbf{A}_k)^{\Br}$4 has trivial $V(\mathbf{A}_k)^{\Br}$5 if and only if $V(\mathbf{A}_k)^{\Br}$6 is torsion-free, thereby linking two prominent but a priori distinct invariants.

Implications and Applications

The results have several ramifications:

• Criterion for Equivalence of Obstructions: Under the stated hypotheses, the presence or absence of Brauer-Manin obstructions to the Hasse principle (or weak approximation) is fully governed by the behavior under Weil restriction. Thus, one can efficiently transfer problems about rational points or weak approximation from a base field to an extension or vice versa.
• Structural Insight: The techniques illustrate that for a large class of varieties (notably, geometrically simply-connected varieties and varieties with free Picard group), the arithmetic complexity introduced by Weil restriction does not create or eliminate obstructions detectable by the Brauer group.
• Limitations: The discussion emphasizes that the result is not expected to be universal; in particular, when the Picard group has torsion, the direct identification can fail or become more subtle. Future research might determine what additional structure (e.g., relating to the total torsion in Picard or the unramified Brauer group) would suffice for analogous theorems, or classify behaviors in the presence of torsion.

Speculations for Further Investigation

• Higher Descent and Nontrivial Torsion: The propagation of other descent-type obstructions (beyond the finite abelian and étale) under Weil restriction remains an open theme for exploration.
• Behavior for Open Varieties: The authors note their results persist for open subsets with small complements, but the full boundary case might admit refined obstructions.
• Automorphisms and Invariant Theory: The identification of isomorphisms between various cohomological sets via naturality of the Weil restriction could be exploited in the study of arithmetic automorphic forms and equivariant descent.
• Generalization to Nonabelian or Noncommutative Obstructions: Investigating if similar identifications exist for more nuanced nonabelian cohomological obstructions associated to the arithmetic fundamental group may deepen the understanding of failures of the local-global principle.

Conclusion

This paper establishes precise isomorphisms of Brauer-Manin sets under Weil restriction for certain classes of varieties, providing a rigorous answer to a question of Colliot-Thélène and Poonen in these settings. By combining étale cohomology, torsor theory, and Galois module analysis, it clarifies the functoriality of obstructions to rational points under a central algebro-arithmetic construction, with implications for the study of rational points and arithmetic duality on higher-dimensional varieties.

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Reference: "Remarks on Brauer-Manin obstruction for Weil restrictions" (2604.10498)