Hasse norm principle for extensions of prime squared degree

Technical Summary: Hasse Norm Principle for Extensions of Prime Squared Degree

Problem Statement
The paper addresses the validity of the Hasse norm principle (HNP) for finite separable field extensions $K/k$ of global fields. Specifically, it seeks an equivalent condition for the HNP to hold when the degree of the extension is $d = p^2$, where $p$ is an odd prime. The obstruction to the HNP is measured by the group $X(K/k) := (k^\times \cap N_{K/k}(A_K^\times)) / N_{K/k}(K^\times)$. The HNP holds if and only if $X(K/k) = 1$. While the problem is fully resolved for prime degrees, degree 4, degree 6, and degrees $8 \le d \le 15$, the case of odd prime squared degrees ($d=p^2$) remained incomplete, with only partial results available (e.g., for "adequate" extensions).

Methodology
The author employs a cohomological approach linking the arithmetic of field extensions to the group theory of their Galois groups. The core methodology involves:

1. Cohomological Translation: Utilizing the isomorphism between the obstruction group $X(K/k)$ and the Tate–Shafarevich group of the norm one torus $T_{K/k}$. This is further related to the second cohomology group of the Galois group $G = \text{Gal}(\tilde{K}/k)$ with coefficients in a specific $G$-lattice, the Chevalley module $J_{G/H}$, where $H = \text{Gal}(\tilde{K}/K)$. Specifically, the study focuses on the group $X^2_\omega(G, J_{G/H})$, defined as the kernel of the restriction map from $H^2(G, J_{G/H})$ to the direct sum of cohomology groups over all cyclic subgroups of $G$.
2. Group-Theoretic Classification: The paper analyzes transitive groups of degree $p^2$. It relies on the classification of such groups by Dobson and Witte, distinguishing cases based on the structure of the $p$-Sylow subgroup $P$ of $G$.
3. Case Analysis:
   • $p$-groups: The author computes $X^2_\omega(G, J_{G/H})$ for $p$-groups of degree $p^2$. This involves detailed analysis of the exponent of the group (whether $p$ or $p^2$), utilizing exact sequences of $G$-lattices, Mackey decomposition, and properties of the Bogomolov multiplier. A key technique involves lifting the problem to a wreath product to handle cases where restriction maps on third cohomology are not injective.
   • General Case: For non-$p$-groups, the paper utilizes the structure of transitive groups to reduce the problem to either a product of transitive groups of degree $p$ or a semi-direct product of the form $(C_p)^2 \rtimes G^\dagger$, where $G^\dagger$ is a subgroup of $\text{GL}_2(\mathbb{F}_p)$.
4. Generalized Representation Groups: To compute the cohomology for the specific semi-direct product structure, the author constructs generalized representation groups (central extensions) and applies results by Drakokhrust and Macedo–Newton regarding focal subgroups and the structure of outer automorphism groups of extraspecial $p$-groups.

Key Contributions and Results
The paper provides a complete characterization of the Hasse norm principle for extensions of degree $p^2$ ($p$ odd).

• Theorem 1.3 (Main Result): Let $K/k$ be a finite separable extension of degree $p^2$ with Galois closure $\tilde{K}$, Galois group $G$, and stabilizer $H$.
  • If $X(K/k) \neq 1$, then $G$ must be isomorphic to $(C_p)^2 \rtimes_\phi G^\dagger$ and $H \cong \{1\} \rtimes_\phi G^\dagger$, where $G^\dagger$ is a subgroup of $\text{SL}_2(\mathbb{F}_p)$ and $\phi$ is induced by the standard representation.
  • If this structural condition holds, then $X(K/k) \cong \mathbb{Z}/p$ if no decomposition group of $\tilde{K}/k$ contains a subgroup isomorphic to $(C_p)^2$; otherwise, $X(K/k) = 1$.
• Recovery of Previous Results: The theorem recovers the result of Drakokhrust and Platonov, which states that the HNP holds for $k$-adequate extensions of degree $p^2$. The paper provides a new proof of this fact without relying on the classification of subgroups of $\text{SL}_2(\mathbb{F}_p)$, instead deriving the necessary conditions directly from cohomological computations.
• Norm One Tori and Weak Approximation: The paper establishes an equivalent condition for the non-vanishing of the unramified Brauer group $H^1(k, \text{Pic}(X))$ of a smooth compactification of the norm one torus $T_{K/k}$. Furthermore, it determines the structure of the defect of weak approximation $A_k(T_{K/k})$, showing it is isomorphic to $\mathbb{Z}/p$ under the same structural conditions on $G$ and the absence of $(C_p)^2$ in decomposition groups.
• Cohomological Computations: The technical core (Sections 4 and 5) provides explicit computations of $X^2_\omega(G, J_{G/H})$ for transitive groups of degree $p^2$. It proves that this group vanishes unless $G$ has the specific semi-direct product structure involving a subgroup of $\text{SL}_2(\mathbb{F}_p)$.

Significance
The paper claims to resolve Question 1.1 (determining equivalent conditions for the HNP) completely for all prime squared degrees $d=p^2$. By combining the results for odd primes with the previously known case $d=4$, it offers a complete solution for arbitrary prime squared degrees. The work is significant for:

1. Completing the Classification: It fills a gap in the literature regarding the HNP for degrees $p^2$ where $p$ is an odd prime.
2. Methodological Rigor: It provides an alternative, computer-free proof for the $p=3$ case (degree 9), which was previously established using computational methods.
3. Unification: It unifies the study of the HNP, the unramified Brauer group of norm one tori, and the defect of weak approximation under a single cohomological framework dependent on the Galois group structure.

The author notes that the proof does not rely on the exhaustive classification of subgroups of $\text{SL}_2(\mathbb{F}_p)$, thereby offering a more structural and self-contained derivation of the necessary and sufficient conditions.