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Vertical unramified Brauer groups of Galois normic bundles

Yufan Liu

TL;DR

The paper addresses computing the vertical unramified Brauer group for normic bundles given by $N_{K/k}(oldsymbol{z})=P(x)$ with a finite Galois extension $K/k$. It extends Wei's partial compactification approach to arbitrary Galois extensions, using group-cohomological tools such as Shapiro's lemma, Mackey's restriction formula, and corestriction to obtain an explicit combinatorial description. The main result provides an isomorphism for $ ext{Br}_{ ext{vert}}(X)/ ext{Br}(k)$ in terms of dual Galois data $igl( ilde G_i'igr)$ and the relations $igl( orall iigr) extstyle igl( extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle extstyle igr)$, namely $ extstyle igl( orall iigr) l_i ext{Cor}_i( ilde ext{G}_i')=0$ in $ ilde G$ with a quotient by the image of $ ext{H}^1(k, ilde T)$. This yields an explicit, combinatorial computation of the vertical Brauer group tied to the irreducible factors of $P(x)$ and the Galois structure of $K/k$, enabling detailed Brauer-Manin analyses for normic bundles in broad settings.

Abstract

We compute the vertical unramified Brauer group of the Galois normic bundles, which are given by $\mathrm{N}_{K/k}(\mathbf{z})=P(x)$. Our main result gives combinatorial formulas for the vertical unramified Brauer groups in terms of the Galois group structure of $K/k$ and the irreducible factors of $P(x)$.

Vertical unramified Brauer groups of Galois normic bundles

TL;DR

The paper addresses computing the vertical unramified Brauer group for normic bundles given by with a finite Galois extension . It extends Wei's partial compactification approach to arbitrary Galois extensions, using group-cohomological tools such as Shapiro's lemma, Mackey's restriction formula, and corestriction to obtain an explicit combinatorial description. The main result provides an isomorphism for in terms of dual Galois data and the relations , namely in with a quotient by the image of . This yields an explicit, combinatorial computation of the vertical Brauer group tied to the irreducible factors of and the Galois structure of , enabling detailed Brauer-Manin analyses for normic bundles in broad settings.

Abstract

We compute the vertical unramified Brauer group of the Galois normic bundles, which are given by . Our main result gives combinatorial formulas for the vertical unramified Brauer groups in terms of the Galois group structure of and the irreducible factors of .
Paper Structure (6 sections, 6 theorems, 29 equations)

This paper contains 6 sections, 6 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Notation and conventions
  4. preliminaries
  5. Review of Wei's results on normic bundles
  6. Computation of vertical Brauer groups

Key Result

Lemma 2.1

Let $G$ be a profinite group, $H$ an open subgroup of $G$, and $M$ a discrete $H$-module. Then for any integer $i\geq 0$, we have Moreover, if $M$ is a discrete $G$-module with trivial $G$-action, then we can write $\textup{Ind}_H^G M$ as $\bigoplus g_i\otimes M$ where $g_i$ are representatives of the left cosets of $H$ in $G$. Define the map Then for every $i\geq 0$, the composition of the foll

Theorems & Definitions (11)

  • Lemma 2.1: Shapiro's lemma
  • Lemma 2.2
  • proof
  • Lemma 2.3: Mackey's restriction formula
  • Remark 2.4
  • Theorem 3.1: wei2022+
  • Theorem 3.2: wei2022+
  • Remark 3.3
  • Theorem 4.1
  • proof
  • ...and 1 more