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Product of Brauer--Manin obstruction for 0-cycles over number fields and function fields

Diego Izquierdo, Yongqi Liang, Hui Zhang

TL;DR

The paper proves that for smooth varieties X and Y over a number field or function field, the Brauer--Manin obstruction to the existence of 0-cycles of degree 1 on the product Z = X\times_k Y exists if and only if the obstruction exists on both X and Y simultaneously. It develops a product map for adelic 0-cycles, constructs universal n-torsors, and obtains a decomposition of the Brauer group of Z that links the Brauer--Manin pairing on Z with the Poitou--Tate pairing on the factors. By relating the local and global obstructions through these cohomological tools, the work reduces the problem on Z to obstructions on X and Y and extends the framework to function fields of C((t)) curves. This provides a cohomological mechanism to test and understand obstructions for 0-cycles on products via their factors, with potential implications for rational points and arithmetic geometry.

Abstract

It is conjectured that the Brauer--Manin obstruction is expected to control the existence of 0-cycles of degree 1 on smooth proper varieties over number fields. In this paper, we prove that the existence of Brauer--Manin obstruction to Hasse principle for 0-cycles of degree 1 on the product of smooth (non-necessarily proper) varieties is equivalent to the simultaneous existence of such an obstruction on each factor. We also prove an analogous statement for smooth varieties defined over function fields of $\mathbb{C}((t))$-curves.

Product of Brauer--Manin obstruction for 0-cycles over number fields and function fields

TL;DR

The paper proves that for smooth varieties X and Y over a number field or function field, the Brauer--Manin obstruction to the existence of 0-cycles of degree 1 on the product Z = X\times_k Y exists if and only if the obstruction exists on both X and Y simultaneously. It develops a product map for adelic 0-cycles, constructs universal n-torsors, and obtains a decomposition of the Brauer group of Z that links the Brauer--Manin pairing on Z with the Poitou--Tate pairing on the factors. By relating the local and global obstructions through these cohomological tools, the work reduces the problem on Z to obstructions on X and Y and extends the framework to function fields of C((t)) curves. This provides a cohomological mechanism to test and understand obstructions for 0-cycles on products via their factors, with potential implications for rational points and arithmetic geometry.

Abstract

It is conjectured that the Brauer--Manin obstruction is expected to control the existence of 0-cycles of degree 1 on smooth proper varieties over number fields. In this paper, we prove that the existence of Brauer--Manin obstruction to Hasse principle for 0-cycles of degree 1 on the product of smooth (non-necessarily proper) varieties is equivalent to the simultaneous existence of such an obstruction on each factor. We also prove an analogous statement for smooth varieties defined over function fields of -curves.
Paper Structure (4 sections, 9 theorems, 33 equations)

This paper contains 4 sections, 9 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Existence of universal $n$-torsors
  4. A Decomposition of $\mathrm{H}^{2}(Z,\mu_n)$

Key Result

Theorem 1.1

Let $k$ be either a number field or a function field, let $X$ and $Y$ be smooth and geometrically integral varieties over $k$ and $Z=X\times_kY$ be the product of $X$ and $Y$, Denote $p$ and $q$ the projections Then the section map \xymatrix{ \mathrm{Z}_{0,\mathbf{A}}^{1}(X)\times\mathrm{Z}_{0,\mathbf{A}}^{1}(Y)\ar[r]^{\qquad\Phi}&\mathrm{Z}_{0,\mathbf{A}}^{1}(Z)\ar@{->>}[r] &\mathrm{Z}_{0,\m

Theorems & Definitions (19)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 3.2
  • proof
  • Definition 3.3
  • Definition 3.4
  • Lemma 3.5
  • ...and 9 more