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Brauer-Manin Obstruction on Generalized Kummer Varieties

Eric Zhu

TL;DR

The paper studies the Brauer-Manin obstruction for generalized Kummer varieties $X$ arising from an abelian variety $A$ over a number field $k$ with a prime-order automorphism $\\zeta$ of order $p>2$. It proves that the obstruction is governed entirely by the $p$-primary part of the Brauer group, by constructing $X$ as a smooth resolution of a quotient $Y'/\\langle\\zeta\\rangle$ with $Y$ a torsor of $A$ coming from $H^1(k,T)\to H^1(k,A)$ with $T=A[1-\\zeta]$, and then showing that $\\mathrm{Br}(X)/\\mathrm{Br}_0(X)$ is finite and the obstruction can be reduced to $\\mathrm{Br}(X)[p^\\perp]$. The analysis uses toric resolutions, Picard group computations, and Hochschild-Serre spectral sequences to relate Brauer groups across the various models, including twists $Y_\\xi$, establishing a canonical $p$-primary decomposition that captures all obstructions. An application to a cubic surface constructed from a genus one torsor demonstrates the absence of a Brauer-Manin obstruction in that family, and, conditional on Skorobogatov’s conjecture for K3 surfaces, suggests a local-global principle for the whole family. The results facilitate practical obstruction checks by reducing to a finite and computable part of the Brauer group.

Abstract

Given an abelian variety $A$ over a number field, we consider the generalized Kummer varieties of $A$ coming from quotients of $A$ by an automorphism of prime order $p > 2$. We prove that the Brauer-Manin obstruction on these generalized Kummer varieties only can come from the $p$-primary part of the Brauer group. This is applied to show that certain families of such varieties have no Brauer-Manin obstruction to the local-global principle.

Brauer-Manin Obstruction on Generalized Kummer Varieties

TL;DR

The paper studies the Brauer-Manin obstruction for generalized Kummer varieties arising from an abelian variety over a number field with a prime-order automorphism of order . It proves that the obstruction is governed entirely by the -primary part of the Brauer group, by constructing as a smooth resolution of a quotient with a torsor of coming from with , and then showing that is finite and the obstruction can be reduced to . The analysis uses toric resolutions, Picard group computations, and Hochschild-Serre spectral sequences to relate Brauer groups across the various models, including twists , establishing a canonical -primary decomposition that captures all obstructions. An application to a cubic surface constructed from a genus one torsor demonstrates the absence of a Brauer-Manin obstruction in that family, and, conditional on Skorobogatov’s conjecture for K3 surfaces, suggests a local-global principle for the whole family. The results facilitate practical obstruction checks by reducing to a finite and computable part of the Brauer group.

Abstract

Given an abelian variety over a number field, we consider the generalized Kummer varieties of coming from quotients of by an automorphism of prime order . We prove that the Brauer-Manin obstruction on these generalized Kummer varieties only can come from the -primary part of the Brauer group. This is applied to show that certain families of such varieties have no Brauer-Manin obstruction to the local-global principle.
Paper Structure (5 sections, 10 theorems, 24 equations, 1 figure)

This paper contains 5 sections, 10 theorems, 24 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Construction of Generalized Kummer Varieties
  3. Proving that $\mathrm{Br}(X)/\mathrm{Br}(X_0)$ is finite
  4. Brauer-Manin Obstruction
  5. Application

Key Result

Theorem 1.1

Let $A$ be an abelian variety of dimension $g \geq 2$ over a number field $k$ with an automorphism $\zeta$ of prime order $p$ defined over $k$. Further assume that $\zeta$ only has finitely many fixed points in $A(\overline{k})$. Let $X$ be a generalized Kummer variety associated to $A$ and $\zeta$,

Figures (1)

  • Figure :

Theorems & Definitions (22)

  • Theorem 1.1
  • Example 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Corollary 3.3
  • ...and 12 more