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Homogeneous spaces over an abelian variety

Margot Bruneaux

TL;DR

The paper addresses the Colliot-Thélène–Iyer question on the existence of rational sections in families of homogeneous spaces over abelian varieties after finite étale base change. It develops a Jacobian-motive filtration via Moonen–Polishchuk, and leverages Beilinson–Lichtenbaum and Pardon spectral sequences to control unramified cohomology and Witt groups, enabling a positive answer in characteristic zero for reductive groups whose root data lack $E_8$ factors (with dim $A$ constraints or $k$ algebraically closed). It further reformulates and extends the problem to torsors and Borel varieties through Demazure’s framework and Rost invariants, obtaining generically trivial or locally trivial base-changed torsors in broad cases. The results connect to broader themes on nef tangent bundles and the Campana–Peternell conjecture, providing a robust framework for understanding rational sections in families of homogeneous spaces over abelian bases and beyond.

Abstract

In this paper, we study a question of Colliot-Thélène and Iyer concerning the existence of rational sections in families of homogeneous spaces over an abelian variety, after base change by a suitable étale isogeny of the abelian variety. Assuming characteristic zero and that the homogeneous spaces arise from connected reductive groups, the problem is reformulated in terms of torsors under reductive groups over an abelian variety $A$. Building on work of Moonen and Polishchuk, we construct a filtration on the motive of a Jacobian variety to analyze the action of isogenies on unramified cohomology and Witt groups. This approach allows for a positive response to the question for reductive groups whose root data do not contain a factor of type~$E_8$ when $\dim A > 2$ and $\mathrm{cd}(k) \leqslant 1$, and for all reductive groups when $\dim A = 2$ and $k$ is algebraically closed.

Homogeneous spaces over an abelian variety

TL;DR

The paper addresses the Colliot-Thélène–Iyer question on the existence of rational sections in families of homogeneous spaces over abelian varieties after finite étale base change. It develops a Jacobian-motive filtration via Moonen–Polishchuk, and leverages Beilinson–Lichtenbaum and Pardon spectral sequences to control unramified cohomology and Witt groups, enabling a positive answer in characteristic zero for reductive groups whose root data lack factors (with dim constraints or algebraically closed). It further reformulates and extends the problem to torsors and Borel varieties through Demazure’s framework and Rost invariants, obtaining generically trivial or locally trivial base-changed torsors in broad cases. The results connect to broader themes on nef tangent bundles and the Campana–Peternell conjecture, providing a robust framework for understanding rational sections in families of homogeneous spaces over abelian bases and beyond.

Abstract

In this paper, we study a question of Colliot-Thélène and Iyer concerning the existence of rational sections in families of homogeneous spaces over an abelian variety, after base change by a suitable étale isogeny of the abelian variety. Assuming characteristic zero and that the homogeneous spaces arise from connected reductive groups, the problem is reformulated in terms of torsors under reductive groups over an abelian variety . Building on work of Moonen and Polishchuk, we construct a filtration on the motive of a Jacobian variety to analyze the action of isogenies on unramified cohomology and Witt groups. This approach allows for a positive response to the question for reductive groups whose root data do not contain a factor of type~ when and , and for all reductive groups when and is algebraically closed.
Paper Structure (11 sections, 56 theorems, 181 equations)

This paper contains 11 sections, 56 theorems, 181 equations.

Table of Contents

  1. A filtration on the dual motive of a Jacobian variety
  2. The inclusion of in and the induced filtration
  3. Motivic interpretation
  4. Unramified cohomology and unramified Witt group of an abelian variety
  5. Étale cohomology with finite coefficients
  6. Unramified cohomology
  7. Unramified Witt group
  8. A question of Colliot-Thélène and Iyer
  9. The question for torsors
  10. The question for Borel varieties
  11. Some generalisation beyond the case of an abelian variety

Key Result

Theorem 2

[Theorem Theorem_homogeneous] Let $A$ be an abelian variety over a field $k$ of cohomological dimension at most $1$, of characteristic $0$, and containing a square root of $-1$. Let $U \subset A$ be an open subset that contains all points of codimension $1$, with $0 \in U$. Let $X \longrightarrow U$ there exists an étale isogeny $f$ of $A$ such that the pullback admits a rational section.

Theorems & Definitions (109)

  • Theorem 2
  • Proposition 1.1.1
  • proof
  • Theorem 1.1.2
  • Remark 1.1.3
  • Definition 1.2.1
  • Remark 1.2.2
  • Proposition 1.2.3: Manin Principle
  • Corollary 1.2.4
  • Lemma 1.2.5
  • ...and 99 more