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Divided powers on abelian varieties

Bruno Kahn

TL;DR

The work develops a robust integral framework for divided powers and Fourier transforms in étale Chow groups of abelian varieties, proving the existence of integral DM (Deninger–Murre) Chow–Künneth projectors away from the characteristic up to 2-torsion. It shows that étale Chow groups admit canonical divided powers on the augmentation ideal, and extends these constructions to abelian schemes via deformation arguments, elliptic-case reductions, and careful control of torsion phenomena. A central achievement is lifting the Fourier transform to an integral étale correspondence and deriving integral analogues of Beauville’s identities, Scholl’s formula for DM projectors, and Suh’s formulas for Jacobians, all modulo 2-torsion. The paper also clarifies the behavior of étale motives modulo 2-torsion, establishes functoriality for Picard/Albanese-type constructions, and provides a deformation-based path to removing 2-torsion obstructions in the global setting. Overall, the results reinforce that étale motivic and integral-étale frameworks are well-behaved for abelian varieties, enabling deep links between Chow groups, motives, and Fourier dualities with explicit algebraic and cohomological control.

Abstract

We prove the existence of divided powers in étale Chow groups of abelian varieties over a separably closed field, and hence of an integral lift of the Fourier transform, away from the characteristic and up to $2$-torsion. The method is to lift the Deninger-Murre Chow-Künneth projectors to integral ones, and draw consequences. Several techniques used here are new.

Divided powers on abelian varieties

TL;DR

The work develops a robust integral framework for divided powers and Fourier transforms in étale Chow groups of abelian varieties, proving the existence of integral DM (Deninger–Murre) Chow–Künneth projectors away from the characteristic up to 2-torsion. It shows that étale Chow groups admit canonical divided powers on the augmentation ideal, and extends these constructions to abelian schemes via deformation arguments, elliptic-case reductions, and careful control of torsion phenomena. A central achievement is lifting the Fourier transform to an integral étale correspondence and deriving integral analogues of Beauville’s identities, Scholl’s formula for DM projectors, and Suh’s formulas for Jacobians, all modulo 2-torsion. The paper also clarifies the behavior of étale motives modulo 2-torsion, establishes functoriality for Picard/Albanese-type constructions, and provides a deformation-based path to removing 2-torsion obstructions in the global setting. Overall, the results reinforce that étale motivic and integral-étale frameworks are well-behaved for abelian varieties, enabling deep links between Chow groups, motives, and Fourier dualities with explicit algebraic and cohomological control.

Abstract

We prove the existence of divided powers in étale Chow groups of abelian varieties over a separably closed field, and hence of an integral lift of the Fourier transform, away from the characteristic and up to -torsion. The method is to lift the Deninger-Murre Chow-Künneth projectors to integral ones, and draw consequences. Several techniques used here are new.
Paper Structure (41 sections, 62 theorems, 161 equations)

This paper contains 41 sections, 62 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. Étale motivic cohomology of abelian varieties
  3. Proof of Proposition \ref{['t1']}
  4. Esnault's example, refined
  5. The torsion bimodule
  6. $1$-cocycles
  7. Hochschild cohomology
  8. Some numerology
  9. Cocycles and coboundaries
  10. A weak form of Theorem \ref{['t3']}
  11. A weak version of Theorem \ref{['t3']} a) (I)
  12. Another lifting method
  13. A weak version of Theorem \ref{['t3']} b)
  14. A weak version of Theorem \ref{['t3']} a) (II); self-conjugate projectors
  15. Proof of Corollary \ref{['c4']}
  16. ...and 26 more sections

Key Result

Proposition 1.1

Let $x\in CH^i_{\mathrm{\acute{e}t}}(A)$. Then $x^r$ is divisible by $(r!)_p$ in $CH^{ir}_{\mathrm{\acute{e}t}}(A)$ for any $r\ge 1$, where $p$ is the exponential characteristic of $k$ and $(n)_p$ denotes the largest divisor of an integer $n$ which is prime to $p$.

Theorems & Definitions (133)

  • Proposition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Remark 1.5
  • Definition 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 123 more