Divided powers on abelian varieties

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.

💡 Research Summary

The paper investigates the existence of divided power operations in the étale Chow groups of abelian varieties over a separably closed field, and uses this to construct an integral lift of Beauville’s Fourier transform away from the characteristic p and up to 2‑torsion. The author’s strategy consists of two main steps: first, establishing a weak divided power structure in CH⁎_ét(A) by showing that for any class x∈CHⁱ_ét(A) the r‑th power xʳ is divisible by the largest divisor of r! that is prime to p (Proposition 1.1). This follows almost immediately from known results on the divisibility of powers in continuous ℓ‑adic cohomology and an integrality lemma (Corollary 2.4). However, these weak operations are not uniquely determined because of torsion, and a stronger, canonical system of operations γₙ is needed.

The author introduces a canonical divided power structure on the ideal of positive‑degree elements of CH⁎ét(A) after killing the 2‑torsion subgroup (Theorem 1.2). The operations γₙ satisfy the usual axioms (γ₀=1, γ₁=id, λ‑scaling, binomial formula for sums, compatibility with products, and the iterated formula γₙ(γₘ(x)) = (mn)!/(m!ⁿ·n!)·γ{mn}(x)). The presence of 2‑torsion prevents a full integral lift; nevertheless, after quotienting by the 2‑torsion ideal, the divided powers become canonical and functorial with respect to homomorphisms of abelian varieties.

The second, deeper part of the work concerns the Deninger–Murre Chow–Künneth projectors π_i. These projectors are known rationally (in CH⁎(A)⊗ℚ) and give a decomposition of the motive of A. The main result (Theorem 1.7) shows that the rational projectors lift to a system of étale integral projectors π_i∈CH^g_ét(A×A)