Divided powers in Chow rings and integral Fourier transforms

TL;DR

This paper establishes a PD-structure on Chow rings of monoid schemes and constructs an approximate integral Fourier transform for Jacobians of hyperelliptic curves, revealing torsion phenomena and limitations in integral Fourier transforms.

Contribution

It introduces a natural PD-structure on Chow rings of monoid schemes and constructs an approximate integral Fourier transform for Jacobians of hyperelliptic curves, highlighting torsion constraints.

Findings

PD-structure on CH_{>0}(M) for monoid schemes

Construction of an integral Fourier transform up to 2^N-torsion for hyperelliptic Jacobians

Demonstration of the unavoidable factor 2 in properties of integral Fourier transforms

Abstract

We prove that for any monoid scheme M over a field with proper multiplication maps from M x M to M, we have a natural PD-structure on the ideal CH_{>0}(M) of CH(M) with regard to the Pontryagin ring structure. Further we investigate to what extent it is possible to define a Fourier transform on the motive with integral coefficients of the Jacobian of a curve. For a hyperelliptic curve of genus g with sufficiently many k-rational Weierstrass points, we construct such an integral Fourier transform with all the usual properties up to 2^N-torsion, where N is the integral part of 1 + log_2(3g). As a consequence we obtain, over an algebraically closed field, a PD-structure (for the intersection product) on 2^N A, where A is the augmentation ideal of CH(J). We show that a factor 2 in the properties of an integral Fourier transform cannot be eliminated even for elliptic curves over an algebraically closed field.