Zero-cycles over zero-dimensional cusps

TL;DR

This paper generalizes the Manin-Drinfeld theorem to higher-dimensional modular varieties by proving rational equivalence of boundary points over 0-dimensional cusps in most classical cases, with specific results for Picard modular varieties.

Contribution

It extends the understanding of algebraic cycles on modular varieties by proving rational equivalence of boundary points in higher dimensions, strengthening previous theorems.

Findings

Rational equivalence of boundary points over 0-dimensional cusps in classical modular varieties.

The difference of special boundary points in Picard modular varieties is torsion.

Generalization and strengthening of the Manin-Drinfeld theorem in higher dimensions.

Abstract

We prove that all points of a toroidal compactification lying over 0-dimensional cusps are rationally equivalent in the integral Chow group for most classical modular varieties (Siegel, Hilbert, orthogonal, Hermitian, quaternionic). This gives a generalization, and even strengthening, of the Manin-Drinfeld theorem in higher dimension from the viewpoint of algebraic cycles. The same result no longer holds for Picard modular varieties, but for them we prove that the difference of any two "special" boundary points, which are dense in the boundary, is torsion.