# Rational equivalence of cusps

## TL;DR

This paper proves that cusps of the same dimension in certain modular varieties are rationally linearly dependent, extending the Manin-Drinfeld theorem to higher dimensions and generalizing modular units as higher Chow cycles.

## Contribution

It establishes rational equivalence of cusps in higher dimensions and introduces higher Chow cycles as a generalization of modular units.

## Key findings

- Cusps of the same dimension are linearly dependent in the rational Chow group.
- Higher dimensional analogue of the Manin-Drinfeld theorem is proven.
- Higher Chow cycles generalize modular units in this context.

## Abstract

We prove that two cusps of the same dimension in the Baily-Borel compactification of some classical series of modular varieties are linearly dependent in the rational Chow group of the compactification. This gives a higher dimensional analogue of the Manin-Drinfeld theorem. As a consequence, we obtain a higher dimensional generalization of modular units as higher Chow cycles on the modular variety.