# On a question of Colliot-Th\'{e}l\`{e}ne on Chow group mod $n$

## TL;DR

This paper introduces the Tate-Shafarevich and Selmer groups for Chow groups of abelian varieties over number fields, proving the finiteness of zero cycles modulo n, thus answering Colliot-Thélène's question.

## Contribution

It defines new arithmetic groups for Chow groups and proves the finiteness of zero cycles modulo n over any number field, advancing understanding in algebraic geometry.

## Key findings

- Finiteness of Chow group of zero cycles mod n over any number field
- Introduction of Tate-Shafarevich and Selmer groups for Chow groups
- Positive answer to Colliot-Thélène's question

## Abstract

In this note we define the notion of Tate-Shafarevich group and Selmer group of the Chow group of an abelian variety defined over a number field. In this context we give positive answer to the question of Colliot-Th\'{e}l\`{e}ne that the Chow group of zero cycles modulo $n$ is finite over any number field.