# Selmer group associated to the Chow group of certain codimension two cycles

## TL;DR

This paper investigates the Galois-invariant algebraically trivial cycles on certain surfaces over number fields, demonstrating that a specific quotient related to these cycles is a torsion group, thus contributing to the understanding of their arithmetic properties.

## Contribution

It proves that the Galois-invariant cycles' quotient group is torsion for surfaces with geometric genus and irregularity zero over number fields.

## Key findings

- The quotient of Galois-invariant cycles is torsion.
- The result applies to surfaces with geometric genus and irregularity zero.
- Provides insight into the arithmetic of algebraically trivial cycles.

## Abstract

Let $X$ be a surface with geometric genus and irregularity zero which is defined over a number field $K$. Let $\mathscr{X}$ denote a smooth spread of $X$ over the spectrum of a Zariski open subset in the spectrum of the ring of integers and $A^2$ stands for the group of algebraically trivial cycles on schemes modulo rational equivalence. If $j^*: A^2(\mathscr{X})\to A^2(X)$ be the flat pull-back corresponding to the embedding $j:X\hookrightarrow \mathscr{X}$ then we prove that $\im(j^*)(K)/A^2(\mathscr{X})(K)$ is a torsion group. Here $\im(j^*)(K)$, $A^2(\mathscr{X})(K)$ stand for the cycles fixed under the action of the absolute Galois group.