Zero-cycles on quasi-projective surfaces over $p$-adic fields

TL;DR

This paper investigates a conjecture about zero-cycles on smooth projective surfaces over p-adic fields, proving it for certain classes of surfaces using rational maps and homological methods.

Contribution

It extends the validity of Colliot-Thélène's conjecture to surfaces dominated by products of curves, employing rational maps and Suslin homology techniques.

Findings

Proves the conjecture for surfaces dominated by products of curves.

Shows the conjecture's stability under generically finite rational maps.

Utilizes Suslin's singular homology to study zero-cycles on open subvarieties.

Abstract

A conjecture of Colliot-Th\'{e}l\`{e}ne predicts that for a smooth projective variety $X$ over a finite extension $k$ of $\mathbb{Q}_p$ the kernel of the Albanese map $\text{CH}_0(X)^{\text{deg}=0}\to Alb_X(k)$ is the direct sum of a divisible group and a finite group. In this article we show that if $\pi:X\dashrightarrow Y$ is a generically finite rational map between smooth projective surfaces and the conjecture is true for $X\otimes_k L$ for every finite extension $L/k$, then it is true for $Y$. Using work of Raskind and Spiess, this proves the conjecture for surfaces that are geometrically dominated by products of curves, under some assumptions on the reduction type of the Jacobians. The method involves studying similar questions for an open subvariety $U$ of a projective surface $X$ by replacing the Chow group of $0$-cycles with Suslin's singular homology $H_0^{\text{sus}}(U)$.