Non-injectivity of the cycle class map in continuous $\ell$-adic cohomology

TL;DR

This paper demonstrates that the integral cycle class map in continuous ll-adic cohomology is not injective in general, providing examples that impact the understanding of coniveau filtration and Abel-Jacobi maps.

Contribution

It presents explicit examples showing the non-injectivity of the integral cycle class map, addressing a question raised by Jannsen and Schreieder.

Findings

Examples of non-injectivity of the integral cycle class map

Implications for coniveau filtration on Chow groups

Impact on the transcendental Abel-Jacobi map

Abstract

Jannsen asked whether the rational cycle class map in continuous $\ell$-adic cohomology is injective, in every codimension for all smooth projective varieties over a field of finite type over the prime field. As recently pointed out by Schreieder, the integral version of Jannsen's question is also of interest. We exhibit several examples showing that the answer to the integral version is negative in general. Our examples also have consequences for the coniveau filtration on Chow groups and the transcendental Abel-Jacobi map constructed by Schreieder.