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Jannsen asked whether the rational cycle class map in continuous $\ell$-adiccohomology is injective, in every codimension for all smooth projectivevarieties over a field of finite type over the prime field. As recently pointedout by Schreieder, the integral version of Jannsen's question is also ofinterest. We exhibit several examples showing that the answer to the integralversion is negative in general. Our examples also have consequences for theconiveau filtration on Chow groups and the transcendental Abel-Jacobi mapconstructed by Schreieder.

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