Geodesic
Noninjectivity of the cycle class map in continuous $\ell $-adic cohomology
Forum of Mathematics, Sigma, Tome 11 (2023)

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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. 
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     author = {Federico Scavia and Fumiaki Suzuki},
     title = {Noninjectivity of the cycle class map in continuous $\ell $-adic cohomology},
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Federico Scavia; Fumiaki Suzuki. Noninjectivity of the cycle class map in continuous $\ell $-adic cohomology. Forum of Mathematics, Sigma, Tome 11 (2023). doi: 10.1017/fms.2023.1

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