The Gersten conjecture for $p$-adic \'etale Tate twists and the $p$-adic cycle class map

We prove the Gersten conjecture for $p$-adic \'etale Tate twists for a smooth scheme $X$ in mixed characteristic in the Nisnevich topology. Our main observation is that, while $p$-adic \'etale Tate twists are not $\mathbb A^1$-invariant, for the proof of the Gersten conjecture it suffices that they satisfy the $\mathbb P^1$-bundle formula. This fits nicely with the emphasis on the projective bundle formula in non $\mathbb A^1$-invariant motivic cohomology recently developed by Elmanto-Morrow and Annala-Hoyois-Iwasa. Furthermore, identifying $p$-adic \'etale Tate twists with the syntomic cohomology defined by Bhatt-Morrow-Scholze, the result generalises the Gersten conjecture for logarithmic deRham-Witt sheaves due to Gros-Suwa to arbitrary characteristic. In the second part of the article, we revisit the cycle class map from thickened zero-cycles on the special fiber of $X$ to \'etale cohomology with coefficients in $p$-adic \'etale Tate twists previously studied in [27]. This cycle class map is important in the study of zero-cycles on smooth projective varieties over local fields and the approach to the cycle class map which we use in this article is more conceptual and, in contrast to the approach in loc. cit., works for arbitrary finite residue fields.