$q$-de Rham cohomology and topological Hochschild homology over ku

Abstract: Hodge-filtered derived de Rham cohomology of a ring $R$ can be described (up to completion and shift) as the graded pieces of the even filtration on $\mathrm{HC}^-(R)$. In this paper we show a deformation of this result: If $R$ admits a spherical $\mathbb{E}_2$-lift, then the graded pieces of the even filtration on $\mathrm{TC}^-(\mathrm{ku}\otimes\mathbb{S}_R/\mathrm{ku})$ form a certain filtration on the $q$-de Rham cohomology of $R$, which $q$-deforms the Hodge filtration. We also explain how the associated Habiro-Hodge complex can be described in terms of the genuine equivariant structure on $\mathrm{THH}(\mathrm{KU}\otimes\mathbb{S}_R/\mathrm{KU})$. As a special case, we'll obtain homotopy-theoretic construction of the Habiro ring of a number field of Garoufalidis-Scholze-Wheeler-Zagier.