A sparse periodic family in the cohomology of the $\mathbb{C}$-motivic Steenrod algebra
Daniel C. Isaksen, Hana Jia Kong, Guchuan Li, Yangyang Ruan, Heyi Zhu
TL;DR
This work identifies a sparse infinite family of indecomposable elements $e_0 g^k$ in the $\mathbb{C}$-motivic Adams $E_2$-page, determining exact existence conditions (when $k=2^n-1$) and the motivic weight of lifts, and relating these elements to both $\tau$-localization and the $\mathbb{C}$-motivic $\mathcal{A}(2)$-mmf-Hurewicz image. It leverages the Burklund-Xu spectral sequence in Chow degree one to isolate a unique family of permanent cycles $q_n h_0^{2^{n-1}-1}$, producing indecomposable elements $x_n$ whose $h_1$-localization corresponds to $v_n$ and whose image in $H^{***}\mathcal{A}(2)$ is $e_0 g^{2^{n-3}-1}$; this identifies the $\mathbb{C}$-motivic $e_0 g^{2^{n-3}-1}$ with a distinguished class in the BX filtration. The results clarify the structure of the motivic Adams $E_2$-page, reveal a sparse $w_1$-periodic pattern among these elements, and provide insight into the algebraic $\mathrm{mmf}$-Hurewicz image, with implications for localization phenomena and $\tau$-localization comparisons to the classical Steenrod algebra. Overall, the paper connects explicit motivic element existence to localization techniques and modular-form–related imagery in the motivic setting.
Abstract
We study a particular family of elements in the cohomology of the $\mathbb{C}$-motivic Steenrod algebra, also known as the $\mathbb{C}$-motivic Adams $E_2$-page. This family exhibits unusual periodicity properties, and it is related both to $h_1$-localization and to the algebraic Hurewicz image of the motivic modular forms spectrum $\mathrm{mmf}$.