We show that the stable homotopy groups of the C 2 C_2 -equivariant sphere spectrum and the R \mathbb {R} -motivic sphere spectrum are isomorphic in a range. This result supersedes previous work of Dugger and the third author.

We compute some R$\mathbb {R}$‐motivic stable homotopy groups. For s−w⩽11$s - w \leqslant 11$, we describe the motivic stable homotopy groups πs,w$\pi _{s,w}$ of a completion of the R$\mathbb {R}$‐motivic sphere spectrum. We apply the ρ$\rho$‐Bockstein spectral sequence to obtain R$\mathbb {R}$‐motivic Ext$\operatorname{Ext}$ groups from the C$\mathbb {C}$‐motivic Ext$\operatorname{Ext}$ groups, which are well understood in a large range. These Ext$\operatorname{Ext}$ groups are the input to the R$\mathbb {R}$‐motivic Adams spectral sequence. We fully analyze the Adams differentials in a range, and we also analyze hidden extensions by ρ$\rho$, 2, and η$\eta$. As a consequence of our computations, we recover Mahowald invariants of many low‐dimensional classical stable homotopy elements.

Folsom, Kent, and Ono used the theory of modular forms modulo to establish remarkable "self-similarity" properties of the partition function and give an overarching explanation of many partition congruences. We generalize their work to analyze powers p r of the partition function as well as Andrews's spt-function. By showing that certain generating functions reside in a small space made up of reductions of modular forms, we set up a general framework for congruences for p r and spt on arithmetic progressions of the form m n + δ modulo powers of . Our work gives a conceptual explanation of the exceptional congruences of p r observed by Boylan, as well as striking congruences of spt modulo 5, 7, and 13 recently discovered by Andrews and Garvan.

We compute some R-motivic stable homotopy groups. For s−w ≤ 11, we describe the motivic stable homotopy groups πs,w of a completion of the R-motivic sphere spectrum. We apply the ρ-Bockstein spectral sequence to obtain R-motivic Ext groups from the C-motivic Ext groups, which are wellunderstood in a large range. These Ext groups are the input to the R-motivic Adams spectral sequence. We fully analyze the Adams differentials in a range, and we also analyze hidden extensions by ρ, 2, and η. As a consequence of our computations, we recover Mahowald invariants of many low-dimensional classical stable homotopy elements.

Moss' theorem, which relates Massey products in the E r -page of the classical Adams spectral sequence to Toda brackets of homotopy groups, is one of the main tools for calculating Adams differentials. Working in an arbitrary symmetric monoidal stable topological model category, we prove a general version of Moss' theorem which applies to spectral sequences that arise from filtrations compatible with the monoidal structure. The theorem has broad applications, e.g. to the computation of the motivic slice and motivic Adams spectral sequences.