In this paper, working in motivic stable homotopy theory over a field \(F\) of characteristic \(\neq 2\), the authors study and exploit the \(kq\)-based Adams spectral sequence, where \(kq\) is the very effective cover of Hermitian \(K\)-theory, \(KQ\) [A. Ananyevskiy et al., Math. Z. 294, No. 3–4, 1021–1034 (2020; Zbl 1453.14064)]. They also use \(ksp\), the very effective cover of \(\Sigma^{4,2} KQ\).
The unit map \(S^{0,0} \rightarrow kq\) induces an isomorphism for \(n \in \mathbb{N}\), \(\pi_{n,n}^F(S^{0,0}) \cong \pi_{n,n}^F (kq)\), which thus identifies with Milnor-Witt \(K\)-theory \(K^{MW}_n (F)\). In particular the cofibre \(\overline{kq}\) of the unit is \(1\)-connected in the homotopy \(t\)-structure. This leads to good convergence properties of the \(kq\)-based Adams spectral sequence, which has the form \[ E_1 = \pi_{t,u} ^F (kq \wedge \overline{kq}^{\wedge n}) \Rightarrow \pi^F_{t-n, u}(S^{0,0}). \] For the main results, \(F\) is taken to be algebraically closed of characteristic zero (the authors discuss which results are expected to extend and how). They work with \(2\)-complete cellular motivic spectra and the \(2\)-complete version of the spectral sequence.
A key fact is that the Betti realization \(kq (\mathbb{C})\) identifies with \(bo\), hence realization induces a map of spectral sequences from the \(kq\)-based Adams spectral sequence to the classical \(bo\)-based Adams spectral sequence, allowing Mahowald’s work, e.g. [M. Mahowald, Pac. J. Math. 92, 365–383 (1981; Zbl 0476.55021)], to be input into calculations, together with more recent work such as [A. Beaudry et al., J. Topol. 13, No. 1, 356–415 (2020; Zbl 1469.55013)]. The motivic setting presents additional complications.
To analyse the \(E_1\)-page motivically, in particular \(kq\)-cooperations \(\pi_{*,*} (kq \wedge kq)\), the authors ‘lift’ Mahowald’s approach, working with the motivic Steenrod algebra \(A\) and the appropriate subalgebra \(A (1)\), together with their duals and comodules over \(A (1) ^\vee\). They use motivic integral Brown-Gitler \(A (1)^\vee\)-comodules, \(H \underline{\mathbb{Z}}_j\), for natural numbers \(j\), which are constructed as sub comodules of \((A /\!/ A(0))^\vee\), mimicking the classical construction. They then calculate (up to \(v_1\)-torsion) \(\mathrm{Ext}_{A (1)^\vee} (H \underline{\mathbb{Z}}_j) \) as \(\mathrm{Ext}_{A (1)^\vee} (\mathbb{M}_2)\)-modules, where \(\mathbb{M}_2\) is the coefficient ring of the mod \(2\) motivic homology spectrum \(H \mathbb{F}_2\). For instance, \(\mathrm{Ext}_{A (1)^\vee} (H \underline{\mathbb{Z}}_1) \) is isomorphic to \(\pi_{*,*} (ksp)\).
The motivic integral Brown-Gitler modules provide the splitting of \(A (1)^\vee\)-comodules: \[ (A / \!/ A(1) ) ^\vee \cong \bigoplus_{i \geq 0} \Sigma^{4i,2i} H \underline{\mathbb{Z}}_i , \] so that the motivic Adams spectral sequence converging to \(\pi_{*,*} (kq \wedge kq)\) has \(E_2\)-page \(\bigoplus_{i \geq 0} \mathrm{Ext}_{A (1)^\vee} (\Sigma^{4i,2i} H \underline{\mathbb{Z}}_i)\). Using Betti realization, they show that this spectral sequence collapses at the \(E_2\)-page, thus their results calculate the \(E_\infty\)-page of this spectral sequence up to \(v_1\)-torsion.
They then analyse the \(E_1\)-page of the \(kq\)-based Adams spectral sequence and the differentials. Their main result calculates the \(0\)- and \(1\)-lines of the \(E_\infty\)-page, together with giving a vanishing line. They also prove that the \(E_2\)-page has a vanishing line of slope \(\frac{1}{5}\).
For this, the authors exploit Mahowald’s results and methods using Betti realization; this requires care with treating \(\tau\)-torsion, which vanishes under realization. The proof of the vanishing line for the \(E_2\)-page uses a delicate argument involving a motivic spectrum \(A_1\) that realizes \(A(1)\) in motivic cohomology as an \(A(1)\)-module, and which has the property that \(A_1 \wedge kq \simeq H \mathbb{F}_2\).
The authors give applications of these results. For instance, they recover the calculation of \(\pi_{*,*} (\eta^{-1} S^{0,0})\) due to [M. Andrews and H. Miller, J. Topol. 10, No. 4, 1145–1168 (2017; Zbl 1422.55034)]. They also show that the \(0\)- and \(1\)-lines of the \(E_\infty\)-page account for the \(v_1\)-periodic classes in \(\pi_{*,*}(S^0)\). Moreover, they exhibit a candidate motivic connective image of \(J\)-spectrum, \(j_o\), defined as the fibre in a sequence \[ j_o \rightarrow kq \rightarrow \Sigma^{4,2} ksp \] and prove that the homotopy of \(\pi_{*,*} (j_o)\) is given by these \(0\)- and \(1\)-lines.