The motivic lambda algebra and motivic Hopf invariant one problem

Abstract

We introduce the F -motivic lambda algebra for any field F of characteristic not equal to 2. This is an explicit differential graded algebra whose homology is the E 2 -page of the F -motivic Adams spectral sequence. Using the R-motivic lambda algebra, we compute the cohomology of the R-motivic Steenrod algebra through filtration 3. This is the algebraically universal case, yielding information about the cohomology of the F -motivic Steenrod algebra for any base field F .We then study the 1-line of the F -motivic Adams spectral sequence in detail. In particular, we produce differentials d 2 (h a+1 ) = (h 0 + ρh 1 )h 2 a valid over any base field F , as well as the following computations in the F -motivic Adams spectral sequence for particular base fields F . For F of the form R, Fq with q an odd prime-power, Qp with p any prime, or Q, we determine the 1-line of the E 3 -page of the F -motivic Adams spectral sequence, as well as all higher differentials in stems s ≤ 7; for F = R, we determine all permanent cycles on the 1-line; and for F = Fq or F = Qp with q, p ≡ 1 (mod 4), we determine all differentials out of the 1-line.These computations are stable motivic analogues of the classic Hopf invariant one problem. We also consider the unstable motivic analogue, showing that it reduces to known results with a finite number of exceptions. As an application, we classify which unstable motivic spheres may be represented by smooth schemes admitting a unital product. * , * is a fundamental computational tool in motivic stable homotopy theory. Here, A F is the F -motivic Steenrod algebra, which acts on M F , the mod 2 motivic cohomology of Spec(F ). This spectral sequence converges to π F * , * , the homotopy groups of the (2, η)-completed F -motivic sphere. This spectral sequence has been used frequently over the past decade. Dugger, Isaksen, Wang, and Xu [DI10, Isa19, IWX20a] have made deep computations for F = C, leading to the current best results on classical stable stems [IWX20b]. Belmont, Dugger, Guillou, and Isaksen [DI16a, DI17, GI20, BI20b, BGI21] made extensive computations for F = R, with applications to C 2 -equivariant stable stems. Wilson and Østvaer [Wil16,WØ17] analyzed the spectral sequence for F = F q to study motivic stable stems over finite fields in weight zero.

$v_1$-periodic motivic homotopy over prime fields

“…where ν ranges over all places, is injective for E " kq and E " L. In the terminology of Ormsby-Østvaer [OØ13], kq and L satisfy the motivic Hasse principle. In [BCQ21], Balderrama, Culver, and the second author showed that the E 2 -term of the motivic Adams spectral sequence converging to the motivic stable stems satisfies the motivic Hasse principle. Since kq and L serve as first approximations to the motivic sphere spectrum, it is interesting to wonder if the motivic sphere spectrum itself might satisfy the motivic Hasse principle.…”