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We give an arithmetic count of the lines on a smooth cubic surface over an arbitrary field $k$ , generalizing the counts that over ${\mathbf {C}}$ there are $27$ lines, and over ${\mathbf {R}}$ the number of hyperbolic lines minus the number of elliptic lines is $3$ . In general, the lines are defined over a field extension $L$ and have an associated arithmetic type $\alpha$ in $L^*/(L^*)^2$ . There is an equality in the Grothendieck–Witt group $\operatorname {GW}(k)$ of $k$ , \[ \sum_{\text{lines}} \operatorname{Tr}_{L/k} \langle \alpha \rangle = 15 \cdot \langle 1 \rangle + 12 \cdot \langle -1 \rangle, \] where $\operatorname {Tr}_{L/k}$ denotes the trace $\operatorname {GW}(L) \to \operatorname {GW}(k)$ . Taking the rank and signature recovers the results over ${\mathbf {C}}$ and ${\mathbf {R}}$ . To do this, we develop an elementary theory of the Euler number in $\mathbf {A}^1$ -homotopy theory for algebraic vector bundles. We expect that further arithmetic counts generalizing enumerative results in complex and real algebraic geometry can be obtained with similar methods.

We enrich the classical count that there are two complex lines meeting four lines in space to an equality of isomorphism classes of bilinear forms. For any field k k , this enrichment counts the number of lines meeting four lines defined over k k in P k 3 \mathbf {P}^3_k , with such lines weighted by their fields of definition together with information about the cross-ratio of the intersection points and spanning planes. We generalize this example to an infinite family of such enrichments, obtained using an Euler number in A 1 \mathbf {A}^1 -homotopy theory. The classical counts are recovered by taking the rank of the bilinear forms.

We equate various Euler classes of algebraic vector bundles, including those of [12] and one suggested by M. J. Hopkins, A. Raksit, and J.-P. Serre. We establish integrality results for this Euler class and give formulas for local indices at isolated zeros, both in terms of the six-functors formalism of coherent sheaves and as an explicit recipe in the commutative algebra of Scheja and Storch. As an application, we compute the Euler classes enriched in bilinear forms associated to arithmetic counts of d-planes on complete intersections in $\mathbb P^n$ in terms of topological Euler numbers over $\mathbb {R}$ and $\mathbb {C}$ .

We study a version of the James model for the loop space of a suspension in unstable A 1 -homotopy theory. We use this model to establish an analog of G.W. Whitehead's classical refinement of the Freudenthal suspension theorem in A 1 -homotopy theory: our result refines F. Morel's A 1 -simplicial suspension theorem. We then describe some E 1 -differentials in the EHP sequence in A 1 -homotopy theory. These results are analogous to classical results of G.W. Whitehead's. Using these tools, we deduce some new results about unstable A 1 -homotopy sheaves of motivic spheres, including the counterpart of a classical rational non-vanishing result.

We give a tool for understanding simplicial desuspension in A 1 -algebraic topology: we show that X → Ω(S 1 ∧ X) → Ω(S 1 ∧ X ∧ X) is a fiber sequence up to homotopy in 2-localized A 1 algebraic topology for X = (S 1 ) m ∧ G ∧q m with m > 1. It follows that there is an EHP sequence spectral sequence

Abstract. For x an element of a field other than 0 or 1, we compute the order n Massey productsof n − 2 factors of x −1 and two factors of (1 − x) −1 by embedding P 1 − {0, 1, ∞} into its Picard variety and constructing Gal(k s /k) equivariant maps from π et 1 applied to this embedding to unipotent matrix groups. This method produces obstructions to π 1 -sections of P 1 − {0, 1, ∞}, partial computations of obstructions of Jordan Ellenberg, and also computes the Massey products