An arithmetic count of the lines on a smooth cubic surface

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.