Beschreibung:
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.