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.

我们对任意字段$ k $上的平滑三次曲面上的线进行算术计数，从而得出以下结论：超过$ {\ mathbf {C}} $的行数为$ 27 $，而超过$ {\ mathbf {R}的行数为} $双曲线线的数量减去椭圆线的数量为$ 3 $。通常，这些行是在字段扩展$ L $上定义的，并在$ L ^ * /（L ^ *）^ 2 $中具有关联的算术类型$ \ alpha $。在Grothendieck–Witt组$ \ operatorname {GW}（k）$中，$ k $是相等的， \ [\ sum _ {\ text {lines}} \ operatorname {Tr} _ {L / k} \ langle \ alpha \ rangle = 15 \ cdot \ langle 1 \ rangle + 12 \ cdot \ langle -1 \ rangle，\] 其中$ \ operatorname {Tr} _ {L / k} $表示跟踪$ \ operatorname {GW}（L）\至\ operatorname {GW}（k）$。进行排名和签名可恢复$ {\ mathbf {C}} $和$ {\ mathbf {R}} $上的结果。为此，我们开发了代数向量束的$ \ mathbf {A} ^ 1 $-同伦理论中的Euler数的基本理论。我们期望可以用类似的方法获得进一步的算术计数，这些算术计数可以将枚举结果推广到复杂和真实的代数几何中。