AI 中文总结
本研究通过合并仿射绝对曲线与阿基米德分量构造$\overline{\text{Spec}\mathbf{Z}}$的绝对代数几何，引入奇算术拓扑斯中的全局绝对曲线，揭示了其与扭量结构、霍普夫代数及几何L-函数局部因子的关联。
AI 中文摘要
我们通过将仿射绝对曲线$(\text{Spec}\\, \mathbf{Z})_{\mathbf{F}_{1}}$与定义在$\mathbf{F}_1$的带号扩张$\mathbf{F}_{1^2}$上的阿基米德分量合并，构造了紧化$\overline{\text{Spec}\mathbf{Z}}$的绝对代数几何。通过给绝对射影直线附加一个形式虚单位，我们得到一个具有典范几何反演对称性的等变拓扑斯，该对称性在其复点上诱导出扭量实结构。这一阿基米德几何被整合到一条全局绝对曲线中，该曲线被定义为奇算术拓扑斯的内部对象，与奇正整数的乘法幺半群对偶，并由球面$\mathbf{F}_{1^2}$-代数的内蕴霍普夫结构支配。绝对弗罗贝尼乌斯作用在奇整数上的限制由到$\mathbf{F}_{1^2}$的标量扩张在算术上强制决定。在复点上，由此产生的动力学同时生成实霍奇结构上的Adams运算和复共轭。在范畴层面，奇算术拓扑斯源自周循环范畴，其$λ$-运算为几何L-函数的局部因子提供了概念性解释。
英文摘要
We construct the absolute algebraic geometry of the compactification $\overline{\text{Spec}\, \mathbf Z}$ by amalgamating the affine absolute curve $(\text{Spec}\, \mathbf Z)_{\mathbf{F}_{1}}$ with an archimedean component defined over the signed extension $\mathbf{F}_{1^2}$ of $\mathbf{F}_1$. By adjoining a formal imaginary unit to the absolute projective line, we obtain an equivariant topos endowed with a canonical geometric inversion symmetry, which induces the twistor real structure on its complex points. This archimedean geometry is incorporated into a global absolute curve defined as an internal object of the odd arithmetic topos, dual to the multiplicative monoid of odd positive integers, and governed by the intrinsic Hopf structure of spherical $\mathbf{F}_{1^2}$-algebras. The restriction of the absolute Frobenius action to odd integers is forced arithmetically by the extension of scalars to $\mathbf{F}_{1^2}$. On complex points, the resulting dynamics simultaneously generates the Adams operations and complex conjugation on real Hodge structures. At the categorical level, the odd arithmetic topos originates in the pericyclic category, whose $λ$-operations provide a conceptual interpretation of the local factors of geometric L-functions.
Comments18 pages
