On the Absolute Geometry of $\operatorname{Spec}\mathbf{Z}$

Abstract

A guiding principle in P. Scholze's $p$-adic geometry asserts that the points of $\operatorname{Spec}\mathbf{Z}$ over an algebraically closed perfectoid field of characteristic $p$ are classified, up to equivalence, by its untilts. In this paper, we give a concrete geometric realization and a generalization of this paradigm. We construct the absolute $\mathbf{F}_1$-arithmetic curve $(\operatorname{Spec}\mathbf{Z})_{\mathbf{F}_1}$ by pulling back the $\mathbf{F}_1$-structure sheaf of the arithmetic site to $\operatorname{Spec}\mathbf{Z}$. We demonstrate that $(\operatorname{Spec}\mathbf{Z})_{\mathbf{F}_1}$ provides a common geometric origin for fundamental structures in $p$-adic Hodge theory, complex analytic geometry, and the adelic scaling site. The moduli space of points of $(\operatorname{Spec}\mathbf{Z})_{\mathbf{F}_1}$ over an arbitrary perfectoid field, modulo intrinsic symmetries, canonically parameterizes the space of all perfectoid fields with the same tilt, providing a universal, characteristic-independent geometric realization of Scholze's heuristic. Evaluating the points of $(\operatorname{Spec}\mathbf{Z})_{\mathbf{F}_1}$ over the field $\mathbf{C}$ of complex numbers reveals, at each prime $p$, that the non-trivial points canonically form two principal homogeneous spaces (torsors) over the Weil groups $W_p=\mathbf{Q}_p^\times$ and $W_\infty=\mathbf{C}^\times$. Quotienting the archimedean orbit by the discrete Frobenius symmetries yields the complex Tate curve with modulus $q=p^{-1}$. We show that this elliptic curve canonically decomposes as the product of its real locus, which exactly recovers the adelic periodic orbit $C_p=\mathbf{R}_+^\times/p^\mathbf{Z}$, and a $p$-independent phase space that emerges naturally as a real analogue of the Fargues--Fontaine curve.