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 $\mathbf{F}<em>1$-arithmetic curve $$(\operatorname{Spec}\mathbf{Z})</em>{\mathbf{F}<em>1}$$ by pulling back the $\mathbf{F}_1$-structure sheaf of the arithmetic site to $\operatorname{Spec}\mathbf{Z}$. We demonstrate that $$(\operatorname{Spec}\mathbf{Z})</em>{\mathbf{F}<em>1}$$ 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 $$(\operatorname{Spec}\mathbf{Z})</em>{\mathbf{F}<em>1}$$ 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 $$(\operatorname{Spec}\mathbf{Z})</em>{\mathbf{F}<em>1}$$ 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 $W_p=\mathbf{Q}_p<sup>\times$ and $W</em>\infty=\mathbf{C}<sup>\times$. Quotienting the archimedean orbit by the discrete Frobenius symmetries yields the complex Tate curve with modulus $q=p<sup>{-1}$. We show that this elliptic curve canonically decomposes as the product of its real locus, which exactly recovers the adelic periodic orbit $C_p=\mathbf{R}_+<sup>\times/p<sup>\mathbf{Z}$, and a $p$-independent phase space that emerges naturally as a real analogue of the Fargues--Fontaine curve.

• The paper presents the construction of an F-arithmetic curve over Spec Z as a universal geometric framework for interpreting untilts of perfectoid fields.
• It employs a topos-theoretic approach that bridges p-adic Hodge theory, tropicalization, and complex analytic geometry to merge nonarchimedean and archimedean features.
• Implications extend to moduli problems in arithmetic geometry, providing a new perspective on characteristic one geometry and the analytic structure of perfectoid fields.

Absolute Geometry and the Arithmetic Curve over $\operatorname{Spec}(\mathbf{Z})$

Introduction and Motivation

This work by Connes and Consani investigates the "absolute" geometry of $\operatorname{Spec}\mathbf{Z}$, providing a concrete geometric model in the context of $p$-adic Hodge theory, absolute algebraic geometry, and the Fargues-Fontaine paradigm. The central theme is the construction of an $F$-arithmetic curve, arising as a pullback of the $F$-structure sheaf of the arithmetic site to $\operatorname{Spec}(\mathbf{Z})$, which geometrically subsumes and unifies paradigmatic structures from $p$-adic as well as complex analytic geometry, and provides a universal moduli interpretation for perfectoid fields and their untilts.

A core motivation is to furnish an explicit geometric realization and extension of Scholze’s heuristic: over an algebraically closed perfectoid field $F$ of characteristic $p$, the set of "untilts" of $F$ modulo Frobenius serves as a canonical substitute for the $\operatorname{Spec}\mathbf{Z}$0-valued points of the "absolute curve" over $\operatorname{Spec}\mathbf{Z}$1. This work delivers a topos-theoretic framework giving this principle sharp form and universality.

Construction of the Arithmetic Curve and Abel-Jacobi Theory

The key construction involves defining the $\operatorname{Spec}\mathbf{Z}$2-arithmetic site $\operatorname{Spec}\mathbf{Z}$3, where the topos is the category of presheaves equipped with an action of the multiplicative monoid $\operatorname{Spec}\mathbf{Z}$4, and the structure sheaf $\operatorname{Spec}\mathbf{Z}$5 is the spherical monoid algebra over the "field" $\operatorname{Spec}\mathbf{Z}$6, viewed through universal Frobenius endomorphisms. Points of the topos correspond (up to initial and terminal enlargement) to totally ordered abelian subgroups of $\operatorname{Spec}\mathbf{Z}$7, capturing local arithmetic directions.

This geometry is linked to the arithmetic of $\operatorname{Spec}\mathbf{Z}$8 by pulling back the structure sheaf along a geometric morphism $\operatorname{Spec}\mathbf{Z}$9, which assigns to each prime $p$0 the ordered group $p$1 and to the generic point the trivial group. The resulting $p$2-arithmetic curve $p$3, with $p$4, has stalks at $p$5 given by $p$6, which are perfect with respect to the $p$7-power Frobenius automorphisms, and at the generic point simply $p$8.

The Abel-Jacobi map, familiar from classical theory as encoding divisor classes, is lifted in this framework to a topos morphism compatible with Picard group constructions, and the explicit commutative diagrams of these correspondences rigorously relate points, divisor classes, and their absolute incarnations.

Local Moduli Spaces and Realization of Scholze's Heuristic

A central technical result is that the moduli space of geometric points of the $p$9-curve over a perfectoid field $F$0 (possibly of arbitrary characteristic) is sensitive only to the prime $F$1 associated with the residue characteristic. Specifically,

• At each prime $F$2, the moduli space collapses to a single $F$3-orbit, due to the ultrametric mismatch under the locality condition, induced by the global Frobenius symmetry.
• At the prime $F$4, the moduli space of $F$5-points is canonically isomorphic to the open unit disk $F$6 inside the tilt $F$7 of $F$8; when $F$9 is algebraically closed, modding out by the intrinsic $F$0-symmetries realizes the set of untilts of $F$1 modulo Frobenius—i.e., the closed points of the Fargues-Fontaine curve $F$2, plus the distinguished point associated to $F$3.

This sharpens Scholze’s heuristic, providing both a universal, characteristic-independent geometric model for untilts, and a mechanism wherein the absolute $F$4-curve "geometric sieve" selects, for any test perfectoid field, only the corresponding $F$5-adic geometry as nontrivial.

Analytic Geometry over the Complex Numbers and Archimedean Phenomena

When evaluated over $F$6, the geometry at each prime $F$7 presents two distinguished "places": the nonarchimedean place $F$8 and the archimedean place $F$9.

The continuity (locality) condition in the context of the archimedean place forces homomorphisms to factor through the universal covering $\operatorname{Spec}(\mathbf{Z})$0, parametrizing the local moduli space as a complex analytic manifold. This space forms a principal homogeneous space (torsor) under the Weil group $\operatorname{Spec}(\mathbf{Z})$1; quotienting by discrete Frobenius symmetries $\operatorname{Spec}(\mathbf{Z})$2 yields the complex Tate curve $\operatorname{Spec}(\mathbf{Z})$3 with modulus $\operatorname{Spec}(\mathbf{Z})$4, which carries a canonical real structure.

Strikingly, $\operatorname{Spec}(\mathbf{Z})$5 canonically decomposes as

$\operatorname{Spec}(\mathbf{Z})$6

where $\operatorname{Spec}(\mathbf{Z})$7 is the real locus (an analogue of the adelic periodic orbit from the scaling site) and $\operatorname{Spec}(\mathbf{Z})$8 is a $\operatorname{Spec}(\mathbf{Z})$9-independent phase space—a real projective line endowed with an unramified double cover, providing a real analogue of the Fargues-Fontaine curve. The canonical holomorphic 1-form on $p$0 splits accordingly, intertwining the Haar differentials of the two factors.

In contrast, the nonarchimedean realization in the $p$1-adic topology yields the standard $p$2-adic Tate curve, $p$3, mirroring the stalk structure at $p$4.

Cyclotomic Embeddings, Galois Symmetries, and Quotients

A key aspect of the realization is the moduli interpretation of points as cyclotomic embeddings into untilts, tracking the action of the cyclotomic Galois group $p$5 and discrete Frobenius $p$6. The process of forming geometric quotients by these symmetry groups recovers first the set of untilts, and then, after Frobenius quotienting, the closed points of the Fargues-Fontaine curve as required.

The treatment clarifies that at any prime $p$7, or at the generic point, the arithmetic and topological conditions trivialize the moduli space due to lack of compatible topology, which is only nontrivial at $p$8, supporting the geometric sieve description.

Connection to Characteristic One Geometry and Tropicalization

This absolute geometry is linked to the characteristic one, or idempotent, geometry. The analytic structure underlying the Fargues-Fontaine curve admits a canonical tropicalization via the family of Gauß valuations indexed by a real parameter $p$9. The profiles $F$0 for analytic functions $F$1 are concave, piecewise linear with integer slopes, so their tropicalizations are convex, piecewise affine functions with integral slopes. These are precisely the global sections of the structure sheaf on the scaling site of characteristic one, giving a functorial link from the analytic world to idempotent geometry.

Implications and Future Directions

The paper demonstrates that the absolute $F$2-arithmetic curve provides a universal geometric framework for understanding the interplay between arithmetic, $F$3-adic Hodge theory, and the geometry of perfectoid fields. The construction is robust under varying characteristics, and the local-to-global principles instantiated here tightly couple the absolute geometry of $F$4 with moduli of untilts, reconciling archimedean and non-archimedean perspectives within a single topos-theoretic language.

On the practical side, this framework has implications for the understanding of untilts and moduli problems in $F$5-adic Hodge theory, potentially impacting approaches to motives and Galois representations. The theoretical significance extends to the development of tropical and idempotent geometries as genuine shadows of $F$6-adic and analytic geometry, offering new viewpoints for further investigation of the arithmetic site over $F$7 and the intricate interactions between various "absolute" geometries.

Future research directions include the study of Frobenius eigenspaces under descent to adelic periodic orbits, the detailed categorical and motivic properties of the scaling site, and applications to the explicit classification of Galois representations in arithmetic geometry via geometric models at the absolute base.

Conclusion

This paper establishes a robust and universal geometric realization of Scholze’s heuristic for untilts of perfectoid fields, framed within a topos-theoretic, arithmetic-site context. The $F$8-arithmetic curve over $F$9, with its locally perfectoid structure and intrinsic symmetries, encapsulates both the moduli of untilts and the fundamental structure of the Fargues-Fontaine curve, reconciling and generalizing key phenomena from $p$0-adic Hodge theory, complex geometry, and arithmetic geometry. The absolute geometric perspective advanced here not only unifies existing frameworks but also prompts new questions at the intersection of characteristic $p$1, characteristic one, and arithmetic geometry, offering a coherent setting for further exploration (2606.06604).