Non-Additive Prolegomena (to any future Arithmetic that will be able to present itself as a Geometry)

Abstract: We give a language for geometry which makes curves and number fields look alike.

• The paper introduces a categorical framework that replaces traditional additive operations with generalized multiplicative structures, effectively addressing issues like the real prime and arithmetic plane.
• It employs combinatorial methods using trees and monoids to construct spectra, divisors, and compactifications of Spec(Z) in a non-additive setting.
• The framework provides new insights into bridging arithmetic and geometry, offering potential reinterpretations of classical conjectures such as the Riemann Hypothesis.

Non-Additive Foundations for Arithmetic Geometry: A Synthesis

Context and Motivation

The paper develops a category-theoretic framework for arithmetic geometry that avoids classical additive structures, prompted by historical and structural obstacles in bridging the language of geometry with that of number theory. The author identifies three persistent enigmas at the heart of arithmetic geometry—the real prime, the arithmetical plane, and the absolute point—arguing that they manifest from the inadequacy of additive language in the algebraic underpinnings of arithmetic.

In contrast with the geometric theories of curves over finite fields or over $\mathbb{C}$—where projective techniques, product spaces, and base fields with initial/final objects are naturally implemented—the arithmetic setting resists direct translation due to the entrenched role of addition.

Main Concepts and Algebraic Constructions

The Case Against Addition

The failure of addition in arithmetic is demonstrated:

1. The real prime: In analogy to the projective line for function fields, extending $\operatorname{Spec}(\mathbb{Z})$ to include infinite places is obstructed. Local structures at infinite places are not closed under addition (e.g., $[-1,1]$ under addition).
2. The arithmetical plane: The tensor product $\mathbb{Z} \otimes \mathbb{Z}$ collapses to $\mathbb{Z}$, so arithmetic lacks a nontrivial geometric plane analogous to $\mathbb{A}_k^2$.
3. The absolute point: The nonexistence of a true “point” beneath $\operatorname{Spec}(\mathbb{Z})$, in contrast to $\operatorname{Spec}(k)$ in geometry, motivates a search for a field with one element $\mathbb{F}_1$.

Replacement: Generalized Rings, Monoids, and Trees

The paper introduces a categorical structure called generalized rings built from sequences of sets $A_n$ with operations (multiplication, contraction) parametrized by symmetric groups and functorial diagrams, but crucially without underlying additive structure. This is supported by:

• Monoids and $\mathbb{F}$-rings: Inspired by Kurokawa et al. and Deitmar, base objects are constructed only with multiplicative monoids, omitting addition.
• Functoriality and Trees: Elements of free generalized rings are indexed by combinatorial objects—finite rooted trees—with morphisms, labels, and boundary identifications, where composition and contraction are defined categorically.
• Self-Adjointness and Commutativity: While commutativity can be relaxed, self-adjointness is essential to the development; existing examples are both commutative and self-adjoint.
• Symmetric Group Action: The symmetric group $S_n$ acts as a symmetry in the combinatorics of the structures.

Reconciling the Real Prime and Local Structures

The generalized ring associated to the real place, denoted ${\cal O}_{\eta}$, is realized as an $l_2$-ball, closed under the new non-additive operations (contractions), and its symmetries are imposed by orthogonal groups, unlike Durov’s $l_1$-norm-based approach which gets the symmetry group wrong.

Scheme Theory and Spectral Geometry

Spectra and Ideals in the New Context

• Spectra and Topology: The paper defines the notion of prime, maximal, and stable ideals (as appropriate in the new category), constructs the spectrum with a Zariski-like topology, and proves that spectrum functors yield compact sober spaces.
• Localization and Sheaves: The machinery of localization, local rings at primes, and sheafification (structure sheaf of generalized rings) is generalized to this setting.
• Generalized Schemes: The author develops the full apparatus of scheme theory, including Grothendieck topologies, gluing, and pro-objects, now with generalized rings as local models.

The Compactification of $\operatorname{Spec}(\mathbb{Z})$

A key technical achievement is the construction of $\overline{\operatorname{Spec}(\mathbb{Z})}$ as a pro-object, compactifying Zariski spectra to include real and complex primes along with their local structures.

Nontrivial Fibered Products and Arithmetic Planes

A crucial and previously unattained objective—the construction of a nontrivial arithmetical plane—is achieved by realizing fiber products in the category of commutative self-adjoint generalized rings. Specifically, the object $$\overline{\operatorname{Spec}(\mathbb{Z})} \times_{\mathbb{F}[\pm1]}\overline{\operatorname{Spec}(\mathbb{Z})}$$ is constructed, and its algebra of global sections described combinatorially in terms of labeled and oriented trees modulo commutativity, self-adjointness, and other relations. An explicit surjection onto the classical $\mathbb{Z} \otimes_{\mathbb{F}[\pm1]} \mathbb{Z}$ is described, clarifying the point at which classical additive collapse occurs.

Divisor Theory and Conjectural Applications

The final sections set up a theory of divisors and meromorphic functions within the non-additive framework. Effectiveness, invertibility, and their relations emulate the classical scenario. For the pro-object compactification, divisor groups are computed (with explicit formulas for $$\operatorname{Div}(\overline{\operatorname{Spec}(\mathbb{Z})})$$), and their structure recovers a direct sum over places, including a real (Archimedean) component given by $\log(\mathbb{R}^+)$.

The paper closes with speculative proposals for non-additive intersection theory on the arithmetic plane and for the definition of “Frobenius divisors.” It conjectures the possibility of extending the Riemann-Roch formalism and intersection numbers to this context, ultimately motivating a program aimed at the Riemann Hypothesis for $\zeta$-functions over $\mathbb{Z}$ within a geometric framework.

Numerical and Structural Outcomes

• The tensor product construction in the category of generalized rings yields objects (e.g., the arithmetic plane) that are strictly richer than their additive analogs, avoiding collapse to the diagonal.
• The symmetry group of the real prime is the orthogonal group $O_n$ or unitary group $U_n$, rather than the smaller group obtained by additive generalization.
• Explicit combinatorial enumerations of classes of oriented trees for small cardinality (as provided in the appendix) demonstrate transparent handling of combinatorics beyond the limitations of additive algebra.

Implications and Speculations

This work provides a robust categorical and combinatorial apparatus for approaching longstanding foundational questions at the interface of arithmetic and geometry. Its machinery circumvents the standard additive collapse blocking the construction of “the correct” arithmetic plane and opens avenues for a genuine geometric interpretation of number fields, including a conceptually sound version of “geometry over $\mathbb{F}_1$."

Practically, the framework suggests new invariants, new approaches to Arakelov theory, and possible reinterpretations of classical conjectures (e.g., the Riemann Hypothesis, the ABC conjecture) via intersection numbers and divisors on arithmetic surfaces. The explicit role of trees and combinatorics forecasts connections to quantum groups, representation theory, and $q$-deformations.

While the full scope of the theory remains conjectural, especially regarding intersection theory and the construction of Frobenius divisors, this approach lays the groundwork for re-formulating both arithmetic cohomology and arithmetic intersection theory in genuinely non-additive, geometric terms.

Conclusion

The paper articulates and realizes a categorical, non-additive framework capable of supporting a geometric theory of arithmetic, overcoming the persistent obstacles created by the traditional role of addition. Through the use of generalized rings, monoidal categories, and the language of trees, it assembles a powerful foundation for re-expressing the fundamental objects and phenomena of arithmetic geometry. This paradigm allows for well-behaved compactifications, divisors, and products, thereby rejuvenating the analogy between geometry and arithmetic that has guided the developmental trajectory of algebraic geometry. The program holds significant implications for deep theoretical advances, including a possible geometric approach to the Riemann Hypothesis.