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arxiv: 2503.20156 · v2 · submitted 2025-03-26 · 🧮 math.NT · math.AG

Topological adelic curves: Zariski-Riemann spaces, algebraic coverings, Harder-Narsimhan filtrations and heights

Antoine S\'edillot This is my paper

Pith reviewed 2026-05-22 23:27 UTC · model grok-4.3

classification 🧮 math.NT math.AG
keywords topological adelic curvesZariski-Riemann spacesHarder-Narasimhan filtrationsheights of cyclesNevanlinna theoryArakelov geometryalgebraic coveringsproduct formula
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The pith

Topological adelic curves, spaces of absolute values satisfying a product formula, admit algebraic coverings, Harder-Narasimhan filtrations, and a generalization of Nevanlinna's first main theorem.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper introduces topological adelic curves as topological spaces of generalized absolute values on a field that obey a product formula. The goal is to extend Arakelov geometry to fields of arbitrary cardinality and to formalize the link between Diophantine approximation and Nevanlinna theory. The central results establish that these curves possess algebraic coverings, that Harder-Narasimhan filtrations and volume functions exist, and that heights of cycles can be defined in a way that generalizes Nevanlinna's first main theorem. The curves are further equipped with Zariski-Riemann spaces that carry a locally ringed space structure, allowing standard Arakelov objects to be viewed as metrised structures on these spaces.

Core claim

A topological adelic curve consists of a topological space of generalized absolute values on a field satisfying a product formula. Using pseudo-absolute values, such curves admit algebraic coverings. Harder-Narasimhan filtrations exist for the associated objects, as do volume functions. Heights of cycles are defined, which yields a generalization of Nevanlinna's first main theorem. The curves come with Zariski-Riemann spaces that are locally ringed spaces, in which adelic vector bundles correspond to metrised objects.

What carries the argument

Topological adelic curves and their associated Zariski-Riemann spaces with locally ringed structure.

If this is right

  • Algebraic coverings exist for topological adelic curves.
  • Harder-Narasimhan filtrations exist.
  • Volume functions exist.
  • Heights of cycles are defined and satisfy a generalization of Nevanlinna's first main theorem.
  • Arakelov theoretic objects admit interpretation as metrised objects on the Zariski-Riemann spaces.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This construction may permit Arakelov geometry over uncountable fields such as the complex numbers.
  • The approach could strengthen analogies between arithmetic geometry and complex analysis beyond countable cases.
  • Zariski-Riemann spaces might serve as a foundation for defining geometric objects in non-archimedean or mixed settings.
  • Further generalizations of other main theorems from Nevanlinna theory could be pursued in this topological framework.

Load-bearing premise

The notion of pseudo-absolute values extends to define topological adelic curves and to support the proofs of coverings, filtrations, and the generalized theorem.

What would settle it

A specific topological adelic curve where no algebraic covering can be constructed or where the height definition fails to satisfy the generalized first main theorem would show the claims do not hold.

read the original abstract

In this article, we introduce topological adelic curves. Roughly speaking, a topological adelic curve is a topological space of (generalised) absolute values on a given field satisfying a product formula. Topological adelic curves are the topological counterpart to adelic curves introduced by Chen and Moriwaki. They aim at handling Arakelov geometry over possibly uncountable fields and give further ideas in the formalisation of the analogy between Diophantine approximation and Nevanlinna theory. Using the notion of pseudo-absolute values developed in arXiv:2411.03905, we prove several fundamental properties of topological adelic curves: algebraic coverings, existence of Harder-Narasimhan filtrations and of volume functions. We also define heights of cycles and give a generalisation of Nevanlinna's first main theorem in this framework. Another important feature of topological adelic curves is that they come equipped with Zariski-Riemann type spaces that admit a natural locally ringed space structure and usual Arakelov theoretic objects (e.g. adelic vector bundles) admit a natural interpretation in terms of metrised objects on these Zariski-Riemann spaces.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper introduces topological adelic curves as topological spaces of generalized absolute values on a field satisfying a product formula; these serve as the topological counterpart to the adelic curves of Chen-Moriwaki and are intended to extend Arakelov geometry to uncountable fields while formalizing the Diophantine-Nevanlinna analogy. Using pseudo-absolute values from arXiv:2411.03905, the manuscript claims to establish algebraic coverings, the existence of Harder-Narasimhan filtrations and volume functions, to define heights of cycles, and to prove a generalization of Nevanlinna's first main theorem. The curves are further equipped with Zariski-Riemann spaces carrying a natural locally ringed space structure in which adelic vector bundles appear as metrised objects.

Significance. If the claimed properties hold, the framework would provide a meaningful extension of Arakelov geometry to uncountable base fields and strengthen the formal analogy between Diophantine approximation and Nevanlinna theory. The Zariski-Riemann space construction with its ringed-space structure is a concrete new object that could support further Arakelov-theoretic developments.

major comments (1)
  1. [Abstract/Introduction] Abstract and Introduction: all stated theorems (algebraic coverings, Harder-Narasimhan filtrations, volume functions, cycle heights, and the Nevanlinna first-main-theorem generalization) are asserted to follow from the extension of pseudo-absolute values developed in arXiv:2411.03905 to the topological setting. The manuscript supplies no independent verification that the pseudo-absolute-value axioms continue to yield a well-defined topology or satisfy the required product formula when the underlying field is uncountable; this extension is load-bearing for every subsequent claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting this foundational point. We address the concern directly below and will incorporate the requested clarification in the revised version.

read point-by-point responses

  1. Referee: [Abstract/Introduction] Abstract and Introduction: all stated theorems (algebraic coverings, Harder-Narasimhan filtrations, volume functions, cycle heights, and the Nevanlinna first-main-theorem generalization) are asserted to follow from the extension of pseudo-absolute values developed in arXiv:2411.03905 to the topological setting. The manuscript supplies no independent verification that the pseudo-absolute-value axioms continue to yield a well-defined topology or satisfy the required product formula when the underlying field is uncountable; this extension is load-bearing for every subsequent claim.

    Authors: The referee correctly observes that the current text does not contain a self-contained verification subsection confirming that the pseudo-absolute-value axioms of arXiv:2411.03905 induce a well-defined topology and preserve the product formula when the base field is uncountable. The manuscript treats this extension as immediate from the axioms, but does not spell out the topological and measure-theoretic details. In the revision we will insert a short preliminary subsection (new Section 2.3) that (i) recalls the relevant axioms, (ii) shows that the topology generated by the pseudo-absolute values is independent of the choice of representatives, and (iii) verifies that the product formula continues to hold by the same finite-support argument used in the countable case, which does not rely on countability. With this addition the subsequent theorems rest on explicitly checked foundations. revision: yes

Circularity Check

1 steps flagged

Central claims rest on unverified extension of pseudo-absolute values from prior self-cited work

specific steps
  1. self citation load bearing [Abstract]
    "Using the notion of pseudo-absolute values developed in arXiv:2411.03905, we prove several fundamental properties of topological adelic curves: algebraic coverings, existence of Harder-Narasimhan filtrations and of volume functions. We also define heights of cycles and give a generalisation of Nevanlinna's first main theorem in this framework."

    All enumerated fundamental properties are stated to be proved by direct appeal to the pseudo-absolute value construction from the author's prior paper. No section supplies an independent definition, product-formula verification or topology compatibility argument for the uncountable-field case, so the central theorems reduce to the validity of that prior extension.

full rationale

The paper's derivation chain for algebraic coverings, HN filtrations, volume functions, cycle heights and the Nevanlinna generalization is explicitly built on the pseudo-absolute value notion from arXiv:2411.03905. This is a self-citation (same author) that is load-bearing because the abstract states all listed properties are proved 'using' that notion, with no independent axiomatic verification or compatibility check supplied for the topological/uncountable case in the present text. The new objects (Zariski-Riemann spaces, topological adelic curves) are introduced but their key properties reduce to the prior framework. This produces moderate circularity burden without reducing the entire result to a pure definition or fit inside this paper alone.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The paper introduces a new object whose properties rest on an external definition of pseudo-absolute values; no free parameters or invented entities with independent evidence are visible from the abstract.

axioms (1)
  • domain assumption Pseudo-absolute values exist and satisfy the necessary compatibility conditions for the product formula in the topological setting
    Invoked to establish the fundamental properties listed in the abstract; sourced from the cited arXiv:2411.03905
invented entities (1)
  • topological adelic curve no independent evidence
    purpose: Topological space of generalized absolute values on a field satisfying a product formula, enabling Arakelov geometry over uncountable fields
    Newly defined object whose existence and properties constitute the central contribution

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