Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Topological adelic curves: algebraic coverings, geometry of numbers and heights of closed points

Antoine Sédillot

TL;DR

This work develops a comprehensive framework for topological adelic curves, extending Arakelov-style geometry to possibly uncountable fields through topological data of pseudo-absolute values and integral structures. It introduces topological adelic curves, their algebraic coverings, and a theory of adelic vector bundles with Harder–Narasimhan filtrations in the proper setting, including Arakelov degrees and slopes. The second major pillar builds adelic line bundles and height theory for varieties over topological adelic curves, extending local Monge–Ampère theory, Fubini–Study metrics, and pushforward constructions to a global, family-based context, with Nevanlinna-type analogies guiding the deformation of Arakelov-style invariants. The results unify Diophantine geometry and Nevanlinna theory in an henceforth robust, uncountable-field framework, enabling volume, height, and intersection formalism across families and paving the way for arithmetic intersection theory and higher-dimensional analogues in GVF-like settings.

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 topological counterparts 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 a previous preprint, we prove several fundamental properties of topological adelic curves: algebraic coverings, Harder-Narasimhan formalism, existence of volume functions. We also define height of closed points and give a generalisation of Nevanlinna's first main theorem in this framework.

Topological adelic curves: algebraic coverings, geometry of numbers and heights of closed points

TL;DR

This work develops a comprehensive framework for topological adelic curves, extending Arakelov-style geometry to possibly uncountable fields through topological data of pseudo-absolute values and integral structures. It introduces topological adelic curves, their algebraic coverings, and a theory of adelic vector bundles with Harder–Narasimhan filtrations in the proper setting, including Arakelov degrees and slopes. The second major pillar builds adelic line bundles and height theory for varieties over topological adelic curves, extending local Monge–Ampère theory, Fubini–Study metrics, and pushforward constructions to a global, family-based context, with Nevanlinna-type analogies guiding the deformation of Arakelov-style invariants. The results unify Diophantine geometry and Nevanlinna theory in an henceforth robust, uncountable-field framework, enabling volume, height, and intersection formalism across families and paving the way for arithmetic intersection theory and higher-dimensional analogues in GVF-like settings.

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 topological counterparts 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 a previous preprint, we prove several fundamental properties of topological adelic curves: algebraic coverings, Harder-Narasimhan formalism, existence of volume functions. We also define height of closed points and give a generalisation of Nevanlinna's first main theorem in this framework.

Paper Structure

This paper contains 71 sections, 67 theorems, 170 equations.

Table of Contents

  1. Topological adelic curves: definition and algebraic coverings
  2. Reminders on pseudo absolute values and adelic curves
  3. Pseudo-absolute values
  4. Definition
  5. Extension of pseudo-absolute values
  6. Space of pseudo-absolute values and integral structures
  7. Examples of integral structure in Nevanlinna theory
  8. Pseudo-norms
  9. Adelic curves
  10. Adelic structures over a field
  11. Algebraic coverings of adelic curves
  12. Examples of adelic curves
  13. Topological adelic curves
  14. Definitions
  15. Examples and constructions
  16. ...and 56 more sections

Key Result

Theorem A

Let $S=(K,\phi : \Omega \to V, \nu)$ be a topological adelic curve. Let $K'/K$ be an algebraic field extension.

Theorems & Definitions (189)

  • Theorem A: Propositions \ref{['prop:algebraic_extension_tac']}, \ref{['prop:action_of_Galois_group_algebraic_extension']} and Remark \ref{['rem:uniqueness']}
  • Theorem B: Theorem \ref{['th:HN_general_proper']}
  • Theorem C: Theorem \ref{['th:height_closed_points_proper']}
  • Theorem D: cf. Theorem \ref{['th:height_of_closed_points_families']} for a more general statement
  • Proposition 1.1.1: Sedillot24_pav, Proposition 3.1.2
  • Proposition 1.1.2: Sedillot24_pav, Corollary 3.2.2
  • Proposition 1.1.3: Sedillot24_pav, Propositions 3.3.1 and 3.3.4
  • Proposition 1.1.4: Sedillot24_pav, Proposition 9.3.3
  • Proposition 1.1.5: loc. cit., Proposition 9.4.6
  • Proposition 1.1.6: Sedillot24_pav, Proposition 6.1.3
  • ...and 179 more