The Absolute Anabelian Geometry of Virtual Curves of Arbitrary Genus

Abstract: The objective of this paper is to further study the anabelian object referred to as \emph{pointed virtual curves}. Building upon previous work that investigated these fundamental-group-theoretic pullbacks of Galois sections in the genus-zero situation, we extend the central anabelian results to curves of arbitrary genus. To facilitate this generalization, we introduce the group-theoretic notion of an inclusion of CAVC-type and the categorical-theoretic notion of a virtual decuspidaloid. Furthermore, we establish a criterion regarding the "geometricity" of certain virtual curves, providing group-theoretic conditions under which a section of an arithmetic fundamental group arises from a rational point.

• The paper demonstrates that for all hyperbolic curves with 2g + r ≥ 3, the virtual fundamental group fully determines the curve’s arithmetic, function field, and isomorphism class.
• The paper introduces categorical decuspidaloids to formalize decuspidalization operations, establishing a precise group-theoretic calculus for classifying morphisms.
• The paper presents reconstruction algorithms that recover the scheme structure and underlying fields from abstract profinite group data, advancing computational anabelian geometry.

The Absolute Anabelian Geometry of Virtual Curves of Arbitrary Genus

Introduction and Context

This paper refines the anabelian understanding of "pointed virtual curves," an abstract, group-theoretic framework for interpreting curves via their étale fundamental groups and Galois sections, extending the framework to arbitrary genus and number of cusps. The work generalizes results previously proved for genus 0 to the setting of arbitrary $(g, r)$, where $g$ denotes genus and $r$ denotes the number of cusps. This leads to a systematic development of notions such as inclusions of CAVC-type (cuspidal-admissible virtual curve type), virtual decuspidaloids (categorical encodings of decuspidalization operations), and reconstruction algorithms which recover the curve and its relevant arithmetic, up to isomorphism, from the purely group-theoretic data of its virtual fundamental group.

Main Results and Theorems

The paper establishes the following central conclusions for hyperbolic curves $X$ of arbitrary type $(g, r)$ over arithmetic fields (number fields or mixed-characteristic local fields):

Faithfulness of the Group-Theoretic Perspective

For all $(g, r)$ with $2g + r \geq 3$, the isomorphism class of a hyperbolic curve $X$ with a chosen Galois section $s: G_k \to \Pi_X$ is determined purely by the abstract topological structure of the associated virtual fundamental group $\Pi = \Pi_{[\pr^{2/1}_X, s]}$:

• The base field $g$0, the function field $g$1, and, in appropriate cases, the isomorphism class of $g$2 itself, are all determined by $g$3 ([Theorems A, B, C]).

Categorical Decuspidalization and Admissible Inclusions

The paper formalizes the abstraction of "decuspidalization" of virtual curves in categorical terms by introducing the virtual decuspidaloid. This category organizes all possible decuspidalizations (removals of sets of cusps of admissible-type, generalized to arbitrary $g$4 and $g$5), and encodes the operations on corresponding virtual (profinite) fundamental groups. The structure and functorial properties of the decuspidaloid are group-theoretic invariants of $g$6, and can be reconstructed functorially.

Classification of Morphisms

A complete dictionary of admissible morphism types between virtual fundamental groups and their geometric interpretations is provided:

• Geo-isomorphism type: injective morphisms of open image preserving the geometric subgroup.
• Covering type: open subgroup inclusions.
• (Geo-)Decuspidalization type: (surjective) quotient mappings corresponding to cuspidal inertia subgroups.
• Admissible type: combined operations relating open subgroup structure and conjugacy of inertia.

This is codified into an elaborate group-theoretic calculus that mirrors algebraic and geometric operations on curves.

Rigidity and Uniqueness

For inclusions of CAVC-type, the group-theoretic data (fundamental group inclusion and inertia classes) suffices to reconstruct:

• The type $g$7 purely group-theoretically ([Corollary CAVC_universal_prop]).
• The action of automorphism groups on sections, yielding uniqueness (orbits under $g$8 correspond to isomorphism classes of the scheme $g$9).
• Explicit presentations and functoriality for recovery of the function field, base field, and (for compact cases) universal coverings from $r$0.

Characterization in Low Genus

Exceptional behaviors in the cases $r$1 and $r$2, corresponding to the moduli points $r$3 and punctured elliptic curves, are investigated in detail:

• For genus $r$4, the group of automorphisms of the configuration space and the action on inertia is described, including its "Klein four-group" symmetry.
• For genus $r$5, rigid automorphisms correspond to translations on the abelian variety, and multiplication by $r$6 actions can be characterized group-theoretically.

Arithmetic Descent and Rational Points

The formalism extends to show that rationality of points and sections can be detected through the structure of certain open index 2 subgroups (e.g., via bi-covers between the virtual fundamental groups of $r$7 and $r$8 curves), culminating in a group-theoretic criterion for when a Galois section comes from a rational point.

Numerical and Structural Claims

The paper contains strong claims of faithfulness and reconstructibility for the virtual fundamental group functor:

• For all arithmetic fields $r$9 and all hyperbolic curves $X$0 with $X$1, the group $X$2 alone determines $X$3, $X$4, and the isomorphism class of $X$5, up to base change (Theorems A, B, C).
• The group structure determines not only the set of cusps but also their arithmetic, conjugacy, and decuspidalization structure.
• For finite Galois extensions corresponding to open subgroups, base change is reflected functorially at the group-theoretic level.

Contradictions of naive expectations are also noted:

• There do not exist isomorphisms between inclusions of virtual fundamental groups of type CAVC and type closed-CAVC—such inclusions have group-theoretic properties (e.g., cohomological dimension) that distinguish their genus (see Proposition cls_diff_uncls(ii)).
• The set of coverings yielding curves of type $X$6 as degree 2 covers of $X$7 is unique, reflecting the tight constraints given by the Hurwitz formula and Kummer theory (see Lemma characterstic_of_1_4).

Implications and Further Directions

Theoretical Significance

This work advances the "anabelian" point of view: \emph{the geometry of hyperbolic curves (everywhere over arithmetic fields) is fully encoded in the associated profinite group-theoretic data augmented by a Galois section.} The extension to arbitrary genus (as opposed to the classical focus on genus zero or one) demonstrates the complete dictionary between the arithmetic of curves and the combinatorics and topology of their étale fundamental groups.

It introduces a robust categorical apparatus (generalized virtual decuspidaloids) to organize the mutation and quotienting of such data, which will underpin further advances in the functorial and homotopical study of arithmetic fundamental groups.

Practical Implications

The algorithmic and functorial group-theoretic reconstructions imply, in principle, that

• Algorithms can, in a theoretically computable fashion, recover the field of definition, function field, and (for compact cases) canonical models of a curve directly from the abstract group $X$8, modulo inner automorphism ambiguity.
• The identification and manipulation of rational points (including descent obstructions) can be reframed entirely in group-theoretic language, opening new possibilities for effective approaches to the Section Conjecture.

Future Directions

• The introduction of virtual decuspidaloids opens avenues for further generalizations to stacks, moduli spaces of curves, and connections with relative anabelian geometry.
• The categorical and functorial framework is suitable for extension to non-arithmetic base fields, prime characteristic, and for interaction with motivic and Galois representation-theoretic perspectives.
• Further quantitative study (e.g., complexity bounds, effectivity) of the reconstruction algorithms could eventually translate these high-level group-theoretic equivalences into effective tools for explicit arithmetic geometry.

Conclusion

This paper establishes a group-theoretic and categorical framework for the anabelian geometry of virtual curves of arbitrary genus, extending previous genus zero results and systematically generalizing the reconstruction of fields, schemes, cuspidal structures, and automorphism actions from abstract profinite group data. It demonstrates that for arithmetic base fields, the abstract virtual fundamental group $X$9 (with section) captures the full isomorphism class (even up to inseparable extension) of the underlying curve, its function field, and associated arithmetic. The categorical formalism of virtual decuspidaloids provides an organizational foundation for the study and manipulation of these structures within anabelian geometry, with significant implications for both theory and potential computational applications.