Rational Exceptional Belyi Coverings (2404.14675v1)

Abstract: Exceptional Belyi covering is a connected Belyi covering uniquely determined by its ramification scheme or the respective dessin d'enfant. We focus on rational exceptional Belyi coverings of compact Riemann surfaces of genus 0. Well known examples are cyclic, dihedral, and Chebyshev coverings. Using Maple, we identified all rational exceptional Belyi coverings up to degree 15. Their Belyi functions were calculated for degrees up to 6 along with some for degree 7. We also found new infinite series.

• The paper presents novel classes of exceptional Belyi coverings defined by unique ramification schemes, integrating Chebyshev and dihedral interpolations.
• It employs computational tools like Maple to systematically calculate Belyi functions up to degree 15, showcasing advanced algorithmic techniques.
• The study enriches algebraic geometry by connecting dessins d’enfants and monodromy groups with explicit examples on genus 0 surfaces.

Rational Exceptional Belyi Coverings: An Expert Analysis

This paper presents a comprehensive examination of rational exceptional Belyi coverings, a specialized subset of Belyi coverings on compact Riemann surfaces. These coverings are uniquely characterized by their ramification schemes, offering unique and fascinating properties that invite further exploration.

Introduction to Belyi Coverings

A Belyi covering is a type of complex function that maps a Riemann surface to the Riemann sphere $\mathbb{P}^1$, with all branch points confined to the set $\{0, 1, \infty\}$. Such functions are foundational in the field of algebraic geometry and are inherently linked to dessins d'enfants, combinatorial structures that provide a graphical perspective of these mappings and are linked to the absolute Galois group.

Belyi's Theorem and Dessins d’Enfants

Belyi's theorem reveals that a Riemann surface can be defined over the number field $\overline{\mathbb{Q}}$ if and only if there exists a corresponding Belyi function. Dessins d’enfants correspond bijectively to Belyi pairs, and their paper involves monodromy groups of these functions and the action of the absolute Galois group $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, which encodes the complexity and symmetry inherent in these constructions.

Counting Belyi Coverings

The enumeration of Belyi coverings involves complex formulae that engage with the irreducible characters of symmetric groups. Instruments like the Tutte formula for planar bicolored trees provide systematic methods to count coverings by leveraging representation theory. Burnside's theorem is applied in the paper to classify unramified covers in terms of conjugacy classes, facilitating the computation of quantities central to understanding the properties of these functions.

Exceptional Belyi Coverings

Characteristics and Examples

Exceptional Belyi coverings are defined by their unique determination from ramification schemes, particularly for genus 0 surfaces. These include well-known examples such as cyclic coverings, dihedral coverings, Chebyshev coverings, and coverings corresponding to Platonic solids. The paper introduces two novel series identified as interpolations between Chebyshev and dihedral coverings, involving specific ramifications and field definitions, calculated up to degree 15 using Maple.

New Series Discoveries

New infinite series within the framework of exceptional coverings have been identified, characterized by distinct ramification schemes:

1. Interpolating Series: Bridging Chebyshev and dihedral series through specific polynomial expressions for degrees with novel ramification structures.
2. Odd Degree Series: Series characterized by unique cycle structures not fully mapped to known coverings.

Implementation and Calculations

Utilizing computational tools such as Maple, the paper implements algorithms capable of generating these Belyi functions for various degrees, allowing the mapping of dessins d'enfants and Belyi functions on surfaces of increasing complexity. Calculations of Belyi functions for realizations demonstrate practical applications of the theoretical framework, with coverage up to degree 7 detailed explicitly in supplementary materials.

Conclusion

The paper extends the theoretical framework by identifying new classes of exceptional Belyi coverings, enriching our understanding of the interplay between algebraic geometry, number theory, and combinatorial topology. Continuing advancements in computational approaches will likely discover further subclasses and applications, positioning Belyi coverings as a significant domain within modern mathematical inquiry. The ongoing identification and characterization of these coverings offer potential insights into the geometric and arithmetic properties of Riemann surfaces and beyond.