[Paper Review] Generalized superelliptic Riemann surfaces

Rubén A. Hidalgo, Saúl Quispe+1 more|arXiv (Cornell University)|Sep 30, 2016

Algebraic Geometry and Number Theory18 references3 citations

TL;DR

This paper introduces generalized superelliptic Riemann surfaces as a natural extension of superelliptic curves, characterized by a central, order-$n$ automorphism $τ$ such that the quotient $χ/\langle\tau\rangle$ has genus zero. The key contribution is proving that such surfaces have a unique generalized superelliptic group of level $n$ except for a specific family when $n$ is even, enabling the construction of algebraic models and establishing conditions under which these surfaces are definable over their fields of moduli.

ABSTRACT

A closed Riemann surface $\mathcal X$, of genus $g \geq 2$, is called a generalized superelliptic curve of level $n \geq 2$ if it admits an order $n$ conformal automorphism $τ$ so that $\mathcal X/\langle τ angle$ has genus zero and $τ$ is central in ${ m Aut}(\mathcal X)$; the cyclic group $H=\langle τ angle$ is called a generalized superelliptic group of level $n$ for $\mathcal X$. These Riemann surfaces are natural generalizations of hyperelliptic Riemann surfaces (when $n=2$). We provide an algebraic curve description of these Riemann surfaces in terms of their groups of automorphisms. Also, we observe that the generalized superelliptic group $H$ of level $n$ is unique, with the exception of a very particular family of exceptional generalized superelliptic Riemann surfaces for $n$ even. In particular, the uniqueness holds if either: (i) $n$ is odd or (ii) the quotient $\mathcal X/H$ has all its cone points of order $n$ (for instance, when $\mathcal X$ is a superelliptic curve of level $n$). In the non-exceptional case, we use this uniqueness property of its generalized superelliptic group $H$ to observe that the corresponding curves are definable over their fields of moduli if ${ m Aut}(\mathcal X)/H$ is neither trivial or cyclic.

Motivation & Objective

• To generalize the concept of superelliptic curves to include non-uniform cone point orders in the quotient orbifold.
• To characterize generalized superelliptic Riemann surfaces via their automorphism groups and algebraic curve models.
• To establish conditions under which such surfaces are definable over their fields of moduli.
• To resolve the uniqueness of the generalized superelliptic group of level $n$, identifying exceptional cases when $n$ is even.

Proposed method

• Using Singerman’s list of finitely maximal signatures to classify possible orbifold structures for the quotient $χ/\langle\tau\rangle$.
• Applying group-theoretic techniques to analyze the centrality of the automorphism $τ$ in ${\rm Aut}(\mathcal{X})$ and its implications on the exponents $l_j$ in the algebraic model.
• Constructing algebraic curves of the form $y^n = \prod_{j=1}^r (x - a_j)^{l_j}$ with $a_j$ distinct and $\gcd(n, l_1, \dots, l_r) = 1$ to represent the surfaces.
• Using the reduced group $\overline{G} = {\rm Aut}(\mathcal{X})/H$ to classify the automorphism structure and determine isomorphism classes of the curves.
• Applying the theory of Fuchsian groups and epimorphisms from triangle groups to construct the underlying Riemann surface as a quotient of the upper half-plane.
• Establishing isomorphism criteria via Möbius transformations, permutations, and multiplication by units modulo $n$ to classify equivalent curve models.

Experimental results

Research questions

• RQ1 When is the generalized superelliptic group of level $n$ unique in the automorphism group of a Riemann surface $\mathcal{X}$?
• RQ2 Under what conditions can a generalized superelliptic Riemann surface be defined over its field of moduli?
• RQ3 What is the structure of the automorphism group of a generalized superelliptic surface, particularly when the reduced group $\overline{G} = {\rm Aut}(\mathcal{X})/H$ is non-cyclic and non-trivial?
• RQ4 How do the cone point orders in the quotient $\mathcal{X}/H$ affect the uniqueness and moduli field properties of the surface?
• RQ5 What is the complete classification of generalized superelliptic curves for low genus, and how do they relate to their algebraic models?

Key findings

• The generalized superelliptic group of level $n$ is unique in $\mathcal{X}$ unless $n$ is even and $\mathcal{X}$ belongs to a specific exceptional family.
• For non-exceptional generalized superelliptic surfaces, uniqueness of the group $H$ allows the construction of isomorphism classes via Möbius transformations and unit multiplication on exponents.
• A generalized superelliptic surface is definable over its field of moduli if the reduced group $\overline{G} = {\rm Aut}(\mathcal{X})/H$ is neither trivial nor cyclic.
• If $\overline{G}$ is trivial or cyclic, the surface is still definable over its field of moduli if the quotient $\mathcal{X}/G$ has odd signature.
• The only cases where definability over the field of moduli fails are non-exceptional generalized superelliptic curves with $\overline{G}$ trivial or cyclic and $\mathcal{X}/G$ having even signature.
• The paper provides a complete algorithmic construction of all generalized superelliptic curves of genus $g \geq 2$ using Harvey’s conditions and group epimorphisms from triangle groups.