[Paper Review] Galois groups of co-abelian ball quotient covers

Азнив Каспарян|arXiv (Cornell University)|Dec 30, 2011

Algebraic Geometry and Number Theory6 references3 citations

TL;DR

This paper classifies finite subgroups of the automorphism group of a split abelian surface $A = E \times E$, where $E$ is an elliptic curve, and determines the Kodaira-Enriques types of the quotients $A/H$ via geometric invariant theory. It provides explicit generators and relations for such groups, characterizing when the quotient is hyper-elliptic, ruled with elliptic base, Enriques, or rational, and links these to Galois covers of ball quotients via the action of finite groups on toroidal compactifications.

ABSTRACT

If $X'= ({\mathbb B} / Γ)'$ is a torsion free toroidal compactification of a discrete ball quotient $X_o={\mathbb B} / Γ$ and $ξ: (X', T = X'\setminus X_o) ightarrow (X, D = ξ(T))$ is the blow-down of the $(-1)$-curves to the corresponding minimal model, then $G'= Aut (X',T)$ coincides with the finite group $G=Aut(X,D)$. In particular, for an elliptic curve $E$ with endomorphism ring $R = End(E)$ and a split abelian surface $X = A = E imes E$, $G$ is a finite subgroup of $Aut(A) = \mathcal{T}_A \leftthreetimes GL(2,R)$, where $(\mathcal{T}_A,+) \simeq (A,+)$ is the translation group of $A$ and $GL(2,R) = \{g \in R_{2 imes 2} \| \det(g) \in R^* \}$. The present work classifies the finite subgroups $H$ of $Aut (A = E imes E)$ for an arbitrary elliptic curve $E$. By the means of the geometric invariants theory, it characterizes the Kodaira-Enriques types of $A/H \simeq ({\mathbb B} / Γ)'/H$, in terms of the fixed point sets of $H$ on $A$. The abelian and the K3 surfaces $A/H$ are elaborated in \cite{KN}. The first section provides necessary and sufficient conditions for $A/H$ to be a hyper-elliptic, ruled with elliptic base, Enriques or a rational surface. In such a way, it depletes the Kodaira-Enriques classification of the finite Galois quotients $A/H$ of a split abelian surface $A = E imes E$. The second section derives a complete list of the conjugacy classes of the linear automorphisms $g \in GL(2,R)$ of $A$ of finite order, by the means of their eigenvalues. The third section classifies the finite subgroups $H$ of $GL(2,R)$. The last section provides explicit generators and relations for the finite subgroups $H$ of $Aut(A)$ with K3, hyper-elliptic, rules with elliptic base or Enriques quotients $A/H \simeq ({\mathbb B} / Γ)'/H$.

Motivation & Objective

• To classify all finite subgroups $H$ of $Aut(A = E \times E)$ for an arbitrary elliptic curve $E$.
• To characterize the Kodaira-Enriques types of the quotients $A/H$ using fixed point sets of $H$ on $A$.
• To determine when $A/H$ is hyper-elliptic, ruled with elliptic base, Enriques, or rational, via geometric invariant theory.
• To provide explicit generators and relations for finite subgroups $H \subset Aut(A)$ that yield K3, hyper-elliptic, ruled, or Enriques quotients.

Proposed method

• Uses geometric invariant theory to analyze the structure of $A/H$ and classify its Kodaira-Enriques type based on fixed point sets of $H$ on $A$.
• Applies the classification of finite order linear automorphisms in $GL(2,R)$ via their eigenvalues, where $R = \text{End}(E)$.
• Reduces the classification of finite subgroups $H \subset Aut(A)$ to classifying finite subgroups of $GL(2,R)$, leveraging the semidirect product structure $Aut(A) \simeq \mathcal{T}_A \ltimes GL(2,R)$.
• Analyzes the action of $H = \Gamma / \Gamma_0$ on toroidal compactifications $X' = ({\mathbb{B}}/\Gamma_0)'$, where $\Gamma_0$ is torsion-free and $\Gamma$ is a lattice in $SU_{2,1}$.
• Uses the fact that $\Gamma_0$ is a normal subgroup of finite index in $\Gamma$, so $H = \Gamma / \Gamma_0$ acts on the boundary divisor $T = X' \setminus X_0$, yielding a compactification $\overline{{\mathbb{B}}/\Gamma} = X'/H$ with cyclic quotient singularities.
• Establishes that the minimal model of $\overline{{\mathbb{B}}/\Gamma}$ is birational to $A/H$, and computes invariants via resolution of singularities and fixed point analysis.

Experimental results

Research questions

• RQ1 Which finite subgroups $H \subset Aut(A = E \times E)$ yield quotients $A/H$ of Kodaira-Enriques type hyper-elliptic, ruled with elliptic base, Enriques, or rational?
• RQ2 What are the necessary and sufficient conditions on the fixed point sets of $H$ for $A/H$ to be of a given Kodaira-Enriques type?
• RQ3 How can the finite subgroups of $GL(2,R)$ be classified by their eigenvalues and orders, particularly for $R = \text{End}(E)$ with $E$ an elliptic curve?
• RQ4 What are the explicit generators and relations for finite subgroups $H \subset Aut(A)$ such that $A/H$ is a K3, hyper-elliptic, ruled, or Enriques surface?
• RQ5 How do the Galois groups of co-abelian ball quotient covers relate to the automorphism groups of toroidal compactifications $X'$ and their quotients $X'/H$?

Key findings

• The finite group $G = \text{Aut}(X,D)$ coincides with $G' = \text{Aut}(X',T)$, where $X' = ({\mathbb{B}}/\Gamma_0)'$ is the toroidal compactification of a torsion-free ball quotient $X_0 = {\mathbb{B}}/\Gamma_0$, and $T = X' \setminus X_0$ is the boundary divisor.
• For any finite subgroup $H \subset \text{Aut}(A = E \times E)$, the quotient $A/H$ is birational to a minimal surface $Y$ that is either hyper-elliptic, ruled with elliptic base, Enriques, rational, or K3, depending on the fixed point structure of $H$.
• The classification of finite subgroups of $GL(2,R)$ is achieved via their eigenvalues, with possible orders $s \in \{2,3,4,6\}$ for elements of finite order in $GL(2,R)$, and the structure of such groups is fully determined by their linear and translation parts.
• Explicit generators and relations are provided for finite subgroups $H \subset \text{Aut}(A)$ such that $A/H$ is of Kodaira-Enriques type K3, hyper-elliptic, ruled with elliptic base, or Enriques, using the semidirect product structure and fixed point analysis.
• The action of $H = \Gamma / \Gamma_0$ on the toroidal compactification $X' = ({\mathbb{B}}/\Gamma_0)'$ is well-defined and preserves the boundary divisor $T$, yielding a compactification $\overline{{\mathbb{B}}/\Gamma} = X'/H$ with at worst isolated cyclic quotient singularities.
• The minimal model of $\overline{{\mathbb{B}}/\Gamma}$ is birational to $A/H$, and its numerical invariants (e.g., $K^2$, $\chi$, $p_g$) are computed via geometric invariant theory applied to the action of $H$ on $A$.