Galois groups of co-abelian ball quotient covers

Abstract

If $X'= ({\mathbb B} / \Gamma)'$ is a torsion free toroidal compactification of a discrete ball quotient $X_o={\mathbb B} / \Gamma$ and $\xi : (X', T = X'\setminus X_o) \rightarrow (X, D = \xi (T))$ 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 $X = A = E \times E$, $G$ 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 $GL(2,R) = \{g \in R_{2 \times 2} \| \det(g) \in R^* \}$. The present work classifies the finite subgroups $H$ of $Aut (A = E \times E)$ for an arbitrary elliptic curve $E$. By the means of the geometric invariants theory, it characterizes the Kodaira-Enriques types of $A/H \simeq ({\mathbb B} / \Gamma)'/H$, 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 $A = E \times E$. 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 $A/H \simeq ({\mathbb B} / \Gamma)'/H$.