Non-arithmetic ball quotients from a configuration of elliptic curves in an abelian surface (Q1989595)

The present paper is devoted to investigate an affine crystallographic complex reflection group \(G\). The linear part of \(G\) is a group isomorphic to \(\mathrm{GL}(2,3)\), exactly the group \(G_{12}\) in the list of finite groups generated by complex reflections obtained by \textit{G. C. Shephard} and \textit{J. A. Todd} in [Can. J. Math. 6, 274--304 (1954; Zbl. 0055.14305)]. The subgroup of translations of \(G\) is given by the lattice \(\Lambda = (\mathbb Z\oplus i \sqrt 2\mathbb Z)^2\). The author shows that the quotient \(X = G\setminus\mathbb C^2\), which can be seen as the quotient of the abelian surface \(A = \mathbb C^2 / \Lambda\) by the group \(G_{12}\), has two singular points of types \(\frac{1}{3}(1,2)\) and \(\frac{1}{8}(1,3)\). Besides, the map \(A \rightarrow X\) ramifies with order 2 along a rational curve in \(X\). The branch locus is a curve \(M\), which has four singular points, two ordinary cusps, a point with multiplicity 4, and another one with multiplicity 6. Then, the weighted projective plane \(X\) is isomorphic to \(\mathbb P(1,3,8)\) and the curve \(M\) is an irreducible curve of homogeneous degree 24. The main results of the paper concern the pairs \((X, (1-\frac{1}{p})M)\). This pair is an orbifold only when \(p=2\). However, for \(p=3,4,6\) or \(\infty\), there is a model birational to \((X, (1-\frac{1}{p})M)\) that is an orbifold. These results are stated in Theorem 5.5, where the respective ball quotient orbifolds are described.