Open Collections

Description

A group acting on an elliptic curve must have order N = 1, 2, 3, 4, or 6. We call the quotient an elliptic orbifold. Certain branched covers of the order N elliptic orbifold are in bijection with tiled surfaces, and form a lattice in the moduli space of N-ic differentials on Riemann surfaces. The enumerative theory of these branched covers suggests a phantom "elliptic orbifold" for all integers N. I will discuss work-in-progress with Peter Smillie proposing a definition for the Hurwitz theory of this non-existent object, and attempts to relate it to quasi-crystals in the moduli space of quintic differentials and the enumeration of Penrose-tiled Riemann surfaces.