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Title:
The elliptic orbifold of order 5
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Abstract:
A group acting on an elliptic curve must have order 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. Following Eskin-Okounkov, generating functions of these Hurwitz numbers can be computed as the q-trace of operators on Fock space. These operators naturally generalize to all orders N>1, suggesting a phantom "elliptic orbifold of order 5." I will discuss work-in-progress with Peter Smillie, proposing a definition for the Hurwitz theory of this non-existent object, and relating it to quasi-crystals in the moduli space of quintic differentials and the enumeration of Penrose-tiled Riemann surfaces.
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