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Per here the first hints of moonshine appeared around 1974 when Andrew Ogg noticed that quotienting the hyperbolic plane by normalizers of the Hecke Congruence subgroups $\Gamma_{0}(p)$ has genus zero iff p is a divisor of the order of the monster group.

A natural question then to ask is can we try to extend the result to genus 1? That is:

  1. For which $p$ is the quotient of the hyperbolic plane by the normalizer $\Gamma_{0}(p)^+$ of the hecke congruence subgroup $\Gamma_{0}(p) \subset \text{SL}(2,\mathbb{R}) $, a riemann surface of genus 1?
  2. If its a finite list of p, does that set of $p$ divide anything interesting?
  3. If its an infinite list, does that sequence of $p$'s divide some other interesting sequence?

Has there been any prior work on (1)?

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  • $\begingroup$ Agreed the teichmuller theory tag might be misplaced but I learned first of quotienting surfaces in a teichmuller theory textbook so i figured adding it would attract the right audience. Feel free to remove and replace with something more relevant. $\endgroup$
    – Sidharth Ghoshal
    Commented Sep 13, 2022 at 18:51
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    $\begingroup$ Do you mean to quotient by the normalizer rather than $\Gamma_0(p)$ itself in "1."? $\endgroup$
    – Will Sawin
    Commented Sep 13, 2022 at 21:17

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