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Moonshine for all finite groups

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Abstract

In recent literature, moonshine has been explored for some groups beyond the Monster, for example the sporadic O’Nan and Thompson groups. This collection of examples may suggest that moonshine is a rare phenomenon, but a fundamental and largely unexplored question is how general the correspondence is between modular forms and finite groups. For every finite group G, we give constructions of infinitely many graded infinite-dimensional \(\mathbb {C}[G]\)-modules where the McKay–Thompson series for a conjugacy class [g] is a weakly holomorphic modular function properly on \(\varGamma _0({{\mathrm{ord}}}(g))\). As there are only finitely many normalized Hauptmoduln, groups whose McKay–Thompson series are normalized Hauptmoduln are rare, but not as rare as one might naively expect. We give bounds on the powers of primes dividing the order of groups which have normalized Hauptmoduln of level \({{\mathrm{ord}}}(g)\) as the graded trace functions for any conjugacy class [g], and completely classify the finite abelian groups with this property. In particular, these include \((\mathbb {Z}/ 5 \mathbb {Z})^5\) and \((\mathbb {Z}/ 7 \mathbb {Z})^4\), which are not subgroups of the Monster.

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Notes

  1. At this stage, the existence of \(\mathbb {M}\) was still entirely conjectural.

  2. We note that the levels of modular functions in \(V^\natural \) are technically \(h{{\mathrm{ord}}}(g)\) where \(h|(12,{{\mathrm{ord}}}(g))\). Our proof of Theorem 1.1 may be easily altered to handle other levels, so for simplicity we only prove it for the case when the graded trace of g is strictly on level \({{\mathrm{ord}}}(g)\).

  3. It is necessary to consider congruences of this form because there will always be congruences among \(T_1,T_2,\ldots ,T_{25}\) modulo \(p^N\) given by multiplying a congruence modulo a lower power of p by the appropriate power of p; however, mandating that the first coefficient is 1 excludes such cases.

  4. Code available at https://github.com/nvafa/moonshine-congruences.

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Acknowledgements

The authors would like to thank Ken Ono and John Duncan for advising this project and for their many helpful conversations and suggestions. We also thank Hannah Larson for helpful conversations and edits, and Robert Wilson for answering our question about subgroups of the Monster. Finally, we thank Emory University, Princeton University, and the NSF (via Grant DMS-1557690) for their support.

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Authors and Affiliations

  1. Department of Mathematics, UNC-Chapel Hill, Chapel Hill, NC, 27514, USA

    Samuel DeHority

  2. Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA

    Xavier Gonzalez & Neekon Vafa

  3. Department of Mathematics, Princeton University, Princeton, NJ, 08544, USA

    Roger Van Peski

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  1. Samuel DeHority
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Correspondence to Roger Van Peski.

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Appendix: Hauptmodul congruences

Appendix: Hauptmodul congruences

Here, we list the maximal moduli of congruences obtained to prove Theorem 1.3. Note that here \(T_N\) denotes the normalized Hauptmodul for \(X_0(N)\).

$$\begin{aligned} 0&\equiv T_1 - T_2 \qquad \qquad \,\,\,\qquad \pmod {2^{16}}\\&\equiv T_1 - T_4 \qquad \qquad \,\,\,\qquad \pmod {2^8}\\&\equiv T_1 - T_8\qquad \qquad \,\,\, \qquad \pmod {2^4}\\&\equiv T_1 - T_{16}\qquad \qquad \,\,\,\qquad \pmod {2^2}\\&\equiv T_1 - T_3 \qquad \qquad \,\,\,\qquad \pmod {3^9}\\&\equiv T_1 - T_9 \qquad \qquad \,\,\,\qquad \pmod {3^3}\\&\equiv T_1 - T_5 \qquad \qquad \,\,\,\qquad \pmod {5^5}\\&\equiv T_1 - T_{25} \qquad \qquad \qquad \,\,\, \pmod {5^1}\\&\equiv T_1 - T_7 \qquad \qquad \quad \qquad \pmod {7^4}\\&\equiv T_1 - T_{13} \qquad \qquad \,\,\,\qquad \pmod {13^2}\\&\equiv T_1 - T_2 - T_3 + T_6 \qquad \pmod {2^4 3^3}\\&\equiv T_1 - T_4 - T_3 + T_{12} \qquad \pmod {2^2 3^2}\\&\equiv T_1 - T_2 - T_9 + T_{18}\qquad \pmod {2^2 3^1}\\&\equiv T_1 - T_2 - T_5 + T_{10} \qquad \pmod {2^3 5^2} \end{aligned}$$

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DeHority, S., Gonzalez, X., Vafa, N. et al. Moonshine for all finite groups. Res Math Sci 5, 14 (2018). https://doi.org/10.1007/s40687-018-0133-5

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