Quasimodular moonshine and arithmetic connections
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Abstract:
We prove the existence of a module for the largest Mathieu group, whose trace functions are weight $2$ quasimodular forms. Restricting to the subgroup fixing a point, we see that the integrality of these functions is equivalent to certain divisibility conditions on the number of $\mathbb {F}_p$ points on Jacobians of modular curves. Extending such expressions to arbitrary primes, we find trace functions for modules of cyclic groups of prime order with similar connections. Moreover, for cyclic groups we give an explicit vertex operator algebra construction whose trace functions are given only in terms of weight $2$ Eisenstein series.References
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Additional Information
- Lea Beneish
- Affiliation: Department of Mathematics, Emory University, Atlanta, Georgia
- MR Author ID: 1058765
- Email: lea.beneish@emory.edu
- Received by editor(s): November 24, 2018
- Received by editor(s) in revised form: April 5, 2019
- Published electronically: July 1, 2019
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 8793-8813
- MSC (2010): Primary 11F03; Secondary 11F22, 17B69, 20C34
- DOI: https://doi.org/10.1090/tran/7874
- MathSciNet review: 4029712