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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Quasimodular moonshine and arithmetic connections


Author: Lea Beneish
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
Published electronically: July 1, 2019
MathSciNet review: 4029712
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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.


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Additional Information

Lea Beneish
Affiliation: Department of Mathematics, Emory University, Atlanta, Georgia
Email: lea.beneish@emory.edu

DOI: https://doi.org/10.1090/tran/7874
Received by editor(s): November 24, 2018
Received by editor(s) in revised form: April 5, 2019
Published electronically: July 1, 2019
Article copyright: © Copyright 2019 American Mathematical Society