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arXiv · 2306.16414

Class Numbers, Congruent Numbers and Umbral Moonshine

Miranda C. N. Cheng·John F. R. Duncan·Michael H. Mertens
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Abstract

In earlier work we initiated a program to study relationships between finite groups and arithmetic geometric invariants of modular curves in a systematic way. In the present work we continue this program, with a focus on the two smallest sporadic simple Mathieu groups. To do this we first elucidate a connection between a special case of umbral moonshine and the imaginary quadratic class numbers. Then we use this connection to classify a distinguished set of modules for the smallest sporadic Mathieu group. Finally we establish consequences of the classification for the congruent number problem of antiquity.

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BibTeXRIS

Miranda C. N. Cheng, John F. R. Duncan, Michael H. Mertens. 2023-06-28. Class Numbers, Congruent Numbers and Umbral Moonshine. https://arxiv.org/abs/2306.16414

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