Work

Class numbers, congruent numbers and umbral moonshine

Year: 2025

Type: article

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 a connection between our classification and the congruent number problem of antiquity.

Source: Journal of Number Theory

DOI: 10.1016/j.jnt.2025.02.007

Authors: Miranda C.N. Cheng, John F.R. Duncan, Michael H. Mertens

Institutions: Institute of Mathematics, Academia Sinica, Institute of Physics, Academia Sinica, University of Amsterdam, RWTH Aachen University

Language: English

Topic: Advanced Algebra and Geometry

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,550