  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.
