# O'Nan moonshine and arithmetic

## TL;DR

This paper constructs a new module for the O'Nan group linking moonshine to arithmetic, revealing modular forms related to class numbers and elliptic curve invariants, and establishing new congruences.

## Contribution

It proves the existence of a moonshine module for the O'Nan group with modular forms connected to arithmetic invariants, a novel link in moonshine theory.

## Key findings

- Existence of a graded module for O'Nan with modular forms of weight 3/2.
- Coefficients expressed via class numbers, singular moduli, and L-function values.
- Congruences between group characters and arithmetic invariants of elliptic curves.

## Abstract

Answering a question posed by Conway and Norton in their seminal 1979 paper on moonshine, we prove the existence of a graded infinite-dimensional module for the sporadic simple group of O'Nan, for which the McKay--Thompson series are weight $3/2$ modular forms. The coefficients of these series may be expressed in terms of class numbers, traces of singular moduli, and central critical values of quadratic twists of weight 2 modular $L$-functions. As a consequence, for primes $p$ dividing the order of the O'Nan group we obtain congruences between O'Nan group character values and class numbers, $p$-parts of Selmer groups, and Tate--Shafarevich groups of certain elliptic curves. This work represents the first example of moonshine involving arithmetic invariants of this type.