O'Nan moonshine and arithmetic

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