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.
- Pub Date:
- February 2017
- Mathematics - Number Theory;
- Mathematics - Representation Theory;
- 40 pages, 12 tables. v3: The McKay--Thompson series are all modular in this version. A connection to the cohomology of the O'Nan group is explained. Some clarifications and corrections have been made based on referee advice. v4: Some further clarifications, corrections and edits. This is the final version, accepted for publication in the American Journal of Mathematics