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.

摘要：

回答 Conway 和 Norton 在他们 1979 年关于 Moonshine 的开创性论文中提出的一个问题，我们证明了 O'Nan 的散发单群的分级无限维模的存在，McKay-Thompson 级数对此的权重为 $3/2$模块化形式。这些级数的系数可以用类数、奇异模迹和权重 2 模$L$-函数的二次扭曲的中心临界值来表示。因此，对于素数 $p$ 除 O'Nan 群的阶数，我们得到 O'Nan 群特征值和类数、$p$-Selmer 群的部分和某些椭圆曲线的 Tate-Shafarevich 群之间的同余. 这项工作代表了第一个涉及此类算术不变量的月光示例。