奥南私酒和算术
O'Nan moonshine and arithmetic
摘要：
回答一个问题所提出的康威和诺顿在其开创性的1979年的论文月光，我们证明了存在一个分次无限维模块的零星单群的O '南，麦凯-汤普森系列的重量$3/2$模形式。这些级数的系数可以用类数、奇异模的迹和权为2的模$L$ -函数的二次扭转的中心临界值来表示。因此，对于素数$p$划分O 'Nan群的阶，我们得到O' Nan群特征值与类数之间的同余关系，Selmer群的$p$ -部分，以及某些椭圆曲线的Tate-Shafarevich群.这项工作代表了第一个例子，月光涉及算术不变量的这种类型。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
