let $g$ be a finite group‎. ‎we say that $g$ has emph{spread} r if for any set of distinct non-trivial elements of $g$ $x:={x_1,ldots‎, ‎x_r}subset g^{#}$ there exists an element $yin g$ with the property that $langle x_i,yrangle=g$ for every $1leq ileq r$‎. ‎we say $g$ has emph{exact spread} $r$ if $g$ has spread $r$ but not $r+1$‎. ‎the spreads of finite simple groups and their decorations ha...

We characterize all infinite-dimensional graded virtual modules for Thompson's sporadic simple group whose traces are weight 32 weakly holomorphic modular forms satisfying certain special properties. then use these to detect the non-triviality of Mordell–Weil, Selmer, and Tate-Shafarevich groups quadratic twists elliptic curves.

The monster sporadic group is the automorphism of a central charge \(c=24\) vertex operator algebra (VOA) or meromorphic conformal field theory (CFT). In addition to its stress tensor T(z), this contains many other vectors smaller charge; for example, it admits 48 commuting \(c=\frac{1}{2}\) whose sum T(z). Such decompositions allow one construct new CFTs from CFT in manner analogous Goddard-Ke...