We characterize all infinite-dimensional graded virtual modules for Thompson's sporadic simple group whose graded traces are weight $\frac{3}{2}$ weakly holomorphic modular forms satisfying certain special properties. We then use these modules to detect the non-triviality of Mordell–Weil, Selmer, and Tate-Shafarevich groups of quadratic twists of certain elliptic curves.

中文翻译：

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椭圆曲线和汤普森零星的简单群

我们表征了汤普森的零星简单组的所有无穷级渐变虚拟模块，其渐变轨迹为权重 $\frac{3}{\mathrm{2个}}$满足某些特殊性质的弱全纯模块化形式。然后，我们使用这些模块来检测某些椭圆曲线的二次扭曲的Mordell-Weil，Selmer和Tate-Shafarevich群的非平凡性。