We prove that the average size of the 3‐Selmer group of a genus‐2 curve with a marked Weierstrass point is 4. We accomplish this by studying rational and integral orbits in the representation associated to a stably $\mathbb{Z}/3\mathbb{Z}$‐graded simple Lie algebra of type $E_8$. We give new techniques to construct integral orbits, inspired by the proof of the fundamental lemma and by the twisted vertex operator realisation of affine Kac–Moody algebras.

中文翻译：

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E8和尖锐的Gens-2曲线的Jacobian的3-Selmer组的平均大小

我们证明带有明显Weierstrass点的2类曲线的3-Selmer组的平均大小为4。我们通过研究与稳定相关的表示形式中的有理轨道和积分轨道来实现这一点。 $\mathbb{ž}/3\mathbb{ž}$类型的分级简单李代数 ${Ë}_8$。受基本引理的证明以及仿射Kac-Moody代数的扭曲顶点算子实现的启发，我们提供了构造积分轨道的新技术。