论文标题
分级谎言代数，压缩的雅各布人和算术统计
Graded Lie Algebras, Compactified Jacobians and Arithmetic Statistics
论文作者
论文摘要
简单的dynkin图分别使用简单奇异性和Vinberg理论的变形，在$ \ mathbb {q} $和核心表示上产生了曲线的家族。索恩（Thorne）猜想并部分证明了这些曲线的算术与这些表示的理性轨道之间的牢固联系。在本文中，我们完成了Thorne的图片，并表明每个家庭中光滑曲线的jacobians的$ 2 $ Selmer元素可以通过相应表示的整体轨道来参数。使用数量的技术，我们根据这些曲线的算术得出统计结果。我们以统一的方式证明了这些结果。这恢复了Bhargava，Gross，Ho，Shankar，Shankar和Wang的结果。主要创新是：使用Colliot-Thélène和Grothendieck-serre猜想的结果对仿生空间的扭转分析，使用Bialynicki-birulaDemompos来的压缩雅各布人的几何特性的研究，以及一般的集成范围代表的一般结构。
A simply laced Dynkin diagram gives rise to a family of curves over $\mathbb{Q}$ and a coregular representation, using deformations of simple singularities and Vinberg theory respectively. Thorne has conjectured and partially proven a strong link between the arithmetic of these curves and the rational orbits of these representations. In this paper, we complete Thorne's picture and show that $2$-Selmer elements of the Jacobians of the smooth curves in each family can be parametrised by integral orbits of the corresponding representation. Using geometry-of-numbers techniques, we deduce statistical results on the arithmetic of these curves. We prove these results in a uniform manner. This recovers and generalises results of Bhargava, Gross, Ho, Shankar, Shankar and Wang. The main innovations are: an analysis of torsors on affine spaces using results of Colliot-Thélène and the Grothendieck--Serre conjecture, a study of geometric properties of compactified Jacobians using the Bialynicki-Birula decomposition, and a general construction of integral orbit representatives.
