Vinberg’s representations and arithmetic invariant theory
Vinberg’s representations and arithmetic invariant theory
复制标题
温伯格的表示和算术不变理论
DOI：
10.2140/ant.2013.7.2331
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发表时间：
2013
影响因子：
1.3
通讯作者：
J. Thorne
中科院分区：
文献类型：
--
作者：
J. Thorne
最近，Bhargava和其他人已经证明了非常惊人的结果的平均规模的塞尔默集团的雅可比代数曲线超过Q，因为这些曲线是不同的，通过某些自然家庭。他们的方法围绕着在共正则表示中计算积分点的想法，其有理轨道可以被证明与这些代数曲线的雅可比行列式的伽罗瓦上同调类有关。本文对每一个单缀Dynkin图构造了一个共正则表示（G，V）和一族几何商VG上的代数曲线。我们证明了这些曲线的雅可比行列式的运算与G的有理轨道的运算有关。在A2型情形下，我们恢复了Birch和Swinnerton-Dyer以及后来Bhargava和Shankar在他们关于Q上椭圆曲线的2-塞尔默群的工作中所使用的轨道与Galois上同调类之间的对应关系.
Recently, Bhargava and others have proved very striking results about the average size of Selmer groups of Jacobians of algebraic curves over Q, as these curves are varied through certain natural families. Their methods center around the idea of counting integral points in coregular representations, whose rational orbits can be shown to be related to Galois cohomology classes for the Jacobians of these algebraic curves. In this paper we construct for each simply laced Dynkin diagram a coregular representation (G, V ) and a family of algebraic curves over the geometric quotient V G. We show that the arithmetic of the Jacobians of these curves is related to the arithmetic of the rational orbits of G. In the case of type A2, we recover the correspondence between orbits and Galois cohomology classes used by Birch and Swinnerton-Dyer and later by Bhargava and Shankar in their works concerning the 2-Selmer groups of elliptic curves over Q.