Root systems and elliptic curves
Root systems and elliptic curves
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DOI：
10.1007/bf01390167
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发表时间：
1976-02
影响因子：
3.1
通讯作者：
E. Looijenga
中科院分区：
文献类型：
--
作者：
E. Looijenga
本文提出了一种新的根系不变量理论。我们简要地概述了主要的几何应用;精确的结果，我们请读者有关定理。设R是一个根系，Rv是它的对偶，QV是由Rv生成的格。如果E是tr上的椭圆曲线，则A：= QV|是R的Weyl群W作用于其上的阿贝尔簇。很容易证明QV上存在W-不变对称双线性型I，使得1在最小长度的根上取值2。从Appell-Humbert定理可以得出，存在A上的”Chern类”-I的线丛5 O，使得W对A的作用提升到W对5的作用，并且具有W使0 cA上的纤维保持固定的性质。则5 1是充分的，W-不变截面的代数oo
This paper describes a new invariant theory for root systems. We briefly outline the main geometric applications; for precise results we refer the reader to the relevant theorems. Let R be a root system, R v its dual and QV the lattice generated by R v. If E is an elliptic curve over tr, then A:= QV| is an abelian variety on which the Weyl group W of R acts. It is easily shown that there is a W-invariant symmetric bilinear form I on QV such that 1 takes the value 2 on the roots of smallest length. It follows from the theorem of Appell-Humbert that there exists a line bundle 5O over A of" Chern class"-I such that the action of W on A lifts to an action of W on 5 with the property that W leaves the fibre over 0cA fixed. Then 5 1 is ample and the algebra of W-invariant sections oo