论文标题
自动形态小组对克莱因四分之一曲线II的雅各比式的行动：不变的theta函数
Action of the automorphism group on the Jacobian of Klein's quartic curve II: Invariant theta functions
论文作者
论文摘要
伯恩斯坦·史克瓦尔兹曼（Bernstein-Schwarzman）猜想，反射产生的不可还原的复杂晶体学组的复杂仿射空间的商是一个加权的投射空间。 Schwarzman和Tokunaga-Yoshida在维度2中证明了这一猜想，几乎所有此类群体都证明了Coxeter类型的所有晶体学反射组，由Looijenga，Bernstein-Schwarzman和Kac-Peterson在任何维度上证明了这一点。我们证明，对第3个晶体学反射组的猜想是正确的，该晶体图3是Klein的简单订单168阶的简单组。在这种情况下，商是3维加权的投影空间，具有权重1、2、4、7。证明的主要成分是对Invariant theta punctions vuctions and aLgebra的计算。与Coxeter情况不同，不变的代数不是自由的多项式，这是主要的绊脚石。
Bernstein-Schwarzman conjectured that the quotient of a complex affine space by an irreducible complex crystallographic group generated by reflections is a weighted projective space. The conjecture was proved by Schwarzman and Tokunaga-Yoshida in dimension 2 for almost all such groups, and for all crystallographic reflection groups of Coxeter type by Looijenga, Bernstein-Schwarzman and Kac-Peterson in any dimension. We prove that the conjecture is true for the crystallographic reflection group in dimension 3 for which the associated collineation group is Klein's simple group of order 168. In this case the quotient is the 3-dimensional weighted projective space with weights 1, 2, 4, 7. The main ingredient in the proof is the computation of the algebra of invariant theta functions. Unlike the Coxeter case, the invariant algebra is not free polynomial, and this was the major stumbling block.
