An effective “Theorem of André” for CM-points on a plane curve
An effective “Theorem of André” for CM-points on a plane curve
复制标题
平面曲线上 CM 点的有效“安德烈定理”
DOI：
10.1017/s0305004112000461
复制
发表时间：
2012
影响因子：
0.8
通讯作者：
U. Zannier
中科院分区：
文献类型：
--
作者：
Y. Bilu;D. Masser;U. Zannier
Y.André(André-Oort猜想的一个基本特例)的一个著名结果是，含有无穷多个坐标为CM-不变量的点的不可约代数平面曲线是一条水平线或垂直线，或者是一条模曲线Y0(N)。安德烈的S证明是部分无效的，这是由于使用了(西格尔的)类数估计。这里我们观察到他的论点可以被修改以得到一个有效的证明。例如，对于对角线X1+X2=1或双曲线X1X2=1，可以很快地证明不存在具有j(τ1)+j(τ2)=1或j(τ1)j(τ2)=1的虚二次τ1，τ2，其中j是经典模函数。
Abstract It is a well known result of Y. André (a basic special case of the André-Oort conjecture) that an irreducible algebraic plane curve containing infinitely many points whose coordinates are CM-invariants is either a horizontal or vertical line, or a modular curve Y0(n). André's proof was partially ineffective, due to the use of (Siegel's) class-number estimates. Here we observe that his arguments may be modified to yield an effective proof. For example, with the diagonal line X1+X2=1 or the hyperbola X1X2=1 it may be shown quite quickly that there are no imaginary quadratic τ1,τ2 with j(τ1)+j(τ2)=1 or j(τ1)j(τ2)=1, where j is the classical modular function.