Indivisibility of class numbers of imaginary quadratic fields and orders of Tate-Shafarevich groups of elliptic curves with complex multiplication
Indivisibility of class numbers of imaginary quadratic fields and orders of Tate-Shafarevich groups of elliptic curves with complex multiplication
复制标题
复乘椭圆曲线虚二次域类数和Tate-Shafarevich群阶的不可分性
DOI：
10.1007/s002220050290
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发表时间：
1999
影响因子：
3.1
通讯作者：
K. Ono
中科院分区：
文献类型：
--
作者：
W. Kohnen;K. Ono
自高斯以来，虚二次域的理想类群一直是许多研究的焦点，最近有许多关于椭圆曲线的TateShafarevich群的研究。在这两种情况下，文献相当广泛，但鲜为人知。整个D表示二次域的一个基本判别式。设CL(D)表示Q（√D）的类群，h(D)表示它的阶，即具有判别式D的原始正二元二次型的通常类数。其中一个主要问题涉及CL(D)的结构，因此人们自然研究h(D)的可被质数整除。这里我们考虑虚二次域。高斯的属理论精确地决定了h(D)的宇称性，但h(D)被奇素数的可整除性却知之甚少。鉴于这些困难，Cohen和Lenstra [C-L]给出了描述CL(D)的“预期”行为的启发式方法，特别是他们预测了h(D)的概率为
Since Gauss, ideal class groups of imaginary quadratic fields have been the focus of many investigations, and recently there have been many investigations regarding TateShafarevich groups of elliptic curves. In both cases the literature is quite extensive, but little is known. Throughout D will denote a fundamental discriminant of a quadratic field. Let CL(D) denote the class group of Q( √ D), and let h(D) denote its order, i.e. the usual class number of primitive positive binary quadratic forms with discriminant D. One of the main problems deals with the structure of CL(D), and so one naturally studies the divisibility of h(D) by primes. Here we consider imaginary quadratic fields. Gauss’ genus theory precisely determines the parity of h(D), but the divisibility of h(D) by odd primes ` is much less well understood. In view of these difficulties, Cohen and Lenstra [C-L] gave heuristics describing the “expected” behavior of CL(D), and in particular they predicted that the probability that ` h(D) is