Divisibility of Class Numbers of Imaginary Quadratic Fields
Divisibility of Class Numbers of Imaginary Quadratic Fields
复制标题
虚二次域类数的整除性
DOI：
10.1112/s0024610700008887
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发表时间：
2000
期刊：
影响因子：
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通讯作者：
K. Soundararajan
中科院分区：
文献类型：
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作者：
K. Soundararajan
设d是无平方数，CL（−d）表示虚二次数域Q（−d）的理想类群。进一步让h（−d）= #CL（−d）表示类数。对于整数g <$2，我们定义Ng（X）为无平方数d <$X的个数，使得CL（−d）包含一个阶为g的元素。高斯的亏格理论证明，如果d至少有两个奇素因子（特别是对几乎所有d），则CL（−d）包含Z2作为子群。因此N2（X）<$6X/π2。Ng（X）的行为对于g的任何其他值都不被理解。我们相信，对于某个正常数Cg，Ng（X）<$CgX。对于奇素数g，H. Cohen和H. Lenstra [3]证明了Cg=6π2（1-Πi=1∞（1 - 1gi））
Let d be a square‐free number and let CL(−d) denote the ideal class group of the imaginary quadratic number field Q(√−d). Further let h(−d) = #CL(−d) denote the class number. For integers g ⩾ 2, we define Ng(X) to be the number of square‐free d ⩾ X such that CL(−d) contains an element of order g. Gauss' genus theory demonstrates that if d has at least two odd prime factors (in particular, for almost all d) then CL(−d) contains Z2 as a subgroup. Thus N2(X) ∼ 6X/π2. The behaviour of Ng(X) is not understood for any other value of g. It is believed that Ng(X) ∼ CgX for some positive constant Cg. For odd primes g, H. Cohen and H. Lenstra [3] conjectured that Cg=6π2(1‐Πi=1∞(1‐1gi))