Kolyvagin’s System of Gauss Sums
Kolyvagin’s System of Gauss Sums
复制标题
科利瓦金的高斯和系统
DOI：
10.1007/978-1-4612-0457-2_14
复制
发表时间：
1991
期刊：
影响因子：
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通讯作者：
K. Rubin
中科院分区：
文献类型：
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作者：
K. Rubin
最近在b[6]中，Kolyvagin引入了一个显著的归纳过程，它改进了stikel berger定理和Thaine[6]关于切环场理想类群的结果。对于每一个Dirichlet特征X模1素数p， Kolyvagin能够确定QV-cr的理想类群的p部分的X分量的阶数。从Mazur和Wiles的工作中，我们已经知道了这些顺序，但是Kolyvagin的证明要简单得多。Kolyvagin的方法还根据Stickel berger理想确定了这些理想类群的阿贝尔群结构。根据字符X是偶数还是奇数，证明自然分为两种情况。当X是偶数时，证明使用包含Q （Pp）的无限阿贝尔域族中的环切单位；在b[4]中给出了该方法的说明和对Q （Ppoo）的推广（Mazur-Wiles定理，Iwawawa的“主猜想”）。当X是奇数时，证明依赖于包含Q （Pp）的无限阿贝尔域族中的高斯和，斯蒂克尔伯格定理的证明也是如此。在本文中，我们给出了柯利瓦金在奇字符情况下的结果，为了简单起见，我们只处理域Q （Pp）。主要结果见定理4.3和4.4。
Recently in [1] Kolyvagin introduced a remarkable inductive procedure which improves upon Stickel berger's theorem and results of Thaine [6] on ideal class groups of cyclotomic fields. For every Dirichlet character X modulo a prime p Kolyvagin was able to determine the order of the X-component of the p-part of the ideal class group of QV-cr). These orders were already known from the work of Mazur and Wiles [3], but Kolyvagin's proof is very much simpler. Kolyvagin's method also determines the abelian group structure of these ideal class groups in terms of Stickel berger ideals.The proof divides naturally into two cases, according as to whether the character X is even or odd. When X is even the proof uses cyclotomic units in an infinite family of abelian fields containing Q (Pp); an account of this method and a generalization to Q (Ppoo)(the Mazur-Wiles theorem, Iwawawa's" main conjecture") are given in [4]. When X is odd, the proof relies on Gauss sums in an infinite family of abelian fields containing Q (Pp), as does the proof of Stickel berger's theorem. In this paper we give an exposition of Kolyvagin's results in the case of odd characters, dealing for simplicity only with the field Q (Pp). For the main results see Theorems 4.3 and 4.4.