Local units modulo circular units
Local units modulo circular units
复制标题
局部单位模圆单位
DOI：
10.1090/s0002-9939-1983-0706497-1
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发表时间：
1983
影响因子：
2.5
通讯作者：
R. Coleman
中科院分区：
文献类型：
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作者：
R. Coleman
在他的论文《分圆域理论中的一些模块》[2]中，岩泽得到了一个显着的定理：p进分圆单元与圆单位完备化的商同构于群环由斯蒂克伯格理想得到的商。然后他用它推导出了一些有趣的全局结果，其中最引人注目的是在第一层类群是循环的假设以及正则性假设下对解析类数公式的加号部分的解释。在本文中，我们将展示如何利用我们的论文“局部域中的除值”[1] 中的结果，现在可以对上述定理给出更简单的证明。我们还简要描述了如何从该定理中获得 Iwasawa 的各种全局结果，这些结果未包含在 Lang 的 [3] 或 Washington 的 [4] 书中。一、符号。设 Qp((T)) 表示 Qp 上具有有限极点的洛朗级数环。设 Qp((T)) 和 Qp[[T]] 表示 Qp((T)) 的由幂级数组成的子环，分别收敛于穿孔开单位球和开单位球上。令Zp((T)) 表示具有整数系数的Qp((T)) 的子环，Zp[[T]] 表示环Zp((T)) n Qp[[T]],。令 S 和 DL 分别表示 [1] 中定义的 Qp((T)) 和 Zp((T))* 上的迹算子和范数算子。它们的特征如下： (1) 5(g)(l(I -T)p) = ]Eg(l (l -T)), (2) DL(f)(I (I-T)p) = Ilf(I t(l-T))l 其中；遍历 Unity tp 的第 p 个根。令 [a](T) = 1 -(1-T) 9g go [p], D (I T) dT f Df for g FE Qp[[T]] l andf E Z ((T))* 然后我们有恒等式 (3) SLog(f) Log(Lf), (4) SDg = pD$g, (5) 58(h) = p89(h), (6) ST(g) = Pm编辑于 1982 年 9 月 20 日收到，并于 1982 年 12 月 20 日修订。 1980 年数学学科分类。小学 12A35。 （1983 年美国数学会 0002-9939/82/0000-1451 /$03.00
In his paper, Some Modules in the Theorv of Cyclotomic Fields [2], Iwasawa obtained the remarkable theorem that the quotient of the p-adic cyclotomic units by the completion of the circular units is isomorphic to the quotient of the group ring by the Stickleberger ideal. He then used this to deduce some interesting global results, the most striking of which is an explanation of the plus part of the analytic class number formula under the assumption that the class group at the first layer is cyclic, together with a regularity assumption. In this note, we will show how with the results in our paper, Division Values in Local Fields [1], it is now possible to give a substantially simpler proof of the above theorem. We also describe, briefly, how one can obtain various global results of Iwasawa from this Theorem, which are not included in either Lang's [3], or Washington's [4] books. I. Notation. Let Qp((T)) denote the ring of Laurent series with finite poles over Qp. Let Qp((T)), and Qp[[T]], denote the subrings of Qp((T)) consisting of power series which converge on the punctured open unit ball and on the open unit ball respectively. Let Zp((T)) denote the subring of Qp((T)) with integer coefficients and Zp[[T]] the ring Zp((T)) n Qp[[T]],. Let S and DL denote the trace and norm operators defined in [1] on Qp((T)), and Zp((T))* respectively. They are characterized by the formulas (1) 5(g)(l(I -T)p) = ]Eg(l (l -T)), (2) DL(f)(I (I-T)p) = Ilf(I t(l-T))l where ; runs over the pth roots of unity tp. Let [a](T) = 1 -(1-T) 9g go [p], D (I T) dT f Df for g FE Qp[[T]] l andf E Z ((T))* We then have the identities (3) SLog(f) Log(Lf), (4) SDg = pD$g, (5) 58(h) = p89(h), (6) ST(g) = Pm Received by the editors September 20, 1982 and, in revised form, December 20, 1982. 1980 Mathematics Subject Classification. Primary 12A35. (1983 American Mathematical Society 0002-9939/82/0000-1451 /$03.00