Characteristic elements in noncommutative Iwasawa theory
Characteristic elements in noncommutative Iwasawa theory
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DOI：
10.1515/crll.2005.2005.583.193
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发表时间：
2003-11
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影响因子：
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通讯作者：
O. Venjakob
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文献类型：
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作者：
O. Venjakob
设p是一个素数，为简单起见，我们总是假定它是奇数。在岩泽理论的椭圆曲线E在一个数字领域k一个必须区分曲线做或不承认复杂的乘法（CM）。对于CM椭圆曲线，它们的深算术性质以及它们的塞尔默群与它们的Hasse-Weil L-函数的特殊值之间的联系，不仅可以用对应于k的分圆Zp-扩张kcyc的（一元）主猜想来描述，而且可以用对应于扩张k∞ = k（Ep∞）的（二元）主猜想来描述，而扩张k∞ = k（Ep∞）是由邻接E的p-幂除点Ep∞而产生的.此外，Rubin [36]在k是虚二次的情况下证明了这两个定理，并且E通过k的整数环Ok具有CM。同样对于非CM椭圆曲线，人们希望至少在平凡化扩张k∞上形成一个主要猜想，但由于缺乏代数和解析的p-adic L-函数，这还没有实现。本文的目的是在一定条件下，证明Galois群G = G（k∞/k）的通常Iwasawa代数Λ = Λ（G）的局部化ΛT的第一个K-群K1（ΛT）<$= ΛT /[Λ × T，Λ × T ]中存在一个代数p-adic L-函数.这里，对于环R，我们用R×它的单位群表示。通过Weil配对，kcyc包含在k∞中，我们设H = G（k∞/kcyc），Γ = G（kcyc/k）.进一步地，我们把标准满射环同态的核记为m（H
Let p be a prime number, which, for simplicity, we shall always assume odd. In the Iwasawa theory of an elliptic curve E over a number field k one has to distinguish between curves which do or do not admit complex multiplication (CM). For CMelliptic curves their deep arithmetic properties and the link between their Selmer group and special values of their Hasse-Weil L-functions are not only described by the (one-variable) main conjecture corresponding to the cyclotomic Zp-extension kcyc of k, but also by the (two-variable) main conjecture corresponding to the extension k∞ = k(Ep∞) which arises by adjoining the p-power division points Ep∞ of E. Moreover, both conjectures are proven by Rubin [36] in the case that k is imaginary quadratic and E has CM by the ring of integers Ok of k. Also for non-CM elliptic curves one would like to at least formulate a main conjecture over the trivialzing extension k∞, but for lack of both an algebraic as well as analytic p-adic L-function this has not been achieved. The aim of this paper is to establish, under certain conditions, the existence of an algebraic p-adic L-function, viz as an element of the first K-group K1(ΛT ) ∼= ΛT /[Λ × T ,Λ × T ] of a localization ΛT of the usual Iwasawa algebra Λ = Λ(G) of the Galois group G = G(k∞/k). Here, for a ring R, we denote by R× its group of units. By the Weil-pairing, kcyc is contained in k∞ and we put H = G(k∞/kcyc) and Γ = G(kcyc/k). Furthermore we write m(H) for the kernel of the canonical surjective ring homomorphism