Iwasawa Theory for $p$-adic Representations
Iwasawa Theory for $p$-adic Representations
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DOI：
10.2969/aspm/01710097
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发表时间：
1989
期刊：
影响因子：
--
通讯作者：
R. Greenberg
中科院分区：
文献类型：
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作者：
R. Greenberg
几年前，Mazur和Wiles证明了岩泽的一个基本猜想，该猜想给出了Riemann Zeta函数(更广泛地说，Dirichlet L-函数)的临界值与某些割圆域的理想类群之间的精确联系。这种联系的第一个暗示可能是Kummer著名的素数不规则性判据。在岩泽的理论中，人们为每个素数p定义了岩泽代数A上的某些模(我们将在第一节中描述)。岩泽猜想则将这些A模的结构与久保田和Leopoldt构造的p元L函数联系起来，这些p元L函数对Dirichlet L函数的临界值进行插补。Mazur意识到，遵循岩泽的模型，人们可以对椭圆曲线E(定义在Q上)和任何素数p提出类似的猜想，其中E具有良好的、普通的约化。Mazur和Swinnerton-Dyer构造了依附于E的p元L函数(假设E是一条Weil曲线)。与Mazur猜想有关的岩泽模是用E的Selmer群定义的，同样也是在分圆域的塔中。这一次，这种关系的线索是Birch和Swinnerton-Dyer猜想。·现在还有其他几种情况下，复L-函数的p-进类似函数被构造出来--例如，Manin的p-进L-函数依附于经典的模形式。因此，在更一般的情况下寻找合适的岩泽模似乎是值得的，这也是我们在本文中的目的。我们将考虑Q上的一维相容系统V={Vi}。因此，Vi是q上的有限维向量空间，其上的维d=dv，Ga=Gal(Q/Q)起作用。(对于任何域k，Gk表示Gal(f/k)，其中Le是k的代数闭包。)如果Q是任何素数，则人们还可以通过选择Q上Q的一个位置Lj_并用该位置的分解群来标识Ga，从而将Vi视为Gaq的表示空间。我们通常会假设p1是Vin的一个“普通”素数，如下所示-
Several years ago Mazur and Wiles proved a fundamental conjecture of Iwasawa which gives a precise link between the critical values of the Riemann zeta function (and, more generally, Dirichlet L-functions) and the ideal class groups of certain towers of cyclotomic fields. Probably the first hint of such a link is Kummer's well-known criterion for irregularity of primes. In Iwasawa's theory one defines for each prime p certain modules over the Iwasawa algebra A (which we will describe in Section 1). Iwasawa's conjecture then relates the structure of these A-modules to the p-adic L-functions constructed by Kubota and Leopoldt which interpolate critical values of Dirichlet L-functions. Mazur realized that, following Iwasawa's model, one could formulate a similar conjecture for an elliptic curve E ( defined over Q) and for any prime p where E has good, ordinary reduction. Mazur and Swinnerton-Dyer constructed p-adic L-functions attached to E for such p (assuming Eis a Weil curve). The Iwasawa modules which Mazur's conjecture relates to these p-adic L-functions are defined in terms of Selmer groups for E, again in towers of cyclotomic fields. This time the hint of such a relationship is the Birch and Swinnerton-Dyer conjecture. · There are now several other cases where p-adic analogues of complex L-functions have been constructed-for example, Manin's p-adic L-functions attached to classical modular forms. It seems worthwhile then to search for appropriate Iwasawa modules in a much more general context and that is our purpose in this paper. We will consider a compatible system of 1-adic representations V={Vi} over Q. Thus Vi is a finite dimensional vector space over Qi (the 1-adic numbers) of dimension d=dv on which Ga=Gal(Q/Q) acts. (For any field k, Gk denotes Gal(f/k), where le is an algebraic closure of k.) If q is any prime, then one can also consider Vi as a representation space of Gaq by choosing a place lj_ of Q over q and identifying Ga, with the decomposition group for that place. We will usually assume that pis an "ordinary" prime for Vin the follow-