Rational points of abelian varieties with values in towers of number fields
Rational points of abelian varieties with values in towers of number fields
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DOI：
10.1007/bf01389815
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发表时间：
1972-12
影响因子：
3.1
通讯作者：
B. Mazur
中科院分区：
文献类型：
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作者：
B. Mazur
K= kckki C " " cKn~'" cK~= L= U Kn n= O (cf. w 1 (C))，使得Gal （KJK）是pn阶的循环。集合F= GaI （L/K）。设4是K上的一个阿贝尔变数，它对除p的所有素数都有很好的普通约简。本文提出的理论的问题是：有理点群A (L)是有限生成的吗？经典的modell - weil定理保证了A （Kn）对于每一个n是有限生成的，但是当我们考虑关于有理点群随着数域的变化而渐近增长的问题时，没有给出任何指示。这里有一些理由希望我们的问题有一个肯定的答案：假设，4是曲线C的雅可比矩阵，那么，4 (L)与最小正则算术曲面的N6ron-Severi群密切相关，该曲面是L上整数环上C的模型。因此，Iwasawa对L与曲线上的有理函数场的绝妙类比：有限域的代数闭包可能导致人们期望a (L)具有类似于有限域代数闭包上曲面的N6ron-Severi群的结构；但这种曲面的N6ron-Severi群是有限生成的。我发现，如果一个人更广泛地研究阿贝尔变量，4/K和任何满足下面（6.1）假设的f -扩展L/K，则上述问题的公理化可以更清楚地得到关注。我们认为这样的一对（信用证，a）可以接受。表示A （Kn）的渐近增长和Shafarevitch-Tate群（4,p ~///A (K~)）的p-初级分量的大部分信息都包含在一个具有p-进系数的多项式中，我们通过w中的一个基本构造来定义这个多项式，它取决于拓扑生成器~ F的选择。
K= KocKI C''" cKn~'" cK~= L= U Kn n= O (cf. w 1 (c)) such that Gal (KJK) is cyclic of order pn. Set F= GaI (L/K). Let, 4 be an abelian variety over K which has good, ordinary reduction at all primes dividing p. The question motivating the theory presented in this paper is the following: Is the group of rational points A (L) finitely generated?The classical Mordell-Weil theorem guarantees that A (Kn) is finitely generated for each n, but gives no indication of what to expect when one considers questions concerning asymptotic growth of the group of rational points as one varies the number field. Here is some reason for hoping that our question has an affirmative answer: Suppose that, 4 is the Jacobian of a curve C. Then, 4 (L) is closely related to the N6ron-Severi group of the minimal regular arithmetic surface which is a model for C over the ring of integers in L. Consequently, Iwasawa's magnificent analogy between L and the rational function field of a curve over the: algebraic closure of a finite field might lead one to expect that A (L) has a structure similar to that of the N6ron-Severi group of a surface over the algebraic closure of a finite field; but the N6ron-Severi group of such a surface is finitely generated. I have found that the axiomatics of the above problem can be kept more clearly in focus if one works more generally with an abelian variety, 4/K and any F-extension L/K satisfying the hypotheses of (6.1) below. We call such a pair (L/K, A) admissible. Much of the information expressing the asymptotic growth of A (Kn) and the p-primary component of the Shafarevitch-Tate group of, 4, p~///A (K~) is contained in a certain polynomial with p-adic coefficients that we define by means of an essential construction made in w This polynomial, which depends upon a choice of topological generator~ F,