The norm map for ordinary abelian varieties
The norm map for ordinary abelian varieties
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普通阿贝尔簇的范数图
DOI：
10.1016/0021-8693(78)90271-5
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发表时间：
1978
影响因子：
0.9
通讯作者：
M. Rosen
中科院分区：
文献类型：
--
作者：
Jonathan D. Lubin;M. Rosen
令 K 为 Q 的有限扩展，其中残差类字段为 k。设 A 是 K 上的 d 维阿贝尔簇，具有良好的普通约简，u 是 A 的扭曲矩阵（u 的精确定义将在第 1 节中给出）。假设 L 是 K 的完全分支 & 扩展，并令 M&A (L) 为群 NKnlKA (Kn) hw ere KCK, CL 和 [K,: K]= p” 的交集。那么我们有一个精确的序列。这是 [l] 中的 Mazur 命题 4.39。由于我们的方法，我们可以快速证明以下关于扭转点的结果。假设 4、EK 和 L/K 是那么 A (L) 的 p 幂挠率小于或等于 pt+ s，其中 pl= card A (k)，并且 s= ord, det (1-u) 我们的证明方法是首先将问题简化为关于形式群的问题，然后考虑某个伽罗瓦模。被认为是“初级的”。
Let K be a finite extension of Q, with residue class field k. Let A be a d-dimensional Abelian variety over K with good ordinary reduction and let u be a twist matrix of A (the precise definition of u will be given in Section 1). Suppose L is a totally ramified &-extension of K and let M&A (L) be the intersection of the groups NKnlKA (Kn) hw ere KC K, CL and [K,: K]= p”. Then we have an exact sequenceThis is Mazur’s Proposition 4.39 in [l]. As a consequence of our approach we can give a quick proof of the following result about torsion points. Suppose 4, EK and that L/K is the cyclotomic i&-extension. Then the p-power torsion of A (L) is finite of order less than or equal to pt+ s, where pl= card A (k), and s= ord, det (1-u). Our method of proof is to reduce the question first to a problem about formal groups and then to the consideration of a certain Galois module. Our approach avoids the use of class field theory and the theory of pro-algebraic groups and may thus be considered “elementary.”