On Abelian Varieties with Complex Multiplication
On Abelian Varieties with Complex Multiplication
复制标题
论具有复数乘法的阿贝尔簇
DOI：
10.1112/plms/s3-34.1.65
复制
发表时间：
1977
影响因子：
1.8
通讯作者：
G. Shimura
中科院分区：
文献类型：
--
作者：
G. Shimura
设A是定义在C上的n维阿贝尔簇，其自同态代数包含2n次数域K的同构像.复数乘法的主要定理给出了A在C在某个域K '上的自同构下的行为，域K'由< J>K在A的切空间上的表示确定。本文的目的是研究A在一个非K '上的恒等映射的自同构下的行为。为了以不同的方式解释这个问题，取A的任何极化0，让Mo是极化种类（A，C）的模的场。然后，通过主要定理，我们可以确定K 'M0作为K\上的类域，但我们不知道Mo的大小。这种信息可以由A在一般类型的自同构下的行为提供。如果A是一维的，那么K是虚二次的，并且K ′ = K。在这种情况下，Mo总是有一个真实的阿基米德素数，因此Mo和K在Q上线性不相交。这可以很容易地从A在复共轭下的行为导出。然而，在高维的情况下，似乎不可能有这样的普遍答案。我将以自己的行为来证明这一点。4，C），Mo的大小在很大程度上取决于（A，C）的同构类，而K ′ M0完全由{K，Q>）决定，如果End（J.）是K的最大阶。此外，{K，< D）的伽罗瓦理论结构似乎对我们的问题的答案（如果有的话）至关重要。在本文中，我们将在一般的初步考虑之后，实际研究以下三种情况：（1）Q上K的Galois闭包有一个4 n阶二面体群作为它的Galois群;
Let A be an abelian variety of dimension n defined over C whose endomorphism algebra contains an isomorphic image of a number field K of degree 2n. The main theorem of complex multiplication gives the behaviour of A under an automorphism of C over a certain field K'which is determined by the representation< J> of K on the tangent space of A. The purpose of the present paper is to investigate the behaviour of A under an automorphism that is not the identity map on K'. To explain this problem in a different fashion, take any polarization 0 of A and let Mo be the field of moduli of the polarized variety (A, C). Then, by means of the main theorem, one can determine K'M0 as a class field over K\but one does not know how large or small Mo is. Such information can be supplied by the behaviour of A under an automorphism of general type. If A is one-dimensional, then K is imaginary quadratic and K'= K. In this case, Mo always has a real archimedean prime, and hence Mo and K are linearly disjoint over Q. This can easily be derived from the behaviour of A under the complex conjugation. In the higher-dimensional case, however, no such general answer seems possible. Indeed, we shall show that the behaviour of (. 4, C) and the size of Mo depend, to a great extent, on the isomorphism class of (A, C), whereas K'M0 is completely determined by {K, Q>) if End (J.) is the maximal order of K. Also, the Galoistheoretical structure of {K,< D) seems essential to the answers, if any, to our questions. We shall actually investigate in this paper, after some general preliminary considerations, the following three cases:(1) the Galois closure of K over Q has a dihedral group of order 4n as its Galois group;