The arithmetic of automorphic forms with respect to a unitary group
The arithmetic of automorphic forms with respect to a unitary group
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关于酉群的自守形式的算术
DOI：
10.2307/1971129
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发表时间：
1978
影响因子：
4.9
通讯作者：
G. Shimura
中科院分区：
文献类型：
--
作者：
G. Shimura
利用在各种算术问题以及在研究的分析性质的形式本身。对于希尔伯特模形式和西格尔模形式也可以这样说。将给定的模形式F表示为复变量u，，*，u的函数。在展开式（0.1）F（u，，*. Us）= Ex c exp（2wi. * 1U '），其中系数c.是复数x在一个格子上运行特别重要的是那些F的所有c，是代数，或更严格地说，割圆。它们形成一个可区分的类，该类在所讨论的代数群的元素的变换下是稳定的。C.的代数性，也是必不可少的，如果值的模函数在一个特殊的点是问题，如在理论的复数乘法。一般来说，如果自守形式定义在管域上并且群包含足够多的平移，则自守形式存在（0.1）型展开。然而，在有些情况下，没有这种扩展。对称域提供了一个典型的例子
utilized in various arithmetical problems as well as in the study of the analytic properties of the form itself. The same can be said also for the Hilbert and Siegel modular forms. One expresses a given modular form F as a function of complex variables u,, * * *, u. with an expansion (0.1) F(u,, *. Us) = Ex c exp(2wi. * 1U'), where the coefficients c. are complex numbers and x runs over a lattice. Especially important are those F for which all c, are algebraic, or more restrictedly, cyclotomic. They form a distinguishable class which is stable under the transformation by the elements of the algebraic group in question. The algebraicity of c., is also indispensable if the value of a modular function at a special point is the problem, as in the theory of complex multiplication. In general, an expansion of type (0.1) exists for an automorphic form if it is defined on a tube domain and the group contains sufficiently many translations. There are, however, cases in which no such expansion is available. A typical example is provided by the symmetric domain