On the real points of an arithmetic quotient of a bounded symmetric domain
On the real points of an arithmetic quotient of a bounded symmetric domain
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关于有界对称域算术商的实数点
DOI：
10.1007/bf01432692
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发表时间：
1975
影响因子：
1.4
通讯作者：
G. Shimura
中科院分区：
文献类型：
--
作者：
G. Shimura
As shown in some of our previous papers and also by K. Miyake, the quotient of a bounded symmetric domain by an arithmetic discontinuous group has in many cases a model defined over an algebraic number field and characterized by certain number-theoretical properties. It is natural to ask whether the model has rational points over a given number field. This is obviously a difficult question, but it turns out, as will be shown in this paper, that localization at an archimedean prime brings forth surprisingly strong results in the situations where a more number-theoretical approach such as non-archimedean localization, which looks more natural, may fail. Indeed, some of those models defined over the rational number field have no real points.More generally, given an algebraic variety V defined over C, say projective non-singular, one can ask a few natural questions:(1) Does V have a model over R?(2) When V is actually defined over R, does V have a real point?(3) How many connected components are there in the set VR of all real points of V?
