Automorphic Representations, Shimura Varieties, and Motives. Ein Marchen*-猫眼课题宝

自守表示、志村品种和动机。

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1977

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Ein M ärchen;R. Langlands

1.导论.我的意图是调查志村品种zetafunctions研究所提出的问题。但我太乐观了。这将是一项猛犸的任务，时间和精力的限制大大缩小了本报告的范围。我认为只有两个问题，一个关于志村品种的共轭，和一个在连续上同调域。乍一看，把它们结合起来似乎是不协调的，因为一个是算术，另一个是表示论，但它们都是在研究无穷多处的ζ函数时出现的。共轭问题在第六节中被表述为一个猜想，这是经过长时间的修改才得出的。我以前的尝试都提交给拉波波特批准，发现缺乏。他们太不精确，甚至在原则上不适合证明志村的方法下降。该猜想，因为它的立场是唯一的声明，我可以发现，符合他的批评，是兼容志村的猜想。该声明的猜想必须先有一些建设，其中有影响，已经逃脱了我。当结合德利涅的概念志村品种作为参数品种的家庭的动机，他们建议引进一组，这里称为谷山组，这可能是重要的研究动机的CM型。它在第五节中被定义，在那里它的假设性质被排练。随着动机和谷山集团的介绍，报告呈现出一种原本不打算有的语气。它不再是简单地制定一个或两个具体的结构问题，但我们开始编织一个组织的超自然和假设，和好奇心驱使我们。德利涅的想法是在第四部分审查，但要理解他们必须熟悉至少与tannakian类别的形式主义的基本要素的动机的非理性理论，说，第二章的主要结果[40]。目前的暑期研究所基于这样的信念：自守表现和动机之间存在密切关系。这种关系通常用L-函数来表达，* 首次出现在自守形式、表示和L-函数中。在纯

1. Introduction. It had been my intention to survey the problems posed by the study of zetafunctions of Shimura varieties. But I was too sanguine. This would be a mammoth task, and limitations of time and energy have considerably reduced the compass of this report. I consider only two problems, one on the conjugation of Shimura varieties, and one in the domain of continuous cohomology. At first glance, it appears incongruous to couple them, for one is arithmetic, and the other representationtheoretic, but they both arise in the study of the zeta-function at the infinite places. The problem of conjugation is formulated in the sixth section as a conjecture, which was arrived at only after a long sequence of revisions. My earlier attempts were all submitted to Rapoport for approval, and found lacking. They were too imprecise, and were not even in principle amenable to proof by Shimura’s methods of descent. The conjecture as it stands is the only statement I could discover that meets his criticism and is compatible with Shimura’s conjecture. The statement of the conjecture must be preceded by some constructions, which have implications that had escaped me. When combined with Deligne’s conception of Shimura varieties as parameter varieties for families of motives they suggest the introduction of a group, here called the Taniyama group, which may be of importance for the study of motives of CM-type. It is defined in the fifth section, where its hypothetical properties are rehearsed. With the introduction of motives and the Taniyama group, the report takes on a tone it was not originally intended to have. No longer is it simply a matter of formulating one or two specific conjectures, but we begin to weave a tissue of surmise and hypothesis, and curiosity drives us on. Deligne’s ideas are reviewed in the fourth section, but to understand them one must be familiar at least with the elements of the formalism of tannakian categories underlying the conjectural theory of motives, say, with the main results of Chapter II of [40]. The present Summer Institute is predicated on the belief that there is a close relation between automorphic representations and motives. The relation is usually couched in terms of L-functions, * First appeared in Automorphic forms, representations, and L-functions, Proc. of Symp. in Pure

Automorphic Representations, Shimura Varieties, and Motives. Ein Marchen*

自守表示、志村品种和动机。

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