It so look like that Shimura variety is a modular space that contains classes of zeros of modular functions that have certain symmetry that have structure of Hodge type. Is this true?
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3$\begingroup$ jmilne.org/math/xnotes/svi.pdf $\endgroup$– Stanley Yao XiaoCommented Sep 9, 2020 at 19:16
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7$\begingroup$ In my opinion, to have a reasonable answer, this question needs to include more information about what you already know about Shimura varieties: What examples are you familiar with? Do you understand the pieces of the definition on a formal level and are looking for intuition about how they fit together, or is there some part of the definition you already don't understand? It would also be helpful to expand on terms in your question like "modular space", "contains classes of zeroes", "have certain symmetry". $\endgroup$– Will SawinCommented Sep 9, 2020 at 19:29
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1$\begingroup$ I understood the definition of modular functions well,I also understood the idea of classes of varieties as quotient spaces,I understood the idea of a Hodge structure but I don't see clearly the intuition of Shimura variety as it fit into all these ideas... $\endgroup$– Wahadti VitalisCommented Sep 9, 2020 at 19:37
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1$\begingroup$ Shimura variety. $\endgroup$– abxCommented Sep 10, 2020 at 5:22
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2$\begingroup$ As well as Milne's lecture notes, you could also try this two-page overview article in the AMS Notices, ams.org/notices/201211/rtx121101560p.pdf, whose title is precisely "What is ... a Shimura variety?" $\endgroup$– David LoefflerCommented Sep 10, 2020 at 6:54
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