# what is the motivation of Shimura variety?

Tonight, a friend of mine give me a concise introduction to Shimura variety . I only get some first impression of it. I think the hodge structure is a generalization of the cohomology ring of Kaehler manifold or algebraic manifold , and i think of the Shimura variety an anologue of the analytic familly of complex manifolds , just as introduced in Kodaira's book Complex manifolds.And i suspect that there maybe some anologue theorem's as what Kodaira had done by deformation of complex structures . I'm just doing some imagination unboundedless , do not laugh at me !Heh!

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We had a similar question at mathoverflow.net/questions/14175/… – S. Carnahan Apr 10 '10 at 16:13
I believe the main original motivation was Hilbert's twelfth problem en.wikipedia.org/wiki/Hilbert's_twelfth_problem – Dror Speiser Apr 10 '10 at 19:48

The theory of Shimura varieties was begun by Shimura, and further developed by Langlands (who introduced the name), and is now a central part of arithemtic geometry and of the theories of automorphic forms, Galois representations, and motives.

Shimura varieties are certain moduli of Hodge structures; but that is perhaps not the best point of view to understand why people study them. Rather, the primary motivation is the following: Shimura varieties are attached to (certain) reductive linear algebraic groups over $\mathbb Q$, and the geometry of the Shimura variety is closely linked to the theory of automorphic forms over the corresponding reductive group.

Thus Shimura varieties make a natural test-case for investigating the conjectural relations between motives and automorphic forms, since they are geometric objects with a direct link to the theory of automorphic forms. (I'm not sure that it's useful to be more specific about this in this particular answer, but for those to whom it is meaningful: on the one hand, one has an analogue, for any Shimura variety, of Eichler--Shimura theory, relating the cohomology of modular curves to modular forms; and on the other hand, by thinking of cohomology as being etale cohomology, one obtains Galois representations which one would like to show (and in many cases can show) are related to automorphic forms in a manner analogous to the relationship between the Galois representations on the etale cohomology of modular curves and Hecke eigenforms that was established by Deligne.)

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