Role in The Langlands Program
Shimura varieties play an outstanding role in the Langlands program. The prototypical theorem, the Eichler–Shimura congruence relation, implies that the Hasse-Weil zeta function of a modular curve is a product of L-functions associated to explicitly determined modular forms of weight 2. Indeed, it was in the process of generalization of this theorem that Goro Shimura introduced his varieties and proved his reciprocity law. Zeta functions of Shimura varieties associated with the group GL2 over other number fields and its inner forms (i.e. multiplicative groups of quaternion algebras) were studied by Eichler, Shimura, Kuga, Sato, and Ihara. On the basis of their results, Robert Langlands made a prediction that the Hasse-Weil zeta function of any algebraic variety W defined over a number field would be a product of positive and negative powers of automorphic L-functions, i.e. it should arise from a collection of automorphic representations. However philosophically natural it may be to expect such a description, statements of this type have only been proved when W is a Shimura variety. In the words of Langlands:
| “ | To show that all L-functions associated to Shimura varieties – thus to any motive defined by a Shimura variety – can be expressed in terms of the automorphic L-functions of is weaker, even very much weaker, than to show that all motivic L-functions are equal to such L-functions. Moreover, although the stronger statement is expected to be valid, there is, so far as I know, no very compelling reason to expect that all motivic L-functions will be attached to Shimura varieties. | ” |
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